The Fujita exponent, originally established for the nonlinear heat equation with power nonlinearity, serves as a critical threshold for global existence versus blow-up of solutions. In non-commutative settings, this exponent depends on the global dimension of the underlying group or the sum of dilation weights in the vector field system. For unimodular groups, the Fujita exponent is given by p_F = 1 + 2/D, where D is the global dimension characterizing the volume growth of balls in the group. When D = 0 (compact groups), p_F = ∞ (always blow-up); when D = ∞ (exponential growth), p_F = 1 (always global existence); and for polynomial growth groups, p_F follows the formula. This framework extends to general Hermander systems of vector fields without group structure, where the exponent depends on the sum of dilation weights.
ICMAM Latin America 2026 - Day 5
Added:Okay, Brian, could you please make co-host to Professor Michael?
Hello.
>> Hello, Professor Michael. How are you?
>> Yes. Okay. Good. Yes. How are you?
>> Good. Good. Could you hear me? Well, right.
Okay.
Will you hear me? Well, okay. Let me see. I think the professor Michael is >> Will you hear us, Professor?
>> Yes. Yes, I can hear. Yes.
>> Ah, okay. Okay.
>> Could you hear me? Because maybe it's my microphone.
>> Yeah. No, I can hear everybody.
>> Okay. Okay. Good. Good.
>> Okay. Professor, maybe you can try to share your screen, your your screen, your slides.
Okay, let's let me where is Yeah.
Wait, can you see or what? What the uh Can you see it?
>> Yes, I can see. Mhm. Perfect.
>> Yes, I can see. Okay. Okay. So, I think we can start, right?
So, professor first asphalt, thank you very much for accepting our our invitation.
Okay. So, okay. Our next speaker to start the last day of our conference Latin America conference is with the professor Michael Rousanski.
So it's my great pleasure and distinct honor to introduce professor Michaelanski, one of the world's leading mathematicians in the fields of non-commutative harmonic analysis, microlological analysis and partial differential equations.
Professor Ransk's academic journey is a remarkable example of intellectual excellence and international collaboration.
After completing his studies with these things at St. Peterbs State University, he pursued doctoral research at Ultrich University under the supervision of the legendary mathematician Johannes Dist earning his PhD with a thesis that made significant contributions to the theory of Fier integral operators with complex faces. From there his career led him through distinguished appointments at Johns Hop Hopkins University, the University of Edinburg and Imperial College London where he rose from lecture to professor and continues to serve as an honorary professor. Today he also holds the positions of senior fools professor and senior research professor at Gen University while remaining professor at Queen Mary University of London.
Through his career, professor Rusanski has profoundly influenced modern mathematical analysis. His research expands harmonic analysis, microlocal analysis, so the differential operators, fier integral operators, analysis on le groups and the theory of partial differential equations including hyperbolic dispersive shinger and evolution equations. His work has not only advanced fundamental mathematical theory but has also provided powerful analytical tools that continue to shape contemporary research across several branches of mathematics and mathematical physics.
>> The international mathematical community has recognized this groundbreaking contributions from numerous prestigious distinctions. Among them are the metal laurel chief one of the Belgium her recognition for scientific excellence.
The sales one project award by the research foundation fllanders to feran sier f valanger prizes granted in 2014 and 2018 for outstanding mathematical monographs. the Isaka award, the Daiwa Adrian Prize, the EPS Ursa ER leadership fellowship in the United Kingdom as well has multiple fellowships from the Japan Society for the Promotion of Science and the Lure Trust. Beyond these remarkable achievements, Professor Roans has distinguished himself as an exceptional mentor, collaborator and scientific leader. Through his vision, he has built vibrant international research community, supervised numerous young mathematician who now contribute across the globe and foster collaboration that bridge continents and mathematical disciplines. His influence extend far beyond his own publications, leaving a lasting impact on generations of researchers and on the development of modern analysis. Today we have the privilege of hearing from a scholar who whose work combines extraordinary depth, originality and lasting influence. His contributions continue to inspire mathematicians worldwide and it simplify the highest standards of scientific excellence. Please join me in giving a warm welcome to professor Michael Roanski who will talk about no commutative fua exponent.
Okay, thank you very much for this uh kind introduction and uh [clears throat] it's a pleasure for me to give a talk here. Um I was trying to prepare a different topic um somehow um but uh then I got short on time so I uh decided to talk to you about non-communive for exponent. So maybe some of you heard it before um then my apologies but um anyway so this is the topic now and um so it's about non community fujit a non-commutive fujit exponent uh which means that uh uh well we have fujit exponent uh which is a threshold for the well poisoners for uh similar heat equation with power nonlinearity and then uh non-commutative uh uh here [clears throat] means that we look at it in in in different non-commutative settings.
Uh so the plan of my presentation is the following. I will um talk about Fujit exponent on groups rather general groups namely unimodular groups general uniodular groups um and uh league groups and uh so we have an underlying group structure this problem one uh problem two that I will talk about is uh when we don't have group structure but we have hermanda system of vector fields and uh we consider sum of squares of these operators.
Uh then problem three is when we have uh concave nonlinearity uh namely u to the power p when p is for example between zero and one it's a conc concave function and problem four uh is when we [clears throat] consider the setting of quantum taurus or quantum ucidian space and uh okay this this part actually this is the one that I was trying um add slides about but uh I think that uh I may um uh as I didn't have time then I may just try to focus on the first three problems.
Okay. So uh nonlinear heat equation it's the following equation. Uh as I said it's a heat equation plus u to the power p.
Uh first I look we look at the convex case. So p is strictly bigger than one.
P is equal to one is not so interesting because it's a linear equation. Then uh we have of course global existence. But for uh p strictly bigger than one uh theian situation in this is fundamental result by fjitta who found this exponent here pf.
So n is a dimension and uh pf and it is a threshold for the well posess. So if if uh p is less than this pf then uh uh there are no global in time solutions to this equation. So all solutions blow up. If P bigger than this PF then uh global solutions exist. We look at positive data always global solutions exist and if P is equal to PF this was um the case was done uh after FOA and soon after FOA in this case uh so the critical index belongs to the blowup regime. So in this in this case we don't have global solution. So this is a very fundamental result which is already which is 60 years old now uh by and uh since then people studied this problem in many settings uh uh in the ukidian situation for example u to the power p can be replaced by a function f of satisfying some properties llassian can be replaced by nonlinear llassian uh one can consider this equation in domain with different boundary conditions. Uh one can consider this equation on manifolds remaining manifolds with different volume growth for example polinomial volume growth with uh different assumptions on the curvature.
So uh this has been this problem has been studied. However, when I became interested in this problem, uh uh two observations uh became clear that in the setting of groups uh very little was known. This is for this problem. Uh and the second is this last line here. Uh if you put function of t in front of u to the power p then uh there are results. But um um again uking results and what happens in groups and then then uh nothing was known. So this is uh the motivation for this talk to understand for this research to understand what is happen what happens if you have uh groups and groups of different types and uh can we produce some general results or um they uh may depend on the type of the group and so on.
Uh so just to remind you here so a group uh just to fix a couple of definitions.
So the group is uh we have identity element E in the group G and we have two operations a group operation and inverse operation that u uh fit to each other like here and we have associativity. We don't assume commutativity if if if the group operation is commut commutive then we have a commutive group a special case so we don't we don't assume it here and a league group league group is a smooth manifold which is also a group at the same time and these two group operations so the group operation and inverse operations they are smooth with respect to the smooth structure of the menu okay so then [clears throat] what we have from such groups If uh we don't need le group assumption for the following is if it's just a locally compact group then there are also two measure there are always two measures left measure and right half measure which are uh uniquely determined measures modular constant constants multiplication by constants uh which are invariant under shifts uh so left measure here is invariant uh under shifts to the left and right measure invariant and the shifts to the right. So in general these two measures are different uh but they are uniquely determined up to constant they give us integrals once we have measure we can u uh define integration with respect to this measure and the integral again becomes left invariant to right invariant and the shifts and uh as I said these two measures can uh can be different in general but uh if it happens that they are they are the same then we call the group uni modular.
Okay. So this is basically the only definition which we need here and uh so examples. So there are many examples like ukidian space to all compactly groups, heisenberg groups, stratified groups, graded groups, homogeneously groups groups they are all unimodular uniodularly groups.
So uh okay so this is so these are the groups which will be covered under in this work because we look at the unimod groups uh and uh let me recall a result which I had with my former PhD student already uh a few years ago uh we looked at the for exponent for general new model groups we can assume it's connected uh otherwise we have we may have results on different connected components. So uh it's it's it's not restrictive. We we look so we look at general uni model group uh by definition we have two har measures which coincide.
So effectively it's one har measure which is b invariant. So invariant shifts to the right and to the left. We take Hermandanda system of vector fields. Uh so meaning that satisfies Hermander condition that iterated brackets of this li brackets of these vector fields span the whole tangent space. Uh we have a so-called caroc distance.
uh which means that we measure uh we measure distance between points by joining them by lines oh no by curves with a property that tangent to this curve is also is always in the span of these vector fields. So such distance exist always and it's called caroctory distance. Um it is uh in general uh different from the um remaining distance if you put remaining structure because there we would allow any curves here we allow only so-called horizontal curves. So the tangent vector is in the span of these vector fields.
Uh then we can see the balls with radius r around a point uh v of r. It doesn't depend on x uh by because it's a group uh and uh uh we can look at the volume of the ball how it grows with respect to r and sum of squares of the vector fields is hypoalyptic it's sublassian which is hyperptic by 3. So there are then uh the following properties that we have. Uh first of all for small radius the volume of the ball always grows like r to the power d for some small d. Uh which depends on the group and on the system of vector fields that we choose it's called the local dimension.
uh and for two for large R there are two possibilities and uh they depend only on the group but not on the choice of the system of vector fields which is nice.
Uh so either V of R grows polomally like R to the power D or some capital D and then D is called the global dimension of uh G. So the volume grow so this is polinomial growth uh volume growth or another possibility is exponential volume growth. So there are only so all uni modularly groups they fall into these two categories uh which we have to study we have to look a little bit different uh differently because uh obviously as the volume grows um uh influences what what is happening at infinity.
So uh this is the situation and then uh so our result is the following uh that we had with in this situation. Uh so we found fujit exponent and we found unifying formula for this fujit exponent to be 1 + 2 divided by capital d where capital d is global dimension and uh uh a unifying formula in the sense that if uh uh we have exponential growth group we can formally put capital d to be equal to infinity then pf is equal to one and if it is a compactly group when uh capital D is equal to zero there is no volume growth in which case PF will be equal to infinity so uh there are essentially three types of groups compact when volume doesn't grow uh second when volume grows polinomally and third when volume grows exponentially uh for each of these cases we have capital D and the formula is the same so local dimension does not enter in this formula as you see uh but it is a global dimension and the results the theorem that we have here is uh in blue is uh similar to the classicala result if p is less equal than pf then there is always blow up if p is bigger than pf then there it is existence so for example compactly groups d is equal to zero pf is an infinity so p is always less than PF it means there is always blow up uh there is always blow up uh on comp groups. So polomial growth there is fut threshold as before and exponential growth when d capital d equals infinity pf is equal to 1. So p is always bigger than pf p is always bigger than one. So there is always global existence.
Uh so this is results that we had and then uh so just to and for completeness just let me mention briefly that uh we developed a certain method which actually allowed us to treat not only for um not only this equation but more general one. We can put a function f of u here not just u to the power p and then we have uh uh satisfying just inequality like this uh or lesser equals than u to the power p larger equals than u to the^ p and then we can look not just sol not just equations but inequalities in the critical case result is for the equality but in the non-critical cases uh we can look at the uh just inequality And then not just equality has blow up but also this inequality has blown up.
Uh okay so this is the situation here and then this is polomial volume growth and then also exponential volume growth we have uh also result for for inequality.
Okay. So problem one which um uh so this is an older result and now I move to more recent result. So problem one is when we have a function 5 oft that we put in front of f of u and uh we want to find we want to see whether what can we say about uh global existence of this equation and assumptions that we make on phi are will be quite mild five will be non- negative uh non- negative function um locally integraable later and f here uh because we want to know we want to have interplay between five and f so we don't take u to the power p not to restrict ourselves but we just take a function f which is zero at zero and positive and we look at the mild solutions here so if there is a classical solution then it's also a mild solution uh so we look at the equation in this integral form and again we say that uh global existence if L infinity norm is controlled for all time and uh but if L infinity norm goes to infinity then we talk about blow up.
Uh so our results are the following.
There is sufficient condition uh to have global solution and uh there is converse which is necessary condition uh which we do for Heisenberg groups and uh these necessary and sufficient conditions coincide for uh that we showed in our paper co it was known they coincide on we showed that they coincide in the first group and uh Um but um recently there there is another improvement that is not published yet. I I think it's it's it's um one can show that uh actually we can find this necessary sufficient condition for any group but I will not talk about this now.
Uh so some technical functions we define this uh minor and measurement function don't worry about them uh they will not enter much uh only we have this they enter here in this condition so uh okay it's a technical condition for example power and nonlinearity satisfies this condition for P bigger than one so we don't worry about that so much so G is a general new model loop f is continuous f is nonlinearity zero at zero positive uh we assume that this mapping is non decreasing but uh again if you think of f of v to be v to the power p bigger than one then uh this is some kind of convexity assumption on f and five is the negative locally integral function then our theorem is the following that if g so first if g has polinomial volume grows because uh uh in comparison when with the previous result when we had uh power nonlinearity and fut exponent we could come up with um unifying expression uh for the index which contains different cases of groups. So here situation is different. Uh we look separately polomial volume growth which contains the case of uh compact groups because then we can put capital D equal to zero and then uh the theorem is the following. The first theorem is that u uh in one direction if there is no uh so if there are no solutions so if there is always blow up then this integral here has to be infinite uh also this integral has to be infinite but this is a bit this is harder to check this is the first type of condition that but it's harder to check because we have to evaluate uh f on infinity norms and so on.
Yeah.
Um okay. Yeah. and and and but this condition is uh is easy to check because um uh we just put phi and f and for every theta we check if this integral is finite or infinite and uh if there is blowups then we know that it has to be infinite. So if it's finite for some theta there it means that uh there is no blowouts. It means that uh global solutions exist and exponential growth here we don't have capital D because capital D is infinite that's why this condition we can we don't come up with unifying condition because if capital D is infinite we cannot use this formula but uh then condition is the following and here local dimension uh comes up uh interestingly that this integral has to be infinite but for all liter but also for all new bigger than local dimension.
Okay. So this is simplication in one direction and uh corology of course is uh by logical negation. If the integral is finite for some theta then there is uh there are global uh global solutions.
uh so some consequences of this result uh so if I is equal to one so we don't have this function uh that we plug in there then uh we can recover the uh just results of the for exponent because if I is equal to one in this integral phi disappears and uh you put f of u to be equal to u to ^ P. So then this will be raised to the power P. It doesn't depend on theta.
Theta comes out of the integral. This is just condition on the power of tow for the integraability of this. And what we get there is exactly the fit exponent uh that we had before.
So then uh in this sense it is an extension of results previous results uh although not complete extension because before in that paper I said our results apply to inequalities also here we just deal with um with equation itself.
Uh so Uklidian case was known by uh these guys and we recover their result here but uh we have also some minor improvement of their result because we uh they had more assumptions on functions on the uh admissible class of nonarities f we reduce it so they required some regularity of f for example we just use we just need continuity of f uh but in terms of orders it's it's we recover that result.
So if uh uh so as I said f_subi is equal to 1 f of v is equal to v e to the power p we have for case and we recover exactly for exponent here and for compact groups when d is equal to z also you look at this condition capital d is equal to z there is no toao here it comes out this comes out and it's just condition on integraability of phi so if five is another one then there are global solutions for any nonlinearity.
So remember if five was equal to one then we had always blow up in the compact case. If five was I just [snorts] if there was no five or five is equal to one then there was always blow up but now in that model if you put five which is one then there is always global existence.
So this is uh interesting to to compare these two cases.
Okay. So now I move on a little bit. I will talk about Heisenberg groups. Let me skip let me be very brief because uh I guess most people know what it is. So we have Heisenberg group which is uh R2N + one. There is uh group law which is not commutative because of this quadratic term. But identity element is zero inverse is just minus the element.
Uh again we have carocar distance there.
And um we have dilation structure here.
We have it's a homogeneous group. So we have dilation structure which makes sublacian homogeneous homogeneous operator and which tells us that the global dimension is the same.
The global dimension in this case is different from the topological dimension. Uh the global dimension is just the homogeneous dimension which is 2 n plus2. So the the as as a topological dimension of the heisenber group is 20 + one but the global dimension or the homogeneous dimension is n + 2.
So it's not remaining manifold subremanion and um so these are left invariant vector fields uh that we put here they have they satisfy herandanda condition um about the commutators.
So they generate um uh yeah so so this this they they um generate sublassian uh so we put here the horizontal gradient and horizontal sublassian is sum of squares of this vector fields and it's hyperptic by the Handanda theorem.
So it's it's it's a special case of the previous construction. Uh so graded let me stratified groups let me skip a little bit uh here. So the the uh condition that I presented to you before was the following that uh so the global dimension is as I said is the same as homogeneous dimension is 2 n plus2 and uh so necessary condition for blowup. So the previous theorem which I showed in the case of the heism group it says that if there is blow up then this integral has to be infinite for every theta and the convert theorem which we had in our paper was the following that uh uh if this integral is infinite then uh there is blow up. So it's it's in the opposite direction but the orders n minus one n + one here and 2 n they're different. Okay. So this was uh uh this was the case but uh before I discuss the difference in these orders let me just give you small example that you don't have to follow but just to say that this condition is very easy is is is checkable.
For example, we put f is like this sum of two nonlinearities. We can put five sum of two nonlinearities. We calculate this uh measurant minor function. We check the conditions satisfied and uh the condition in the integral just reduces to gives us condition p. So ide ideally if we have like necessary sufficient condition for example or just necessary sufficient condition then uh by checking the value of this integral we can find values of p and q here p is bigger than q so it's a bigger p which is responsible and if it is smaller than this and we have blow up so these conditions are really um easy to check so on the heisenber However uh H1 let me say that H1 the first group is N is equal to 1.
Then what we have is n + 1 is equal to two and is equal to two. So in this case we found necessary and sufficient condition which is formulated here is that uh so there is blow up if and only if this integral is infinite. This integral which is easy to check is infinite. So necessary and sufficient means that if there is um uh so global existence um uh so it's complete characterization however in um uh so for other Heisenberg groups these conditions do not coincide and uh in the paper we put this conjecture that it is this integral which is responsible uh and I think now so as I said there is pretty likely that uh this con conjecture is proved and u but I will not talk about this now yet anyway. So then I move to the next situation. The next situation is when we have um uh let's forget about the group structure.
Now this problem two is when we have here sum of squares. So uh what happens in this situation? So let's say we are on a we have uh herand system of vector fields. We don't have underlying group structure. So we can think of like operators uh and uh uh what happens in this case. So uh let me describe this setup in more detail. So we have uh Rn but we allow uh not just okay Rn has a usual multiplication has uh no because we don't talk about the group it's it's just usual ukidian space but we put anotropic dilation on it to make it more to to broaden the class of applications as I said for example model cases operators So we have delta lambda which is dilation map which is uh anizotropic. So we have sigma 1 in the first variable sigma n in the last variable and uh we talk about vector fields. So vector fields homogeneous of order l if uh when we apply when we combine five with this with this dilation map we apply x and lambda to the power l comes out. So it means homogeneous of order and we start with a family of vector fields. So Hermanda so system of vector fields which are homogeneous of order one with respect to this dation. So they don't have to be homogeneous of order one with respect to the standard ukidian dilation but dilations but they're homogeneous of order one with respect to this delta lambda. And second condition is condition. It's formulated here a little bit in a fancy way but this is just katana condition that uh about this vector fields we can reduce it to the one point again if you want to check because of this dilation structure but basically it is herandic condition so suboplastian is uh hyptic and example which I mentioned before is this gian vector field so let's say x1 x2 inner two this gamma here uh it is uh Um uh so gamma integer so these are hermando vector fields uh but it's not elliptic operator some of the squares but and and it's degenerate elliptic but uh it's hyper elliptic by theorem and these vector fields we they are homogeneous of one if you put uh if you choose these dilation weights appropriately so these sigas are dilation weights Uh again we have as usual we have carocatorial distance here which I will not u define [snorts] again but the definition is written here. So tangent vector. So we connect two points x and y by curves gamma. We look at the length usual length and the property is that tangent vector to gamma is spanned by these vector fields. Here we look at the balls with respect to this distance and they have nice properties because we have dilation structure. So the if we dilate points then the distance dilates of 41 uh if you dilate the ball then uh everything dilates as well and uh what is important is that the measure if you look at the how the volume of the ball growth grows with respect to dilations.
So if you put instead of radius r we put lambda r then lambda to the power q comes out where q is the sum of the weights. So you see that this is uh a number this will be the number actually important for the forget exponent which is the sum of the dilation of the weights of this dation.
So if you're in the case when all dations are standard one then all sigas are equal to one in this case q is just equal to n the topological dimension but for gian vector fields for example this q is different from two so two two vector fields and r2 the the q will be different will depend on the dations on the delations which makes these vector fields homogeneous for the one okay so then we look the um uh heat equation. So we look at the heat exponent. So this is nonlinear heat equation. So we look at the heat operator. It has a heat kernel.
Uh so it has a heat kernel and uh it has estimates. There are heat kernel estimates. This is known. Moreover, recent results by Baj and Boli that we base our analysis on uh give us so it's a a bit more properties. It's a global heat kernel which satisfies all the properties of the heat kernels that you want. So it's for example non- negative uh causal cosal symmetric in space variables it's uh integral is equal to one and uh and we have uh kind of semigroup property uh at t plus s it's it's found from heat kernel kernels at the times t and this okay so then we look at the nonlinear equation that we're interested in for fuja nine equation and then we look at the mild solution. So by formula you has this this expression and we look at this these kind of solutions.
So our result is the following uh so this result applies also to inequalities not just to equalities. So it says that so if q is sum of these weights and lambda is less than this uh this exponent and of u is larger than u to the power alpha. So for example you can take f of f to be equal to to be equal to u to the power alpha if you want to have this ex exactly pa case then even inequality doesn't have solutions so there is always blow. So if alpha is smaller than this number then there is blow up. If alpha is equal to this number then there is also blow up but uh now for for the equation and if alpha is larger than this number then there is uh there is there are global solutions.
Okay. So this is the situation.
Uh I would say also complete understanding of ua effect or general uh hermander for general hermando vector fields. So this is without group structure.
Okay. So and problem three I think I have some minutes left. Uh problem three is u uh concave case. Concave case. And here we can prove surprisingly very general results uh about underlying space. We don't need it to be a rand. We don't need it to be a group. What is surprising? We just need it to be a measure space only measure space. Uh so x is a measure space. We consider this equation.
Okay. You may ask if we have a measure space what is this L? So we go around it. We say the following that uh we assume that operator A has a kernel integral kernel K and instead of this equation we look at this equation in the integral formulation namely we look at this equation. So in this way we bypass necessity immediately. So we extend generality because we don't have to talk about operator L.
we uh talk about equation solution to this equation in the integral form like this when k we keep in mind that k is a integral kernel of f uh so that such equation they are known as uh and and also here I would say that we it's a more general situation we have this forcing term depending on t and x and here we have a function h uh before was depending only on tier now it can depend on t and x. So if nonlinearity is concave, it's it's nice in a sense that it allows us to obtain results in much more general situation.
Uh so in the literature, these are called uh hammerstein type integral equation and uh on the kernel we will not make much assumptions. We just assume that it's measurable positive like we have it for for cases of llassian and sublassian it's just measurable positive and uh when we look at this integral then it has this estimates and uh basically there are two cases if lambdas are equal to zero then the integral is for example equal to one like we have in the cases of u for example or the previous cases integral of the heat kernel is equal to one.
Uh so this is called stochastic case and uh another situation when there is exponential decay. So lambda minus lambda plus positive we call it substic case. So this is a case if operator L has a spectral gap for example. So your spectral starts from zero then we have stocastic case but spectral gap if you are for example in symmetric spaces or we add some potential operator to uh we like hyperbolic space for example then there is spectral gap then we'll have this exponential decay. So those are these two cases.
Uh we have we will have results for both cases but uh in subst substic case we can prove more general results. We can drop some assumptions uh because we have additional exponential decay. So so what what settings it covers. So we can ituh where it applies to and these are all new results I believe. uh consequences of our result they most of them are new. So for example alislas big drama operator on complete remaining manifold with uh bounded from below [snorts] reach curvature.
Uh this is one case when our results apply another uh remaining manifolds without curvature condition but with volume growth which is quite uh exponential to R squ.
So we can allow very fast exponential growth. Uh this will cover this case for example all symmetric spaces covered by this situation. Uh we can consider sum of squares on general groups.
uh covered by this situation. We can consider sublacians before with remaining manifold but on sub remaining manifold consider sublacian satisfying some so-called curvature dimension inequalities.
I will not go into this but uh uh it is this case and bit of some another abstract condition for for when uh this holds but I don't have time to talk about it. So this is a uh assumptions that we have. So nonlinearity g so g is nonlinearity h is function in front of it but nonlinearity what we assume again very mild assumption g is g of 0 is zero and it's so increasing concave concave function this is a uh technical condition about g uh sometimes we can drop it for example is g strongly concave If under this assumption then uh under this condition then uh this assumption is automatically satisfied. Uh so examples are uh as I said before u to the power alpha where alpha between 0 one also this kind of narities. So they are all concave or strongly concave. So the function g which is a condition which you may think here is a no there is no g here but g here which is g of which is u of zero.
So this is basically the initial condition. Uh u of zero is g of zero.
But here is more general because we allow g of depending on xt for the for the cash problem. It would be just uh uh would not would be just uh cash data.
So we assume it's not negative and sometimes we assume this that it's bounded bounded uh away from zero. But um uh sometimes I will indicate that we don't need to assume it always and so g is no negative bounded function and h the function in front of g it is just a measurable function no negative measurable function. So very general assumptions and this is our theorem with uh colleagues uh who was in Gant University uh for a while and then in Armenia we have the following result that if uh so uh we assume that this h this is a function in front of the lelassian is bounded by some functions of t so it covers the case when it is just a function of t but we can allow depending uh on x also then the result is the following that uh there is there is no forget exponent in this case there is always global existence so there is always uh there exist always global bounded solution of that equation and uh remember that in in in uh when we I I I didn't mention it but when we talk about global existence Forget exponent we talk only about small data global existence for small data. So here we have global existence for any data also for large data. Then we don't have blow up. We always have global existence for large data. Moreover we can do a bit more. We can u and you can see part the idea of the proof from this formulation is that we can have this iterations I uh uh iteration scheme uh coming from integral equation and uh uh the point is that this iterative approximations they converge they give us decreasing sequence which converges to our solution. So this is existence uniqueness.
We have uh uh always uniqueness of solution and uh uh as I said sometimes we assume that g is bounded uh away from zero then if they bounded away from zero then we can also get the rate of convergence of this scheme of these approximations. But uh if you don't assume these conditions then we just have convergence without the rate. Okay. So this is uh uh rather general result and uh I think uh my time is uh finishing. So let me summarize. So uh uh in this talk so what what I what we did we establish uh when we have time dependent nonlinearity then we establish necessary conditions and generally remodeling groups um or necessary conditions for blow up which is equivalent to sufficient conditions for the existence of solutions.
um in the case of Heisenberg group we obtained groups we obtained uh implication in the other direction and uh they coincide in particular cases but in general they up to now according to this results they don't they didn't coincide and uh so heat equations for general sum of squares without group structure uh uh I have presented pujit exponent.
So this problem is resolved. Um I would say uh we have formula for the pjit exponent. Uh we can obtain different uh implications for this. We can also so this was uh nonlinearities I consider convex uh uh like power type but we can consider also logarithmic type. I didn't uh talk about this here. And for finally for concave nonlinearities we have global existence and uniqueness not just for small data for large data and uh without any for threshold because nonlinearity power in this case it's less than one uh so we always have global existence and I think I stop here thank you >> okay thank you thank you professor or Michael for your nice and interesting talk.
Okay, so I think now we have um time for questions.
So let me check the chat on YouTube. If someone has question, you can write on the chat.
Please remember that.
Okay, I am checking.
Okay, looks like uh there are no questions for now.
Uh maybe Jenny or the professor Jenny has a question or the professor Christristiana please you can raise your your virtual hand also okay Jenny Jenny said no thanks everything is clear >> thank Thank you. I'm glad. Yes.
>> Yeah. Yeah. It was very clear.
>> Maybe everything is clear. So, yes.
>> Yeah.
>> On the chat.
>> Okay. Jenny, thank you very much for the nice presentation.
>> Yes.
>> You're welcome.
>> Okay. On the chat, we have a hello from Italy. Okay. So, we are people from Italy at this moment.
Okay. Okay, we have a question. What results do you have if f is not continuous?
Juan Pablo Lopez.
>> Yeah.
Yeah. If f is not continuous then we don't have results but no okay depends if uh concave case concave case we have these results which uh so here uh in the concave case we don't assume continuity. So uh if the conc no no sorry we assume continuity. So yeah, we assume continuity indeed. And uh and in the previous cases we assume continuity then we don't have results for if f is not continuous. Yeah, we don't have results. We we haven't looked at it. Uh but uh maybe this year maybe some this can bring some effects. Uh that's right.
But we don't have results.
>> Okay. Thank you Jul. Tell us. Thank you so much. Tell you, sorry. Thank you so much. Very nice talk.
Okay. Okay. So, I think >> Okay. Thank you.
>> Thank you, professor again.
>> Yes.
>> Okay. So, for now we have a few minutes to the next talk.
D the nail will start in four minutes. Okay.
So I think we will wait.
Um our next speakers is the professor Christristiana de Phillips.
>> Professor Okay. You're here. Great.
>> Yeah. Hi Christian.
>> Okay.
>> Hello. Should I share the slides?
Yeah, you can try.
>> Yeah. No, because in general it takes me a few minutes to do this. [laughter] You have a few minutes.
>> I'm not very good with this. Uh, okay.
Uh, let's do like this.
Come on. Last time it worked. Uh, when do you see it?
Uh, okay.
Oh.
Um uh do you see them now? No.
Okay.
Okay. So, probably [sighs and gasps] Okay. So, maybe it can make sense to do the following.
Um [snorts] Um, give me one second. Okay.
Um uh Chrome top for for some reason it doesn't allow me to share uh to share my slide.
Okay, let's try this. Works, >> it seems. Yes. Okay. So, let me just rename it.
Okay. Fantastic.
Uh, is the visual clean or >> Yes, it's uh not full screen perhaps, but uh >> um >> it's clear. Let me try if because I I have them full uh full screen on my computer but uh um do you know how to share slides differently? I only noticed >> L perhaps >> uh control L.
>> No no nothing happens.
Um window to uh yeah sorry uh window no then uh then it goes like this. I guess it is the easiest way.
So we can uh read presentation mode. Cristiana.
>> Um fantastic.
>> Oh okay. [laughter] >> Great. You spotted it. Thank you a lot.
>> Okay. So perhaps we can start. So it's my pleasure to introduce Cristiana.
Cristiano de facilitis is an Italian mathematician whose research focuses on regularity theory for elliptic and parabolic partial differential equations.
She completed her PhD at the University of Oxford in 2020 under the supervision of Christensen.
After a posttock position at the University of Turin, she joined the University of Parma as an assistant professor in 2021 and now she is a full professor in the same university.
Cristiana has received numerous prestigious distinctions recognizing her outstanding contributions to the analysis of partial differential equation. Among these uh are the CM early career prize in analysis of partial differential equations in 2025.
Then the European mathematical society prize 2024. The Bardi prize 2023. The Joino Yapino prize awarded by the academia national in 2020. while [snorts] in 2023 she was elected as an inaugural fellow of the European Mathematical Society Young Academy. In addition to these honors, she has obtained important competitive research funings including a prestigious European research council starting grant awarding in 2025.
another recogn recognization of the originality and international impact of her research. Today she will uh speak on nonuniform ellipticity and nonlinear potential. Please Cristiano and thanks for your talk.
So thank you a lot for the very generous introduction and my apologies for the usual technical problems. Uh thanks also to the organizer for allowing me to present my results in such a prestigious venue. Okay, I'm going to report some recent regularity results on new oliptic problems.
Okay. Uh what am I talking about? Uh okay, sorry.
What's wrong with this?
Um, okay.
This um why it doesn't stop.
Okay, fantastic. Okay, so what is molyicity?
Uh keep in mind as a key uh model example the nonparametric area integral.
Okay, more general new ellipticity is a very degenerate ellipticity type that fures this configuration. Okay, so you are essentially tracking the say regular autonomous integrant between in the large two negative powers of the gradient variable. Indeed here here mu is larger oral than one and is a possibly nondereasing function. Okay.
Okay. Now, why the regularity theory for this class of problems is pretty delicate.
First of all, because they are intrinsically non-uniformly elliptic.
Indeed, if you uh try to quantify the ratio between the highest and the lowest second value of the action of a molytic integral that is the electricity ratio related to this problem, you see that it immediately blows up on uh for on large values of the gradient variable just because your problem is molytic. do not need to quantify the size of me or whatever.
Okay.
And uh in when whenever you're dealing with uh this class of equations or functionals well the very first fundamental step you want to make in order to secure regularity is proving gradient boundedness that is lip shit's regularity for solutions. This is already noted by Joint using as a test bed precisely the minimal surface equation. Indeed, as soon as you know that the gradient of solutions is bounded, then this object becomes uniformly elliptic quadratic and you can push the regularity as much as you want.
So if all of us agree on the fact that the most important aspect for regularity is being able to show that solutions are lipids continuous then you understand where the difficulty in dealing with molyic problems also relies indeed remember that the only that you are controlling negative powers of the gradient variable meaning that mu electricity plays against the achievement of lipid's regularity.
Indeed, the largest gradient variable becomes the smallest is the amount of information that you control. You lose more and more information. That's why this uh this uh type of equations are very very delicate under the point of view of regularity.
Now a natural question would be if uh malipic problems or more generally new uniform realptic problems lives in maths textbook or they have something to do with reality. Well, they actually pretty much motivated but by the modeling of real objects. For instance, electronological fluids or strongly anisotropic materials like lava, tissues, probably your breakfast like milk and cereals or mud as well or um a mixture of fluids with different hardening properties. Then we already mentioned minimal surfaces that are also um important model in electromagnetism and they also appear in WKM theory. Let me be more spec more more specific under this point of view. Indeed the nonparametric area integral is mu iptic with degree of mipicity equal to three.
Okay. And the very same model can be generalized up to some extent by considering let's say variation on the nonparametric area integrant as well. Or we can look at the lorens brother or the nonparametric area integral that is the electrostatic burning field energy. All of these are nonuniformly elliptic.
But we could also consider uh a log type models whose most prominent example is this one. They play important roles. For instance, in material science in the modeling of brand layering fluids or one can compose more and more logarithms in order to approach linear growth but still remaining quantitatively super linear. Okay. So all of these examples are muyptic and therefore non-uniformly elliptic.
Okay. Now the matter of regularity in nonuniform elliptic problems I I hope I convinced you that it is a rather difficult task indeed it attracted the attention of lots of important names in mathematics before the 70s where very heavy machineries such as global methods barriers classical solutions trans solutions were employed but let's say that significant progress appeared after the 80s in particular particular thanks to the work of Jacob who introduced a series of very specific models that are if you want closer to the uniformly elliptic world rather than the non-uniformly elliptic one but give you lots of insights towards the general theory. Moreover, Chelin and Marelini laid the foundation for the model the modern variational approach. It is the one that nowadays everybody used and therefore allowed to overcome the the use of global methods classical or strong solutions. Okay. So the essentially the possibility of working in a variational setting allows it to work okay precisely where you can prove existence of solutions.
And finally, expos connected the regularity of solutions to nonuniform realtic problems to a classical phenomenon in the calculus of variations that is the occurrence of lentian phenomenon. And as we shall see the two things go are pretty much linked together.
Okay. What about the basics of regularity in the oliptic problems? If we focus for instance on this uh basic model example the joint efforts of uh Bomb Miranda Leon Simon and more recently Beck and Schmid push even further the regularity for this class of autonomous integrals tells us that if mu stay between one and three three remember is the case of minimal surfaces. So it corresponds to pal 2 then solutions have continuous gradient and then you can push it even further if you like. Um keep in mind that this limitation okay mless density is essentially telling you that the ellipticity ratio related to this class of problems cannot cannot blow up too fast. Okay.
In particular, if mu is larger than three, then nothing is known nor on the regularity side nor on the irregularity side. So it is open. Okay. Still uh of course the matter of mpticity can be made even more wild by considering also an isotropic term that force a cube growth from above and reflects on the growth electricity filter of the integrant itself by leading to mucity.
Okay. So you have an even wilder behavior of the action of your integrant from above.
Okay. What about the regularity for this class of mu elliptic problems? Well, first result is due to fon that again prove that if Q is not too large meaning if the electricity ratio doesn't blow up too fast then solutions have continuous gradient and of course the result can be generalized to the vetorial situation of course provided that the integrates depend on the modules of the gradient they can cover the case of iterated logarithms as well as obstacle problems of course for the scalar setting um build our folks later on achieved partial regularity within this setting and uh let's say that the restriction on the blow up rate of the ellipticity ratio was recently improved by back gam shuffner okay what about bonded solutions bounded minima we already saw with that boundedness played a rather consistent role in achieving regularity already in the minimal surface case.
What can be said about mucio electricity? Well, that bonded minima enlarges if not maximizes the uh non-uniformity range and this is a result of build and folks. However, all the results that I presented so far strongly rely on the possibility of differentiating twice your variational literal. one is your oiler lanch equation that is satisfied by minimizers and the second time you need to totally differentiate your oiler lanch equation and then if you're considering bounded minima then you need to combine this with galardon type inequalities okay in order to construct an unbalanced mo iteration this perfectly work if your integrant allow you to allows you to differentiate But this in general not the case indeed if you plug in for instance non- differentiable ingredients you can no longer work this path. Okay. And so all this construction delivered up to now simply breaks down.
Okay.
Uh another important message that uh the the regularity results I shift so far tell you are that not all the values of mu and q are admissible or not all of them guarantee regularity meaning that in all these examples in all these results that I told you the ellipticity ratio cannot blow up too fast and this is a common feature of nonuniformly elliptic problems indeed if we look at polomial uniformity that can be uh in a sense quantified by trapping the action of your integrant between two different powers of the gradient variable.
In this case, the situation was uh clarified around the end of the 80s.
Indeed, Jagint Marelini Jinta Marelini and Dong prove that suitable constraint on the blow up rate of the ellipticity ratio is necessary in sufficient condition for regularity. In particular, if the ellipticity ratio blows up, then unbounded minimizers might appear. What does it mean? Well, this is a rather striking phenomenon typical of non-uniform elliptic problems. Indeed, this happened for strongly convex integrant in the scalar setting.
This is again a pecularity of non-uniformly variational integrals.
Indeed, if you want to see within the same setting irregularity in the uniformly elliptic setting, then you want to look at vectoral problems in order to uh experience formation of singularities.
And so what about non-aututonomous integrals? So what happens if we plug in ingredients?
Well, uh this is a a rather delicate phenomenology because if you already think to the case of harmonic maps I right there is a whale lamb was up to a certain scale you can essentially swap your function with the with a modification of it however as soon as I plug in ingredients let's say other continuous ingredients then the presence of these external components somehow inhibits the self-improving properties of delion and therefore you can only prove and continuity of the gradient for instance this is a classical sher theory therefore the pres therefore if we if we plug in ingredients in non uniformly elliptic problems then the situation becomes pretty much worse and in particular novel phenomenologies and novel interactions between coefficient and gradient variable might occur As a test bed, let's consider this model introduced by Jacov in the setting of homogenization and for studying the possible occurrence of labent phenomenon. Now the structure is pretty simple. You have a pilion plus negative weightlion and what should we expect under the point of view regularity? Well, point by point this object is uniformly elliptic.
Indeed, if a were constant then either if it is positive or uh or zero doesn't matter. Uh minimizers have always solve their continuous gradient.
So one expects that if a is allowed to vary so it is non negative it is let's assume that it is continuous then if it is under continuous I expect continuous gradient. Okay. And this is sharp thanks to our counter example tutorials that tell you that impossibly the generate uh integrals like the pillar plion uh the best you can get is gradient that they're consumed. So the picture is rather clear.
However, the fact that a is allowed to vanish gives to this model a very mild amount of nonuniformity that still allows to escape the classical setting. Indeed, assume for the moment that A is just bounded and let's imagine that we are on this tracker board. We flip a coin and decide if we want to go left or right. So we go either blue or red. Let's say that A is one on the red zone and zero on the blue zone. So we can switch on and off depending on where we decide to go. G2 electricity information.
But this is a very violent interaction that goes beyond the classical you know perturbative behavior of coefficients allowed by uniform elipticity.
Nonuniform elipticity allows for this and indeed this new coefficient solutions interaction gives rise to very um very huge u irregularity phenomena.
Indeed, this is a contra example due to famili the electricity ratio blows up too fast and this is quantified by oops by by this inequalities linking ambient dimension growth electricity exponents P and Q and alpha elder exponent.
Then one can construct a family of dish problems uh driven by this uh um this double face operator such that solutions have set of essential discontinuity points of nearly maximal abs of dimension meaning that solutions are not even better than any of their competitors. Okay, but why this is significant? Again, this is a scholar example.
Remember the pecularity of non-uniform elasticity already in the autonomous setting. You might see um irregular solutions even when in the uniform reality case regularity should be expected. Moreover um this this example holds for minima.
So you are not violating any energetic information.
Uh the boundary atom is lit continuous.
can be taken even smooth no problems. Uh the dial problem is defined on a ball.
So there are there is no weird formation of singularities coming from casps on the domain.
The integrant is radial. So there is no eugenotropy of any kind.
It can be made non- degenerate or non-s singular. So now we killed essentially any type of any any possible source of singularities. Therefore, the real origin of uh singularity is the harsh interaction between coefficient and gradient variable. And I will turn back on this.
Indeed um this result was later on sharp sharpened by balining and that showed that actually the set of essential discontinuity points of minima as soon as Q is too large as soon as they blow up the ellipticity duration blows up too fast then the set of essential discontinuity points of solutions might have maximal as of dimension that is n minus p.
Moreover, the labia phenomenon might occur in the class defined upon smooth boundary data.
Uh the density of smooth maps in a modular convergence might fail while regularity holds whenever Q is less than or equal than P plus alpha. Okay.
Um okay, moving further, we just spotted a classical fact for uniformly elliptic problems that is shoulder theory. If uh coefficients are rendered then the gradient is correspondingly there and an open problem immediately emerges that tells you that what happens in the non-uniform elliptic setting. Does sher theory holds and we already saw that several of the paradigm of classical sher theory might break down as soon as a very little amount of non-uniformity appears as it is the case of uh the doublephase energy. Okay.
Indeed uh this was a problem raised at several stages in the literature. Here indeed we see a classical a comment of Liberman under a classical paper of Jacquinti.
Indeed, uh, Liberman said that the perturbative approach of joint juicy perfectly works in the uniformly elliptic setting but dramatically fails in the nonuniformly elliptic one unless they don't already start with lipids regular solutions. But this is this essentially kills non-uniformity.
Okay, I mean if the gradient is not allowed to blow up then I mean nonuniformity simply doesn't make any sense.
Moreover, another um classical theorem due to dura shows you several limitation of uh the uh previous methodology for uh delivering the in the ununiformly elliptic setting. Indeed they had to assume bounded gradients and moreover all the entries of the nonlinear field driving the equation are supposed to be differentiable but remember in the light of contour example as the ununiform models include possibly the generate models then the best regularity you could expect is a gradient continuity therefore it doesn't make any sense to consider ingredients that are more regular than just elder. Okay.
And in particular, Ianov identified where is the chief difficulty in dealing with this type of results. That was achieving lip spots. That was the key problem. Okay. Because the perturbative methods were no longer functioning.
Indeed once lip shious continuity is there then non uniformity becomes immaterial and one can trace the problem back to the realm of uniformly elliptic problems. Okay.
Moreover, when mulepticity comes into play, there is another a further difficulty that is the problematic compatibility of mulepticity with the classical sher setting that prescribe elder continuous ingredients and this will be addressed. Indeed, an answer to uh to to this issue can be found in these papers. And uh I mean in in this talk I will try to focus on the matter of the validity of sher theory in the molytic setting and in particular on the compatibility of molyptic problems with with other ingredients.
Okay. So uh one of the main novelties of our approach uh was indeed a was indeed the achievement of direct lip estimates. Okay, we prove liit estimates in a direct way.
Uh indeed Ivanov was right. This is the main difficulty to overcome. However, in order to do this, we had to overcome 100 years old paradigm for the achievement of lip estimates in non- differentiable problems. Indeed, if you think to I don't know basic uniform realtic uh PDS with elder ingredients, how do you prove lip regularity?
Well, you first prove that the gradient is elder continuous and then you retain gradient bundledness. This is the usual way.
However, this is no longer feasible in the non-uniformly elliptic setting owing to the deep nonomogenity of all the excess DK estimates involved and therefore one needs to first go to lipids and then try to prove uh continuity continuity of the gradient.
uh so let's have a look to what happened in the classical literature from the oliptic problems. Well, in all the results you can find uh coefficients are always assumed to be citular and this might ring some belt. Indeed, classical counter examples due to jquint and surg tells you that if the ellipticity ratio blows up too fast, okay, then one can construct a very simple model integral with the s regular coefficient even smooth.
such that minimizer fail to belong to W11 and this is particularly problematic.
Okay.
And even uh if you want more disturbing phenomenology appears in the uh classical area setting also in this case if coefficient fails to be C2 then minima might fail to belong might fail to belong to W11. Okay. So you might find meanwhile that just belong to BV and these two counter example highlight how problematic is new electricity with respect to shelder theory. So maybe two uh object with the uh that that might be difficult to com to to put them together. Okay.
Okay. So let's try to see what happens for uh for the regularity of nonautonomous new oliptic problems.
Can we savage sher theory somehow?
I will try to convince you that [clears throat] for at least a reasonably large class of molytic problems flow the is there in an optimal fashion. Okay.
So let's have a look to the case of uh logarithmic type energies does those nearly linear growth meaning that they are super linear at infinity in this sense but grows lower than any power larger than one. This is a sort of limiting configuration before the classical area case. Okay.
Um, our first uh typical model example is the energy that was deeply studied by Fraser Fuks and Seran is rather common indeed in harmonic analysis.
We see that it is uniformly elliptic with a logarithmic type of blow up of the ellipticity ratio.
So classical methods f is non uniformly elliptic indeed. And the same can be said about uh iterated logarithms in which the ellipticity appears again and the ellipticity ratio again blows up even faster.
What do we have here under the point of view of sher theory uh for el type energy? This was settled by myself and later on sharpened by myself Fina Philip and Miranini. What we have here is that if u is a local minimizer of uh this local energy with a coefficient that is that stays away from one and it is better continuous then the classical shoulder implication and of course uh similar gradient continuity holds if we consider it logarithms and of course this result can be reasonably extended to the uh vectoral situation provided that functional not satisfy your back structure meaning that they depend on the modus of the gradient.
So what about the uh the case of of the double phase energy? So let's try to understand what's going on in this limiting configuration and if this can you know give some type of explanation about what what happens for uh for for sher theory when some anizotrop some anotropes is there okay and in particular what is the relation between malipicity and inter continuity okay so let's consider Consider this model with of course iterated logarithm logarithmic variant.
This is of course nonuniformly elliptic.
Therefore any type of perturbative approach immediately fails this situation.
And let's see what we have here. Well, if we [clears throat] take uh local minimizers of this energy and we impose that the ellipticity ratio doesn't blow up too fast, then we can conclude that the gradient is locally continuous in omega.
This is uh this essentially parallels the classical result when polomial nonuniform ellipticity appears in a quantitatively superlinear setting and the same bunch of techniques can be extended to cover various type of logarithmic rowing models.
But let's have a deeper look at the literature and let's see what happens if we consider bounded minima bounded solutions. Okay.
Indeed uh looking back at the previous at the past literature we see that the combination of local boundedness with a suitable constraint on the blow up of the elliptication leads to sharp result.
[snorts] In this is indeed the case. Uh what we have for the basic uh uh for this very basic model is that we take bounded minimum a here is under continuous. We assume that q is less than 1 plus alpha and then the gradient is locally under continuous in omega. I want to try to convince you that this is sharp. Okay.
And in particular this is the right setting where to study sher theory molyptic problems. Of course, uh the one that I showed you is a model result model case of a more general results that allow to consider iterated logarithms as well as obstacle problems. So under classical assumption on uh on on the obstacle what you what we obtain is that if Q is less than 1 plus alpha and uh solutions are bounded then uh the gradient uh the gradient of minima is ether continuous and in particular if you are dealing with non-s singular problems meaning that here has is sorry is strictly positive positive then we can recover at least in the plain logarithmic case we can recover uh alpha continuity for the gradient. Well if mu is larger than one then we have a loss of alpha over two still uh what are the the the main difficulties in dealing with this problem? Well the main difficulty is covering the warn uniformity range. How we do this? Well, the basic observation is that uh the integrant is non- differentiable. So in particular, there are no ways to achieve a second variation.
However, if we read the elder continuity as fractional differentiability, we can observe that the integrant possesses like 1 plus alpha derivative almost.
And therefore one can start thinking to the to to to build up a fractional moderation to get regularity.
However, molyticity hits harder under also under this point of view. Indeed, the the fact that you are dealing with molytic problems forces a lo a loss of differentiability gain that results in a loss of alpha / two in the non uniformity range. Okay. Therefore, one needs to bypass also this obstruction.
The way to do this is to construct a hybrid reverse inequality that interpolates between any arbitrarily high leag norm of the gradient. Here t can be anything between one and infinity of course infinity excluded and the L1 norm of the gradient itself.
However, this constant must be proportional to an arbitrarily small power of the infinity norm of the gradient. And what is most important is that this omega is as small as we like but it does not depend on t.
So at this stage one can say well then you are done right because you just send t to plus infinity and you you have your lip control and then you can reabsorb the small and infinity norm that you have on the right. This is not the case because uh fractional moer type iterations are linear iteration and therefore you have a linear gain in uh in integraability that cannot be killed by sending just just by sending t to plus infinity.
Therefore, what we control is any arbitrarily high leag norm of the gradient via an arbitrarily small infinity norm of the gradient.
Arbitrarily small power of the infinity norm of the gradient. And this will be particularly helpful when plugging in this kind of information in uh in a nonlinear potential theoretic version of the George that will eventually result in uh ellipit regularity subject to the sharpest non-uniformity threshold.
Now other technical novelties that appear in this paper is a novel approach to uh variational obstacle problems that bypasses classical techniques that dates back to the 80s. Okay. Indeed, how do you deal with variational obstacle problems?
Well, the first approach is linearization after folks. Indeed, one can show that minima of varational obstacle problems of this type and keep in mind that the presence of an obstacle is actually a convex constraint.
Therefore, minimum discuss of problems are actually unconstrained local minimizers of the same variational integrals with a force in term which is I not problematic but we should see how the forcing term looks like. Now the forcing term essentially behaves as the divergence of the f of x i which perfectly makes sense whenever this object is differentiable. Okay, which is the case if f is suitably of course suitably differentiable in the gradient variable and coefficient that represent a major obstruction. If the coefficient is merely header, this object does not does not make any sense and there is no way around this.
Another approach is the perturbative one after co and the the idea is essentially uh to uh compare your minimum of the obstacle problem with a force dial problem of this form. So you are again moving the obstacle from a pointwise constraint to a right hand side and then you you compare this forced variational integral to a homogeneous one in order to extract regularity from the frozen problem.
However, here there is an even obstruction that is represented by the perturbative approach that remember strongly require uniform ellipticity.
Okay, if that thing breaks down, there are no ways to adapt this type of approach for in for the non-uniform elliptic setting.
So both these techniques fails for already for our basic model. Indeed uh linear linearization fails because if a is continuous this this quantity is meaningless.
Again perturbation fails because our model is maleic that is non uniformly and so you no longer have homogeneous excess decay estimates.
And what we do is instead again an in hybrid approach that allows to fractionally differentiate the problem and then bypass all this marginal obstruction created by the coexistence of non- differentiable ingredients in mu electricicity.
Now I insisted a lot about the fact that uh our result is sharp in the sense that if the elliptic ratio blows up faster than the one that I showed you then serious irularity phenomena might appear. Let me show you why. Well um let's assume that we are in dimension n larger or equal than two. Let me consider a cube the n dimensional cube and let me assume that Q is larger than 1 plus alpha. As this image suggest counter type fractals would play a major role in this situation.
Okay. So what do we have?
Well, first of all, we can construct a mutilating coefficient A that is alpha continuous a smooth boundary atom and force labrenia phenomenon to occur in the dish class with boundary atom equal to U0. Indeed, we can show that the infyum attained by our log double phase energy in u not plus w11 of q is strictly smaller than the infom attained by the same functional on the set of smooth moot of c infinity with compactly super functions and you zero trace on the boundary.
This is the classical formulation of leentia phenomenon. Moreover, we can simultaneously show that the density of smooth maps with respect to modular convergent fails.
And so we already saw that as soon as the electricity ratio blows up, even basic functional analytic properties or spaces of function defined by the definitess of this energy might immediately break down.
But uh uh what is more significant to me is uh the lack of regularity.
Indeed what we can show is that give me any asylum I can construct moderating enough continuous modulating coefficient and a smooth boundary that such that solutions to this dial problem fails to belong to W1P for all P larger than 1 plus epsylon meaning that of course in this situation sher theory instantaneously fails I mean your your your solution does not belong to W12. For instance, if you take P uh sorry, epsylon sufficiently small.
So in particular, it cannot belong to W1 infinity. It cannot be the gradient cannot be continuous.
Okay, how do we prove this result? Well, this is essentially a story of uh uh malicious competitor that is emitted in the class of competitors for this del problem by a cheating coefficient in a poor hornness boundary dat help helpless in this situation. Now uh I said from jokes what is the idea here?
Well, the idea is that the fact that A is allowed to vanish, okay, permit to certain irregular maps that in principle doesn't do not have W finite W1 Q energy to enter within the finite energy class for this function because you can essentially force A to vanish whenever the gradient of this competitor is non zero. This way you are forcing your competitor to have finite energy and therefore to be admissible for your problem.
So how do you construct it? Essentially you take the regularized version of characteristic functions of double cones with tips on a very dense uh counter uh counterfract. Okay.
Then which is compactly contained inside your unitary cube.
Once you have used your very bad competitor, then you cut it off, smooth it out.
You can make it compatible with a very regular trace at the boundary of the cube.
Then the idea is to prescribe very traces at two parallel faces of the cube with different sign.
So this is uh more or less the picture.
And if you overlap those two images, you see that a precisely hides those region in which the competitor oscillates very widely. Okay.
Okay.
Then how do we conclude? Well, the fact that you have on parallel faces very high traces with different with different side with opposite sign. Okay.
force the MIMO to have too high energy to you know cover this job and therefore it is forced to strongly oscillate on the fractal. This way one can in some sense force the set of essential discontinuity points of view inside the fractal in such a way that the out of dimension of it becomes larger or equal than n minus one - epsylon. Okay. But then basic so theory kicks in and tells you that since the set of of your set of essential discontinuity points is so large is so big.
You cannot belong to W1 P for all P larger than 1 plus epsylon. Okay.
Otherwise this object would have outdoor dimension strictly below n minus one minus epsylon. And this leads to a contradiction.
Okay.
What about general neolytic problems? So how do they uh react to the presence of elder coefficient? In particular, can we make moltricity compatible with shelder theory?
Well, what what we proved is that uh the same uh uh constructions for the counter example works for suitable relaxation of this area type integral. Okay. So there are no hopes of regularity whatsoever as soon as Q is larger than 1 plus alpha.
Now in within this threshold where again you can show that uh minima uh minima will never be continuous in particular provided that this this uh this holds and what what you can see from this threshold any particular what you pay and indeed if we have a look at the the various bunch of techniques that are employed in order to show regularity uh the Q and isotropy should be balanced within this way. Okay, which means in our situation recalling that mu is equal to p + one it means that an inequality of this type should hold. Okay.
But uh but what's the problem here?
Well, the problem is that uh P is larger than one, right?
Therefore, in order to be able to deal with any alpha elder continuous coefficients also with coefficients that are that are that have a very low elder continuity exponent, then you have to take mu very close to one meaning that a compatible configuration with sher theory is the mu ellipticity given by log type models.
Okay. As our double face one.
Okay. Uh so this is uh those are the papers that are the main references for this work and I thank you for the attention and apologize again for the technical inconvenience.
>> So thank you very much Christristiana for your nice and clear uh presentation.
Uh are there some uh question or comments?
You can uh write in the chat for the moment there is nothing. Uh I have just a curiosity Christian. Um so uh you assume that the function a is greater positive okay so it can be also zero but uh does the region where a vanish need to satisfy any particular geometric properties or can be no nothing >> you know it can be whatever you want and this is one of the chief difficulties in dealing with the this class of integrals the fact that you have essentially zero control on uh the zero level set of A you have you only know that a if you are after regularity a is an ender continuous function and you have the zero set of an alpha continuous function which is what it is. You have no no additional information and you actually achieve regularity without having any information on this.
>> Okay. So it can be also zero to the boundary of omega or >> it is free to to do whatever you want.
It is just an elder continuous function that can vanish whatever it prefers.
>> Okay. Thank you very much. I don't know if there are other question.
If uh not I thank again Christristiana.
Thank you very much Cristiana for >> your contribution your talk.
>> Thanks a lot. Let me >> thank you very much. Sorry.
Uh we have perhaps about 10 minutes before the last the the other talk.
So we can make a break of 8 minutes.
>> Hello Jenny.
>> Hi Andre.
I don't know if the next speaker is uh >> it's not here yet but uh it's just minutes before. So we will just wait for her and we will connect everything later.
>> We have nine minutes or so.
>> Yes. [laughter] >> N back.
Okay.
>> Thank you.
>> Thank you.
Hello, Professor Sil.
Hello, Professor Sil. Can you hear us?
video.
Okay.
foreign.
[laughter] [laughter] No, it's just >> [laughter] >> Sorry.
Okay.
It's better.
See now [laughter] [gasps] Fibon.
Okay. See.
Okay.
Okay.
Okay.
[laughter] Thank you.
particip.
YouTube much YouTube.
Okay. Okay. Okay.
>> Okay.
I think we can start now.
Uhto I think.
Okay.
Gracias.
Okay, I think we um Hello everyone. It's a pleasure to introduce Professor Sylvia Pesia. She is a professor at the university of potam and is internationally recognized for her outstanding contributions to global analysis, spectral geometry, pseudo differential operators, non-commutative geometry and mathematical physics. Her research has play a fundamental role in building connections between analysis, geometry and quantum field theory.
Today, professor Pesia will present her talk entitled the lu infinito aloofenito.
Uh before we begin, I would like to mention that professor pesa is not feeling well today but has very kindly made the effort to join us and give this lecture. So we are truly grateful for her commitment and we kindly ask you all for your understanding and patience during the presentation. Professor Pesa, thank you very much again for being with us and uh you you can start. Thank you very much.
>> Thank you very much.
I was now asked to turn to English. So I'll turn to English. I I thought I had to do everything in Spanish. I'd prepared everything in Spanish. But now you'll have the slides in Spanish and the and the talk in English. So thank you very much for your patience if I say things which are not quite correct because my head is not functioning very well. And what I want to talk about is uh based on joint work with two Chinese friends and colleagues um with whom I've been um uh working with for 20 odd years I would say. Uh so their names are on the slide Lu and Bing Jang. And um so had I known it would have been in English I would have told them but it's a bit late for them. I don't think they speak Spanish. Okay. So um maybe um I don't know whether you can interrupt for questions. We'll see what happens. So wait a minute now I'm I'm having problems. Oh yeah. Okay. So what we're going to talk about is how to make the infinite finite. And uh it sounds like a strange question uh but it's in fact at the heart of uh contemporary mathematics and physics how to extract a finite part from an infinite or you could say divergent expression and uh it's as old as Ola even older probably and we'll then jump to fine man we'll see how so what do reman both no Ola was Swiss I was going to say German was Swiss Reman was German and Fineman American. What do they share? Um, so here are their uh portraits and you can see they're from different um different centuries. So what do they have in common?
Um so the uh leading idea is uh that uh there's a common scheme to the three approaches or the three ways to deal with infinities namely extracting a finite part from an infinite or divergent part. So extracting a convergent part from an infinite uh divergent part. Uh so you back to school uh material you probably remember well when you first met the harmonic sum oiler uh sum of k sum with k go running from one to infinity of 1 / k the ja function of reman and if you take s= 1 which you're not allowed to do in principle as it stands here because you have to take the real part of s larger than one but assume you were taking s= 1 you would find again the harmonic sum and this integral um which if you not familiar with physics might look a bit um strange so the convention uh later probably is that this um k which is in Rn the absolute value stands for its norm and some in physics you don't write it so if if ever you see k2 it means norm of k to the square m is the mass and I'm assuming it's non zero because um otherwise I'd have an infrared divergence. So an a divergence around zero and I don't want it here there's a divergence at infinity because the two is smaller than the four more or less.
Okay. So um that's the kind of thing that's a very um uh downto- earthth example of the kind of thing we are looking at. Can we assign to all these expressions a finite value? And then you might ask why? Um and the answer is yes.
uh but yes and no because yes you can assign a finite value but it's not very satisfactory because the finite value depends on the method you use as we'll see but why do you insist on uh extracting a finite part from a divergent expression because uh you could say at least for for for me one of the essential applications I see is uh that you want uh to measure the uh surrounding universe. And for that um this is supposed to represent to a picture um accel um particle accelerator and uh you feed it with some data but you can't just feed it with infinities. You have to feed it with formulas which will spit out some uh finite quantities. So you have to deal with in these infinities which are everywhere. In fact, I think mostly we're confronted with infinities when we want to describe our universe.
So let's go back to the harmonic sum and uh uh I suppose you also remember from a long ago maybe for most of you. I'm sorry about the latte which is not perfect as you see but I did my best. Uh so the red part is supposed to be what you want to get rid of. Why do we want to get rid of this log n? We're summing 1 / k from 1 to n. As a matter of fact, we wanted to sum it from 1 to infinity.
So we want to let n go to infinity. And for that we have a problem because log of n goes to infinity when n goes to infinity. And uh Oiler had already uh spotted that out and he just took it uh just subtracted log of n as I put in the second in the um circled or squared formula. You extract a log of n you you subtract it and what you get a very famous constant the oiler mascaroni constant and um this constant is still very mysterious. So all this is still not completely understood. and Reman uh a century later as we saw um did the same thing but from another point of view uh introducing the famous function which carries his name the za function and uh recalling our uh first university steps we know that this sum k over minus s uh sorry k to the power minus s uh from k= with k running from one to infinity is convergent for real part of s posit uh larger than one and unfortunately we want it for one so that's bad luck for us and when you let the sum uh which you find uh for real part of s larger than one when you do a an expansion a meamorphic expansion at one you find what's written uh below uh at the bottom of the slide and the red part is again what you want to get rid of because we want to evaluate it at one. So 1 / s minus one is is not really a good choice but uh that's okay. I'm sorry I tried to repair this just before the talk. So and it didn't Oh sorry sorry. [clears throat] So wait a minute. So you can't see the last formula but it's a good exercise to guess it. So you take the difference uh theta s -1 over s minus one and you let s go to one and this goes to gamma. So they found the same gamma uh which is uh reassuring but we'll see that it's not that reassuring because everything depends on the scale you use. So if instead of summing from 1 to n I summed from 1 to k n where k is an integer then because the logarithm is additive on products I would get log of n plus log of k [snorts] the log of n should be in color by the way and uh uh oh there's a mistake there's a mistake I hope you see it um then the sum the so we're taking again Uh so there's a the colors are wrong.
The the red should be log of n and not log of k and I should be extracting log of n in the in the squared formula.
Sorry for that. And if I subtract log of n as I did before this the divergent part I unfortunately don't get gamma but I get gamma plus log of k. But it's just as reasonable to uh take the sum from 1 to n as to taking the sum from 1 to kn.
So there's there's this ambiguity we're confronted with. And the same thing for the za function. If I multiply by mu at the power s minus one, then I get a log of mu. Again, this shouldn't really be in red, but okay, doesn't matter. And the divergent part I take away as before 1 / s minus one. And what do I get?
Gamma plus log of mu. So if I take mu equal k I get the same thing uh from oiler's point of view and from um reman's point of view if you if you like.
Okay. I hope it's okay. If there are any urgent questions let me know. like [laughter] why did I put log of k instead instead of log of n that's a mistake now let's go to uh physics and fineman diagrams this is the fineman diagram uninteresting very uninteresting for the for the void uh so um it's just um it's just a circle it's just a loop and every one of these uh introduces a factor 1 / the norm of k^ 2 + m2. I'm assuming we have mass again because I don't want to deal with two types of divergences uh infrared and and ultraviolet.
Infrared being for k equals zero ultraviolet be being for k going to infinity. I don't want to deal with th both because I can't. When you improve one you um you make the other one worse.
So I don't want to u and it's not just me it's a it's a real problem. So uh that's why we put a mass which is non zero so that we don't have two problems to deal with only at infinity as is I don't know whether you see this hand do you see the hand no okay do you see yes okay so uh this uh diverges again for the same reason I said at the beginning because two is smaller than k this is a a exercise analysis first one exercise and now I introduce uh a regularization. This is the way physicists would do it. I put an absylon uh I put a factor uh in the in the denominator k uh norm of k to the power minus epsylon and I'll let epsylon go to zero which will give me one in the numerator and all the rest as well. You see uh I have a 2 pi 4 2 p<unk> 4 minus epsylon when epsylon goes to zero goes to 2 pi 4 and mu to the power epsylon when epsylon goes to zero goes to one. So indeed e [laughter] i absylon m tends to uh I m when epsylon goes to zero.
Now if real part of epsylon why did I introduce epsylon here? If the real part of epsylon which you can assume is complex uh is larger than two then you have convergence and uh you can show that there's a analytic continuation at absylon equals zero I suppose it's written below but now we won't we'll skip that too bad and here it's here is the expression again the first line is just copy paste and the second line. I'm changing variables. I'm doing a change of variable. R equ= MT. That's just to get rid of the M and to go back to maths where we have 1 + T2 instead of M2 plus T2. And then you can recognize if you're familiar with it with a beta function.
You can recognize a beta function at the so it depends on two parameters. It's it's uh it's defined in terms of gamma.
Here it is. B of XY equal gamma X gamma Y over gamma X + Y and um the functions the gamma functions uh admitting uh um meamorphic continuation it's we get a meamorphic continuation of our I absylon function um at epsylon equals zero.
Okay, m is fixed and so we write the lon expansion at epsylon equals z and we find uh we find uh this expression which is maybe not that amanable but uh that's typical of inte fineman integrals they quickly get complicated doesn't really matter what matters to us is what is in red we want to get rid of what is in red the divergence and keep the finite part and the finite part is the good part is the blue part. So I've kept the blue part and uh and as you see I put a mu at the beginning you might have noticed and everything depends on the mu on the mu if I rescale mu I get an additional term so all this is relative to the choice of parameter you make which is not reassuring you wouldn't you don't really want that but physicists are used to deal with dealing with such problems they um the that's the whole uh discussion in renormalization which I won't do I won't uh dwell into right now. So now um I'll seem to be doing something different and but it'll be in fact very very much related is to count integer points on cones.
You might think why do that? uh this I just say a word I won't say the whole story it was inspired by work by Michelle Ver and u Nikl Berlin on uh on toric uh geometry and um on on the um uh Todd genus ftoric varieties so it's inspired by geometry I can tell you the story if we have time afterwards so we take a convex cone. So, it's like a an ice cream, but um in fact, it's not like quite like an ice cream. It's like a it's not a round ice cream. The I mean, the section is not an ellipse. It's um it's a it's a polygon. Uh because if you had for example a circle as a section, it would be very complicated. Um, calculating the number of discrete points in a disc is a complicated problem. That's not our problem. And you can describe a cone. You soon we'll have pictures. You can describe a cone as an intersection of hyperplanes. Uh, this is what I call the H form. uh this so you because so I'm now showing with my hands because you're you have um uh not a round cone but you can you can um we'll see an example you can uh describe it as the intersection of half spaces and uh so there's a closed and open cone according to whether you take or not the boundary of the half space and you can describe it in terms of generating The generators are the vectors um generating the um this what they called the sides of the cones. It's not called like that. I'll I'll find the word. And uh so the a [clears throat] very deep theorem says that any uh polyhedric convex cone can be well let's put it closed that doesn't really matter can be described both in the form H and in the form V. So either as an intersection of h of half planes half spaces or as um generated by by vectors. I think we'll see an example now. So let's take two generators uh one zero we're in R2 with the uh orthonormal basis a canonical orthonormal basis E1 E2 and we take E1 and E1 plus E2 as our two generators and then all I've said before is that we take posit non- negative linear combinations of these two that's in the form V and in the form H you'll have this description that x2 is non- negative and x1 - x2 is non- negative.
Oh dear. And then you can guess the last part. It's very good like that it maintains your attention. Um uh that you can guess that the inequalities will become strict.
Here it is. uh whether I keep uh the green the dark green lines here, this one or this one is whether I take the closed or open cone.
And what we want to uh do is to count the points of the u of the uh latis C intersected with Z K. We could take another Z K is a latis. So I'm saying it wrong. We want to count the latice points of the cone. And for that we take a latis. Here I've taken a very simple latice z k. I could take another latis but it makes it a bit more complicated.
I'm not going to go into that now. And uh let's take a this very simple cone zero plus infinity. So it's it's uh r.
So it's all the non- negative uh real numbers. That's a cone. And so if you take the open cone is the positive real numbers. And if you take the intersection with zed, you have the positive integers.
And uh in two dimensions, uh a nice one is this chen cone. We'll call it chen. You'll see why maybe. I'm not sure we'll have time to see why.
It's generated by E1 and E1 plus E2. And the intersection is uh with the with the latis with zed 2 is u consists of pairs n plus n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n with mn n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n non- negative and now we want to uh count the integer points in C intersected well no okay this sentence is not quite correct which correct but it's redundant I want to count the integer points in C which amounts to counting the points in C intersected with Z K or said in another way, how do I give a meaning to the sum sum over all these points of the expression n to the power zero which is one sum of ones over a set is counting the number of points in the set. Okay, so I'm counting the number of points of integer points in a cone. Okay. And [clears throat] that's our question you and it looks like it's stupid and uh hopeless but it's very deep in fact. So that's our question. What's how to calculate this? And uh we're going to use yet another way uh is the exponential regularization which is very much used in maths and in physics. for example in the heat in index theory in physics if in maths if you're a geometer or a topologist. So um it's based on the fact that e to the power 0 is one and the exponential is continuous at zero.
So e to the power minus epsylon n when epsylon tends to zero tends to one. And now we're in we're replacing remember we want to define the sum over the integer points of the cone. We want to uh define the sum of ones. It's it's hopeless because of course it's infinite. Uh the cone is infinite. So we'll have infinitely many points. we're turning into a less hopeless issue. Namely, uh we're replacing the one by an approximation e to the power minus epsylon n. So that's what we're actually computing. And this sum this sum, you know, this is a geometric series. And so it's a very simple uh expression. And if you're familiar with the Todd function, you might recognize in the background that the Todd function is not too far.
But unfortunately and but that's expected. There's there's um a divergence where where we left it.
Remember we wanted to sum the ones and we replaced we cheated. We replaced one by e to the minus epsylon n. It's okay as long as epsylon that's should be said here. It's okay as long as epsylon is positive. But if epsylon is equal to zero, we have a problem. And again, we have a divergence. We take it away. And I forgot to say FP is finite part. I've already used it several times. It's taking away the divergent part minimally. I could have I could take everything away. For example, I could take this away, subtract this, but then I would get zero or I could take get take get rid of everything. And I like to compare this with um what I imagine uh going to the dentists in the middle ages was like. Um I suppose the dentist dentistry was not very developed and um they would just uh take out the uh the to the teeth that were um spoiled or that were um damaged. So they would take out the damaged teeth and you might as well take out minimally not taking out the uh good teeth. And that's what we're doing here. We're we're like a medieval uh dentist taking out the bad teeth and leaving what's um uh rescuable enablingly the fine this minus one/2. So we get if we do that we get minus one/2.
So that looks very strange because remember we were summing um ones over we were taking infinitely many ones and we get something negative and something non- integer. It's u a bit uh disheartening.
So it [clears throat] looks as though it's getting worse and worse.
And uh now let's do it in two dimensions uh to see what that gives us. At least even if it's very strange, we should have something in two dimensions which seems compatible with what we have in one dimension. So let's see. We're counting the integer points here. Uh so C1 is open. I think I decided it doesn't it's not very clear. I think I should put a zero. We'll see what I do afterwards. I've forgotten. Uh so it's the integer points in here. And I think I leave the ones on the boundary out.
Let's see. Yes. So Z plus means I'm taking out the zero. So here it is. I'm there should be a comma here between those two. And uh if I change variables as I claim here, we get this. And when you do this, you find two independent sums, one over m and one over n. So you find the the ones we were working with before this one and that one and uh so notice the diff difference this depends on two parameters this depends on one okay and so now we remember that we had so I don't know whether you remember we had we had this formula so this is amounts to saying that it's a holomorphic function at zero over absylon if you think of it 2 minutes. Uh simplifying this looking at it from afar comes down to looking to claiming that it's a holomorphic function at zero holorphic germ over epsylon. So we go back to what we've got.
So here it is holomorphic function of epsylon 1 over epsylon 1 holomorphic the same one of of epsylon 1 plus epsylon 2 over epsylon 1 plus epsylon 2 so in the end we don't really care for the moment what we have upstairs the blue I recall is good and the bad the red is bad so the blue is holomorphic at zero that's the idea it's we are only focusing on zero because um because that's where we want the limit in the end. We forget about the rest. So we're talking about germs, not so much functions. And so if you look at the singularities, you see upsylon equals 0, upsylon 1 equals 0, upsylon 1 plus epsylon 2 equals 0. And it's linear in the variables. So we got we get a meamorphic germ with pose linear in zero. Let's remember this.
It's of it's of course the the expression here is not linear as a whole but it's products of linear forms in the epsylon I and that we have to remember.
So let's see what we do with this. So this is a recap. So now no we now we're doing it more in detail. So remember I had this formula I just showed you again of remember that this is s subsylon this is well what did I call it s0 epsylon 1 and this is s0 epsylon 1 plus epsylon 2 not quite there's this factor and if you I'm I'm not sure I think I'm wrong here that's strange I've left a version with mistakes so this should be epsylon 1 plus epsylon 2 there's a mistake here. And so you just get the product of what you had in one variable.
One here, one there. It's the same type of expression. One minus 1/2 minus 1/2.
It's just the variables now have changed. Absylon 1 for this one, epsylon 1 plus epsylon 2 for this one.
And when you do all that, you pro you make you build the product, you expand, you find this. And so you think, "Oh, that looks good. I'm done." I get rid of the red on worms. Okay, this looks rather red. Not quite red because there are things on top, but let's get rid of it. Uh because I'm not sure whether they're uh uh rotten to teeth or whether they're still healthy teeth. So, I'll take them out just in case. And I'll keep one/4.
So, it's still a question mark. Is this legitimate or not? Wait a minute. I'm getting rid of something which doesn't enable me to see what I'm ah [clears throat] uh I can't I can't see what I think you can see it but I can't see. Oh dear. I can't see the bottom formula. Uh just a minute. Uh, it's KI which is wanting to help me but it's it's getting in the way and it's it's hiding this part. So I I assume uh I can't see what what I'm talking about but I assume it's 1/4 plus ex etc. So you're so the finite part would be 1/4.
Um okay I'm not sure what what this is.
Let's see. And then um I maybe it was the question ah okay it's a question is 1/4 really this quantity. So we're now going to be more suspicious and uh we're going to be a little more systematic and use the geometry of codes. So in our group of three um co-authors um there's uh Ligu who's who was a number theorist still is but is also a combinatorist there's Bin Jang who is an algebraic geometer and also much more and he knows he knows cones because and the geometry of cones he's more familiar with it than we are because uh in toric geometry you use that in an essential way.
So, uh yes, uh strongly convex just means uh there's a point, you know, it's it's um it's really a cone. It's not just Yeah, it's it's there's a there's a top or a bottom to the ice cream.
So, now we're going to subdivide. So, that's the geometry we're we're introducing. We're going to subdivide uh the quarter space. The quarter space spanned by E1 E2. I mean this part quarter cone. It's a cone. It's the cone span by E1 E2. We're going to split it into three.
C1 here the diagonal E1 plus E2 and C2.
And uh this is a subdivision. We're subdividing the big cone into three smaller in inverted commas. They're just as infinite cones. And you can either have take into account the diagonal or not taking into the account the diagonal. So here I'm forgetting about the diagonal. It's an open subdivision.
Here it's a closed subdivision where I I don't forget the diagonal. So think of uh the difference between integrating and summing. If you integrate you don't this is this diagonal is one-dimensional whether whereas the cones are two-dimensional and so if you integrate you don't see the one-dimensional part it'll be zero in the integral. If you integrate in two dimensions whereas if you sum discrete things then you will see the diagonal and so it was the opposite I was I should have said so you'll see the diagonal so you that's why we have to keep it keep it in mind whether we do include it or whether we don't include it.
So let's go to um the um laplas transform of the cone. It's a kind of generating function. It's a way to describe the cone. It's a kind of dual function. So it describes the cone as well as the cone itself. It's enough to understand a cone as long as you take it for all epsylon i positive. So this is um this is a a um a scala product epsylon x which is sum of epsylon i x i. So in one dimension um we said at epsylon 1 equals z.
Remember we want to set epsylon equal to zero. That's the idea all over epsylon 1 equal to zero uh we have 1 / upsylon as a pole. Uh at zero we have one over epsylon 1. And uh so that's the pole.
You see the similarity. There's a generator E1 and a pole one over epsylon 1. So there there's a similarity between the two. Now we're going to the two-dimensional one generated by E1 E1 plus E2. It's always the same one we're looking at. And we've got um two um uh two hyperplanes two yes hyperplanes of co of poles. Sorry. Uh epsylon 1 equals 0. epsylon 1 plus epsylon 2 equals zero.
And again, you see the similarity in the structure. So I can't see what you see.
It's hidden by this ki thing, this um AI thing. Um and so I can't see the bottom, but uh it's this there's a similarity between the shape of the singularities and the generators.
Yes. Okay. And then let's see C2. I take another one. Remember I had C1 C2. I had this one and that one. So I'm taking first this one and then that one. This one is generated by E1 E1 plus E2. And the other one is generated by E2 E1 plus E2. And so so I do that and again I'm not surprised. I see the same pole structure here as reflecting the generators of the cone.
Okay.
So when you do this much more in a much more general context, I'm not going into the details. It's a simplicial cone which means that this is a um uh these vectors are linearly independent.
Then you have um this kind of formula.
You only need to remember you see again the generators. These are the generators. They're there again just like we had them before. And this is this is a scalar products.
So there are n generators but each of them is a vector in k in r k and epsylon is has k components. This is a scalar product and there are n of these in this product. So let's go to back to the the this one. You see uh this would be the sum of epsylon. So let me see what I'm saying. Yes, this would be epsylon scala product E1 and this would be epsylon absylon being epsylon 1 epsylon 2 scala E1 plus E2. I hope I'm saying that correctly. I'm not sure now. Anyway, we have poles and uh and okay, this subdivision and you see the geometric subdivision gives a algebraic subdivision.
So, uh LC1, so this is a cone. So, LC1 is this expression for C1. It's it comes from what I wrote before. It's just this. It's the laplas transform. So it looks like this. You have um LC1 of epsylon 1 epsylon 2 here. LC2 of epsylon 1 epsylon 2 here. And [clears throat] the product of these two expressions LC0 epsylon 1 LC0 epsylon 2. This corresponds to this decomposition which we learn in high school in fact and maybe again at university and uh so we have um a new pole. So here you see we just have poles upside one upside two due that's that's if we look at this big code. Okay this are the it corresponds to the generators here are the poles.
Now this decomposition gives us a sum by which two new a new pole arises epsylon 1 plus epsylon 2. And you notice because we were working with an integral there's no the this part doesn't add another term which we'll see very soon. This is because we're working with integrals.
the onedimensional the integral on a one-dimensional space. The integral in two dimensions of a one-dimensional subspace is zero.
So now I'm looking at sums, discrete sums. So now you see I'm a bit more careful.
And you see we have wait a minute. Uh yeah, we have the ones we had before.
These are the ones. This is C1. This is C2, but we have an additional term coming from that because in the sum you can't forget the one-dimensional space uh cone. You can't forget it. And so uh you have a a similar formula. Uh it's again a a geometric subdivision gives you an algebraic subdivision and the finite part accordingly has three terms.
So what do we find? Uh, we find now I can't see a thing. We find zero, I think. Is that right? You have to confirm. I can't see it just now. So, is that Yeah, zero. If you So, it's a shame I can't see it. I don't know why I can't make it disappear. This thing. Uh, so I have to guess. I've forgotten now.
The first one is minus one/2, I think.
No. Oh, no. 1/4 1/4 and minus one/2. So 1/4 we computed at some point if you remember 1/4 this gives you 1/2 minus 1/2 gives you zero. So it's again a bit misleading.
You find that the number of points here this regularize reormalize whatever you want to call it number of points is zero.
So wait a minute now I have to get back to yes uh that's what I wrote here but on the other hand so remember we have this we have this form we have this that was this this subdivision uh property and so on we've computed this side taken the finite part as epsylon goes to zero of this part let's do it on this part and this is more this is simpler so And uh you think okay the finite part should be the product the finite part of a product should be the product. No.
Yeah. The finite part of a product should be the product of finite parts.
You would like it to be and if you do that uh assuming that then this thing gives you minus one/2 to the^ 2 which is 1/4 and of course it's not the same. So from a formula which was correct that was correct as an equality of meamorphic functions you get to a if you do things naively you get namely if you take too many teeth out more or less and not careful to keep the healthy teeth then uh you find yourself with um a very strange formula zero equals 1/4. So that was not the right thing to do and in physics you call that to cure divergences. So that's why I like to compare it with dentists because I really think uh physicists think in terms of in medical terms when they say you cure divergences. So, um, now we're going to try to be a little more subtle and not take out too many teeth. Making the mistake of taking out extracting a healthy teeth among the other uh the other damaged teeth.
So for that and that's where our work with uh Bing Jang and Ligu comes into play inspired by the work of uh Nicole Berlin and Michelle Ver because they they used uh um they used the scalar product um the notion of orthogonality.
Um so uh what we did a mistake we made without realizing is that um we did things like upsylon over epsylon equals 1 which is true but so epsylon over epsylon when epsylon is non zero equals 1 but at when you let epsylon tend to zero it still goes to one But did you really want to keep this? Was it not a slightly rotten tooth or did you want to keep that tooth? Because there's a one over absylon. If I if I recall what we went through earlier on, I said, let's be bold and let's get rid of this. You see, it's a bit like what I've just described. It's something going to zero over something else going to zero. And sometimes this converges. So you don't know whether it's a bad tooth or or a good a good tooth. You don't know. And we just boldly got rid of them, which is maybe better for in the long run, but maybe we should have been more careful and could have kept them. And so this is a way to be a little more subtle and maybe keep more teeth in one's mouth.
And so we start again. We do the same thing. So I have how many? Five minutes or how much do I have?
>> Yes, professor. You have five minutes and then questions.
>> Okay. Thank you. So I won't get at all to the end. Uh I don't know what but it doesn't matter. The main thing is to convey an idea. So we'll convey one idea here. Uh you see um so those are the expressions we had before. Uh this is S0 I think I called it SC0 I think I called it of epsylon one. This is oh so again I'm I think I make a mistake here. I think it's a one period 2. I have to check. It's a good exercise. You can check after it. And so we find this times that. And remember I just showed you we got rid of this. Remember? And now we're going to uh inspect this more carefully. uh in getting rid of that we were left with 1/4th. Now we're going to look into that more carefully. This we get rid of. This is obviously rotten. We just get rid of it. But this was not so obvious and we're going to be more subtle and rewrite it. So I'll write I'll write the whole thing. Uh so we're writing let me see what we're doing.
We're writing epsylon 1 as half of epsylon 1 plus epsylon 2 plus epsylon 1 minus epsylon 2. If you I can't write here but this epsylon 1 I write it as half the sum of this and the sum of that and in doing that I get this. Now I decide these are really rotten teeth.
This is really a rotten tooth because epsylion 2 is um corresponds to the generator E2 which is orthogonal to the generator E1. Here I wrote that and you see E2 is in blue, E1 is in red and this one is orthogonal to that. So in a sense they're not simplifiable.
They're not reducible.
They're not uh you can't do anything about these bad teeth. You have to extract them.
And uh that's why we get So now we get rid of something different. It wasn't it wasn't what we had before. We didn't we don't get rid of this because this is not orthogonal to that. E1 is not orthogonal to E1 plus E2. So we get rid of uh different parts and we find 3/8.
So it's different from what we had before.
And now maybe we could finish on a positive tone. We are relieved because remember we had the naive way when we extracted the wrong teeth. Maybe we could say I'm not sure. we have to think in terms of teeth what it made what it meant. uh if we now uh do the same thing compare this was with the subdivision if you remember subdivision I'll just go back just so that it's uh where is it here we have this subdivision and it led to three terms if you remember and uh now we've dealt with them differently and we've got 3/8 instead of 1/4 before we had 1/4 one quarter now we have 38 38 and minus 1/2 in as a whole it gives you 1/4 and this is indeed what you get on the left hand side it's 1/4 so you're relieved we oh dear there's a mistake here uh there should be an equal where the hand is there should be an equal we've got indeed that the extraction of the finite parts is uh using this scalar product is indeed uh let's say the extraction of the finite part of the product is indeed the finite part of the no I haven't said that right the finite part of a product is indeed the product of finite parts if you're careful as we were here and since I have to stop I'll just say a bit about Michelle V and Nicole Berlin's work that's um so it's it's Yes. Uh so I'll compare it to uh work done by uh Alan Kon and Durkheimmer Alan Kon uh who built the non-commutative geometric geometric school and today they to together they um dealt with expressions like that much more complicated. And what they did was use the same parameter epsylon 1 equals epsylon 2 equals epsylon. So they use the same one. So they find themselves with expressions like upsylon over absylon which we said was one 10 goes to one when upside goes to zero. And so in doing that they extract uh in doing that they left they left teeth which were not really healthy which were half healthy and they had to use a co-algebraic device half algebra co-roducts to undo these mistakes to unwrap these mistakes and correct them by counter terms here following Berlin and We never make these mistakes because in introducing introducing first keeping several variables not just one.
we can we can uh keep in mind where they come from when we when uh following Alan Con and and but not just them all physic most physicists work with dimensional regularization which uses one parameter if you if you forget where when you identify epsylon when with epsylon 2 and call it epsylon then um you identify these parameters And you forget where this one came from. It came from one divergence, one cone. And this one came from another cone. You forget whereas we never forget. And uh and to never forget, you should never forget that they're orthogonal. You could take another scalar product, not the conventional canonical scalar product, and it would depend on on that choice.
Uh but that's the idea. So it's complicated because we have meamorphic functions in several variables which is a is tedious but it's worth the it's worth investing work and using metamorphic functions in several variables because then you don't make mistakes and have to correct them with a very complicated co-algebraic uh very beautiful but complicated co-algebraic device. Here we just it's it's very naive. We take out the rotten teeth. We don't have to think is this really rotten is this not really rotten.
And so uh yes so there's no obstruction and let me go to the last slide. So I I won't look into all this but I'll try to go to the last side or the Yeah, we can conclusion.
Okay. Uh okay. So maybe okay I can't do that it's too no so uh that's maybe the conclusions and I'll stop uh you using meamorphic functions with linear poles in several variables unifies reormalization of discrete sums over cones and fman integrals. You can look at them in the same way. The geometric focus on uh on reormalization is um through the choice of a scal scala product. You can say it's a weakness. We make a choice. It's a bit like choosing coordinates. So uh but it's a geometric way of making a choice. Let's say it's a geometric choice. And then you have a reenormalization group which is indeed like in the spirit of Kanan Kramer a pro-uni potent group I won't say what that is doesn't matter now there are open questions you can ask what happens if you have two different scalar products can you relate them uh can you relate the transformation groups which I haven't had time to talk about it's it's for people who are familiar with it. It's related to renormalization groups. Yes, they are isomorphic. We could show that.
But then how can you relate the finite parts? You find that's more difficult.
Uh how do you relate what we call a reormalization group which involves several variables with the one of Kona which involves one of one variable? We don't really know. We have we have um ideas but we don't really know uh do we are l are meamorphic functions with linear poles linear remember we just had linear combinations of the epsylon eyes are they enough to describe all the divergences we'll come across and recently well now some years we've been working on that we're supposed to finish in the following in the coming weeks. Uh we've been looking at cycllotomic multi-za no se cycl yeah generalizations of cyclomic multiga functions. For those who know, they'll recognize the words.
Uh we're looking at uh cycllotomic multida functions on or sums on cones and uh we saw because of the cyclomic cyclomicity I don't know how you say that because of the complex variables entering the cyclomic multisa uh functions we have uh poly logs. Uh so we have logarithms in the divergences.
So we don't have in it's not enough to look at linear pose. We need to look at logs but that's we're preparing for that and we're working hard. It's quite uh quite hard work. Thank you very much for your attention. Sorry that I talk too long. Thank you.
>> Thank you very much professor uh for your nice talk.
Uh now we are open to questions in YouTube. Uh let me check the YouTube.
Yes, we have some questions here. So I will read them now. Um thank you for your talk. What changes if one replaces set K with an aine latis a set K + A?
>> Ah Z K plus Aine. Yes, you're right.
>> Aine. Yeah, Aine.
>> Aine. Yes, you're right. very very uh pertinent question. Uh so uh we haven't dealt with that but implicitly what we're assuming is what uh Berlin and V also do you can always uh reduce to um um linear latises.
So you can always uh use if you look if you look at the uh generating function it just introduces um multiplicative term if you look at Benin and V's papers you'll see that everywhere it just look int um translating by a vector to go from a linear one to aine one just translates.
So this translation just introduces a multiplicative term which you which just follows all the computations. You just take it with you but it doesn't uh change the um heart of the matter.
>> Okay. Uh there is another question by Miguel Rees. Thank you for your presentation.
uh when counting the points of on the cone did you try to apply generating functions?
>> Well, those are generating functions.
They are those those are exactly the generating or I'm not sure I understand the question. These uh what I call laplas transform maybe I don't understand the question. These laplas transforms are the generating functions.
Is that what uh maybe they uh maybe they I don't know they they were there. So either discrete one uh I mean obtained by a discrete sum discrete laplas transform or obtained by a continuous laplas transform both the two types of generating functions which give you two types of information. uh the continuous laplas transform gives you purely polar information. The it just gives you the poles and the discrete one gives you much more. If you are familiar with the oiler McLaren formula, the two generating functions relate by the Oiler McLaren formula for codes which is in fact the title of one of the papers by Bellin and Van that inspired us.
>> Okay.
We have two more questions. Uh I think I will do it really fast. So perhaps this is just a naive question but do you envision any applications of your work for example in physics?
>> Naive [laughter] it's a I will say something strong. It's a perverse question because it's uh it's saying what is this useful for? Yes. So um so we were inspired in physics by the work of spear oin spear whose work uh who who um uh introduced analytic reormalization in the 60s if I'm not mistaken 60s7s I'm not sure now and uh analytic reormalization is similar to what we do namely uh he used several parameters and uh seeing how difficult it was. It took us it's still taking us a huge amount of time and energy to work with these multi- variable meamorphic functions. I wrote to him one day uh realizing that he was still alive and asked him um you know why why his approach I didn't say it like that I said it more diplomatically but more or less asking him why his approach had been abandoned more or less because people don't use an analytic reormalization so much they use dimensional reormalization which is in one parameter it's just one parameter and and he answered uh I suppose it was too difficult um because of meamorphic functions in several variables but in fact he proves so this answers the question in part maybe he proves that all the amplitudes in a series of papers all the amplitudes uh for um oh wait a minute um I think it's probably um it polomial interactions physics physical systems with polomial interactions. I have to check. Uh he proves that the amplitudes give rise to these meamorphic functions with linear poles and he gives the structure of the poles. In fact, I name him uh here. I think I saw spear. You see spear and maybe I'll say uh yeah. So spear. Yeah. And another person I'd like to um to quote here is Vietnam who's happens to be in Wuhan with me. He in in fact substituted me today for my lecture because I was ill. So uh Vietnam and Bing Jang worked together on fineman integrals on manifolds inspired by the work by spear and by what we did together with Bing Jang and Liu. So uh and you see the linearity is simple.
It's just sums of epsylon eyes. There's no coefficient. Okay? No, there's only coefficient one. And that's typical.
Whereas in Berlin and V at the bottom of the slide for general cones, you have coefficients. So their their point of view is much more general. Uh it it includes kind of fine man integrals. But as I said, I we can't yet compare what we do with what directly we can compare in some ways what we do with what um Durkheimr and Alon did, but we can't compare it directly. We still are missing a direct comparison. So I it's a yes and no uh answer to your question.
>> Okay, professor. Thank you very much. I have a question but I know that you are with a headache so I do not.
>> No but it's okay it's okay when it'll come back afterwards when I stop speaking.
>> Okay. So I will ask my question and then thank you very much. So you mentioned uh at the beginning this thought genius in the historic uh setting in historic geometry is it appearance here related to this reandro or a singer in index theorem or is it different somehow?
>> Uh so I'm not I'm not a specialist of to geometry. Um, wait a minute. O, uh, is it related to an index theorem? I would say not directly.
>> Okay.
>> Uh, wait a minute. I'm not sure. Maybe other people can answer this better than me in the audience. I'm sure somebody can answer better than me. It's a good question, but I don't think so. Wait a minute. Uh, so the Todd the Todd genus is uh is a characteristic. It gives rise to a characteristic class, but it's not related to an index, is it? Uh, someone can say better than me, I'm sure. So, I can't really answer. I think no. I think no, but I'm not sure.
>> Especially now with a headache, I'm even less sure. [laughter] >> Yeah. Okay, professor. Uh, thank you.
Thank you very much for uh attending today even though you are sick. So thank you very much for your time. Um you have a special >> and I would like I since because I was not too well I forgot to uh thank the organizers because I think this initiative is very nice. Thank you. Uh thank you to you. Thank you uh Andrea.
Thank you to Duvan and other people I probably don't know. Uh thank you because I think this is a very nice initiative. Thank you.
>> Yeah. And thank you very much for always u support our initiatives. Uh you have a very special uh place in our hearts. So thank you very much for all >> thank you to you. Thank you to you.
Goodbye. I won't attend the other talks or I'm sorry I didn't attend talks but I couldn't. Thank you. Goodbye. Bye bye.
>> See you professor. Bye.
>> Bye bye.
So to all of you uh our viewers in YouTube also we will uh have uh 40 minutes break uh the next uh so we will start again at 12 p.m. Colombian time and 19 central European time with the talk of professor Simon Donaldson.
After that we will have the talk of Mari Jose Pacificico and at the end we will have our Ekman prize 2026. So you are all welcome to join us in uh 40 minutes. Thank you very much and see you all soon.
You >> see you >> professor Donaldson, are you there?
>> Uh maybe he wants to check his uh slides also. So we can ask him.
If not then we can come back in 40 minutes >> [snorts] >> Hello, Professor.
>> Are you there?
>> Yes. Hello.
>> Hello. Thank you for for being here.
>> Thank you for sending the link for >> Hello, Brian.
>> Hello, Professor Nora.
How are you?
>> Thank you for helping me to join.
>> It is a pleasure.
>> Thank you.
Maybe we can check if if you can share your slides properly >> for the time.
Okay, >> now I'm not >> Oh, I need to share my screen. Sorry, haven't done that. Hang on. Forgot to share my screen.
Hang on. Uh, hang on. Share screen.
Okay, now you're sharing your screen.
>> Perfect.
>> Can you try to to move the slides?
>> I'm sorry.
>> Can you try to move the slides?
Now it's black.
Ah okay it's working perfectly. Thank you professor.
>> Okay that's good.
>> I want to say that we are just a few here but most of the participants are attending the talk via YouTube and we're live stream on YouTube. Is that okay?
>> Okay. Yes. Mhm.
>> Okay. Nice.
So we are going to wait uh five more minutes and we can start with your presentation.
>> Well, very good.
Oops.
Okay, professor, you're ready. Can we start now?
>> Right, I'm ready.
>> Yeah.
>> Yeah. Yeah. Okay, professor, let me introduce your talk. Hello everyone again. It is a great honor to introduce professor Sir Simon Donaldson one of of the most influential mathematicians of our time is in ch he's sharing pure mathematics at Imperial College London and permanent member of the Simon Center for Geometry and Physics at Stonybrook University. His work has profoundly shaped modern differential geometry and topology particularly through the use of gosh theory in the study of fourdimensional maniples.
He received the fields medal in 1986 and was selected a fellow of the royal society in the same year.
His many other distinctions include the crafort prize, the shaw prize and the breakthrough prize in mathematics. He was nited in 2012 for his services to mathematics today. Professor Donaldson will present his talk entitled collapsing a special holomy manifolds and multival functions. Thank you very much for being here with us. Please go ahead of course whenever you're ready.
Thank you Brun for the kind introduction and thank you to the organizers for inviting me to uh take part in this meeting.
[clears throat] This will be a talk about um uh topics in differential geometry and uh this is the overall plan to follow. We'll begin the first part will be um uh best [clears throat] general review of some questions uh in the rev involving convergence of uh Romania manifolds and a particular collapsing phenomena.
Uh then secondly we'll move on to look at a uh particular case of sevendimensional manifolds with uh a very special kind of structure associated to the uh exceptional le group G2.
And in the third part if we reach that we will uh talk a little bit about some interesting questions in analysis that arise here uh and also in another area other areas of differential geometry over the past few years involving what you might in brief describe as multialued functions with certain kinds of singular branching sets.
So let us begin. Uh let's first of all think what does it mean to say a sequence of Romanian manifolds has a limit.
Uh this is an interesting question uh particularly for Romanian manifolds with some special properties. Uh for the purposes of this talk we'll be thinking of Einstein manifolds manifolds with zero reach.
So a very general way of formulating this convergence is um due to Gromoff.
Uh it involves uh the notion of the Gromoff house distance uh which uh is formulated uh in great generality for compact metric spaces.
So if we have two of those X and Y, then this is an obvious notion that you could say of what it means for these to be approximately the same or close is that there is a a homeomorphism between them which only changes the distance functions by a small amount for some small epsidon.
But the Grum of Htor theory is somewhat more subtle than that. uh and in particular allows X and Y not to be homeomorphic.
The basic notion is the gromov house distance between x and y which this is the definition is you take the infe of numbers so that you can find a metric in the sense of metric spaces on the disjoint union which extends the given metric on the two pieces and such that both x and y are epsylon dense. So as to say any point of y is with an absion of some point of x and vice versa.
So you can think of this as saying in place of the in the previous notion for each point x in x we have a point f ofx in y. Now associated to x in x we have have a small set of points in y whose distance is less than epsylon from x.
And similarly for y and y. So we don't have a correspondence between points but we make a point correspond to a small set if the if the if the small if the distance between the thick between the spaces is small.
So with this notion of this distance function we have the notion of convergence of a sequence of metric spaces to some limit.
And in particular, of course, a Romanian manifold is a metric space with uh distances defined by the infom of lengths of paths between points. So we can apply this to sequences of Romania manifolds.
But although we start with Romanian manifolds, in principle, the limiting object could be something quite bizarre, some more um less familiar kind of metric space.
So uh another starting point is a fundamental theorem of Gromoth which um is for the case relevant to us. It says if we have a sequence of manifolds with zero or moderately positive breachy curvature and if they have bounded diameter uh then there is a subsequence which converges in this sense to some um limiting metric space.
This is a fundamental theorem. The proof is not particularly difficult um uh using essential idea is to do coverings by metric balls and the fact that by a well-known facts in differential geometry there is the bishop inequality the hypothesis on the reachi curvature gives some control of the volume of balls.
So let's suppose uh as you say we're discussing um just restrict to the situation of manifolds with zero reach curvature then we can always scale so the diameter is one we're talking about compact manifolds but then there are two cases if throughout our sequence there's some fixed lower volume bound the volumes are bigger than some strictly positive number then a deep theory developed by Cher and Cing around the beginning of the century and developed further by uh Chiger and neighbor more recently gives a good understanding of the properties of the limit space.
So if we started with n-dimensional Romanian manifolds, the limiting object is still an n-dimensional space. It's the Romanian manifold outside a singular set uh which is small in a sense. It has co-dimension at least four.
So this is a well understood situation from a theoretical point of view and this is called the non-olapsed case.
just to think of a um a simple example uh which will introduce some facts and ideas that we'll take up later. Let's consider a complex algebraic surface in three-dimensional complex projected space, a non-s singular complex algebraic surface. And let's suppose the degree is four.
So in that case the um the calabi conjecture proved by yao in the 1970s uh tells us that there is indeed a metric on x with zero reachi curvature.
In fact a kala metric which we'll talk a bit more about later.
So this is for smooth complex algebraic surfaces. But what happens if we uh look take a family of such things so defined by a family of polomials say pt of degree 4 so they're smooth for small nonzero t but uh when t is zero we develop um a singularity. So we just think of the simplest kind of algebra geometric singularity an ordinary double point singularity.
Though to be more explicit if we work in aphine coordinates in um an aphine piece of CP3 and the singularity is at the origin then we could for example have our polomials written in the fine coordinates. We could take the standard model uh nondenerate quadratic form plus t times some generic polomial of higher degree um where q q where q contains only terms of degree three or more. So that the degree there's no linear part uh and the degree 2 part is just given by this standard sum of squares.
So this is the simple example of a family of surfaces uh parameterized by t developing an ordinary double point singularity when t is zero.
>> [clears throat] >> So the the um the uh singular um um aphine quadric if we put z + 2 2 + z 3 2 is zero that can be identified with the quotient of c2 by the back where we take a vector zed to minus zed.
So how do we achieve that? If I take a a point with coordinates za 1 za2 in z2 and then I map that by these three degree 2 polomials into c3 then you will see that this is unchanged. If we change this the sign of the zetas and precisely gives a 2:1 map onto this aphine quadric.
So what happens this is not not obvious but what can be proved to happen is that these Yao globby Yao Einstein metrics on these complex surfaces converge in the limit to something with a single point singularity uh which but it's not a this is a very simple kind of singularity. It's just what's called an orifiled metric. when we lift to the uh to the cover we get a smooth metric on. So this is the simplest kind of singularity here a point appearing inside of four-dimensional spaces and one can also understand in detail the behavior of the metrics for very small t close to this uh nearly singular point as it were but we want really to focus in this talk on the other case the collapsed case as it's when the volume of our sequence of manifolds goes to zero. The diameter is one but the volume goes to diff zero.
So theoretically this is a much less well understood case.
uh but we don't really need um for our purposes here particularly u knowledge of any very general theory uh but more a kind of a general picture of what one expects [clears throat] and that is uh roughly as follows this is somewhat imprecise um what we're saying but I think you'll get the you get the overall idea so what expects is that the limiting object is a space of strictly lower dimension say n minus k and uh when we're close to the limit for large i the manifold has a vibration over this lower dimensional space where the diameter of the fibers is very small. The diameter of the fibers is of order some small number delta I which tends to zero as I tends to infinity.
So here we mean a vibration in a generalized sense uh but outside a closed subset a typically lower dimensional subset of the base then one expects that this is a genuine smooth differentiable vibration. So the fiber is some um different differentiably. So I'm given say k dimensional manifold x.
So it was saying that what one expects is the picture looks roughly like we take a fiber bundle and then we as we approach the limit we're collapsing by scaling down the fibers relative to the scale of the base.
But it's not exactly a fiber bundle because we expect in reality to encounter some kind of singular fibers where picture is more complicated.
Well, how about the metrics? What we expect is that if we stay away from this singular set, we take some fixed point away from it and then we look on a scale in the base comparable to the scale of the fiber and then we rescale that piece of our manifold. So, we take a a delta i neighborhood of our point in the base.
So it's the same uh as the scale of the fiber. We take the pre-image of that and then rescale by multiplying lengths by a factor of 1 / delta I. Then that thing should converge to some product converges I tends to infinity to some product structure on our fiber times ukidian space. So we get essentially we get a metric on the fiber uh that extended just by multiplying by flat directions corresponding to the base.
And since our original metrics were supposed to have zero reaching curvature, the metric we have on this lower dimensional manifold on the fiber should also have zero reaching curvature.
So on a scale of order delta I in the base the structure on the fibers is approximately constant. I mean this could also be 100 times delta I or a thousand times delta I but some fixed multiple of delta I.
But on a on a a large scale, the scale of the diameter of the base, we expect that the structure could vary.
So at each point in the base away from this set delta we have a Einstein metric a metric of zero reaching curvature on X. uh and this way we get a map from the base minus this exceptional set delta into the modulized space of Einstein metrics on X modulo difforphism and going a bit further one can convince oneself that This map should satisfy an equation, a partial dimensional equation which um which uh is somehow the dimensionally reduced version or the asmtotic version of the Einstein equations in n dimensions.
So the the the equation we expect to get in the general situation is that um if we take this map use the induced metric on the base B and a natural metric on the modulized space of Einstein structures defined by the L2 norm on tensors on X then that should be a harmonic map.
And secondly, if we look at the reachi curvature of this induced metric on the base, that should be equal to some something formed by taking the derivative of our map kai and then contracting using the metric on the target space. This derivative is a linear map from the tangent space of B to the tangent space of the moduliz space. uh we can use the metric on the modulized space to contract those indices as it were to get something which is a quadratic form on the tangent space of B which is where the rishi curvature lives.
So that's the kind kind of outline slightly imprecise of the general picture one might hope to see when studying this kind of collapsing phenomena.
But think about that you can imagine an analogy or situation in real life. Supposing we have a a very long hose of length say about a kilometer. Um so the diameter is roughly a centimeter that sort of order.
So the here the the length of the hose is the analog of B and the diameter is the analog of the diameter of the fiber.
So we can imagine that over distances of order 1 meter say this diameter is essentially constant. Essentially we have a perfect hose. So the hose is very close to a product of the zero the interval 01 for the one meter by a circle of some length of the order of 1 cm.
But this is a weak hose. So we can imagine that due to the stresses and strains the the diameter changes slowly as we vary over its whole 1 kilometer length. So we can imagine that if we go say a distance 10 m the diameter might have changed somewhat for example from say half a cm to 2 cm.
So what we expect then is we get a function on uh the interval from n to 1,000 parameterizing the whole length of the hose which is defined by this slowly varying diameter.
And we could imagine that there might be some differential equation which determines this function depending on the physics of the situation. Exactly.
But that would be analogous to this equation we wrote down for governing the asmtoically the coupling between the metric on the base and the uh this map into the modulized space of structures on the fiber. So here the modulized space would just be represented by the single parameter R giving the diameter of the of the hose.
Actually before we go on let's be a bit more precise uh because we'll need this later. Um when we talk about modularized spaces we have to be careful uh in various ways. So to be precise let let us consider not the full difforphism group but the subgroup of difforphisms isotopic to the identity.
Well this is a normal subgroup and the quotient by the full of the full difforphism group by the subgroup is what's called the mapping class group.
This is originally the terminology from the case of reman surfaces.
So then we can form what you might call the tikeke space of Einstein. We take all the Einstein metrics but we divide not by allorphisms by only by these ones isotopic to the identity.
So by construction then the discrete group the mapping class group which we're calling gamma acts on this type of space and the quotient is our moduliz space.
How a better way then to express what we're doing was to say is to say that there is a flat bundle over the complement of this exceptional set in the base. The fiber is this type of space. The structure group is this discrete group gamma. And uh rather than a map we should really think of this kai is being considered as a section of this bundle.
But because this gamma is a discrete group, it's sort of locally essentially the same thing but um more accurate technically.
Well, we talk about compact manifolds of zero reach curvature.
The only examples of those which are known apart from the case of flat manifolds not interesting are manifolds with what are called special holonomy which effectively means that there is some additional algebraic structure on the tangent space of the manifold which is compatible with the Romanian metric in that it's preserved by the parallel transport of the levicha connection.
We'll talk briefly about a well-known and important case out of it's called calabia manifolds.
So we have our real dimension n which is now 2 m. So these have a compatible complex structure. So they're m dimensional complex manifolds.
uh it is compatible in algebraically compatible with the metric. So in the familiar way those two things define a two form little omega uh that would be just what is a kala manifold. The extra thing we want is a nonzero holorphic m form we're calling it theta on the manifold.
So this collection of algebraic data reduces the structure group of the tangent bundle down to the group summ.
If we didn't have theta, we would have the unitary group U of M, which is what preserves the complex structure. Uh putting in this holorphic M form reduces us to the special unitary group.
So uh it's known by um again by Yao's proof the kalabi conjecture that there are many of these things and they can be characterized actually algebra geometrically essentially or in terms of complex geometry for us for for example if we took any complex hypersurface of the appropriate degree in projective space so the degree should be one more than the dimension of the projective space then we get a manifold which it's proved admits such a metric and this is not a metric that one knows explicitly that you could write down explicitly but there is an existence proof which asserts that it exists and if you're doing m= 2 uh so we're talking about complex surfaces then this is goes back to the case we considered four considering surfaces of degree four in three-dimensional complex projector space.
So now for such a for such a um calabia manifold M there are special a special class of submanifolds or special lronian sub manifolds and these can be defined by saying that we have a um first of all they're submanifolds of half the dimension >> [clears throat] >> uh and uh the they're characterized by saying first leian so the restriction of the two form to L vanishes and secondly that we take the real part of this holorphic M form so that's a real M form that should also vanish on the submanifold.
So there is a um famous and long-standing problem or circle of questions uh related to a particular case of this collapsing phenomena in the case of these uh calabi faults. This is what's called the stroga yao zazlo conjecture going back um 30 years it looks like.
So it says that if we consider complex structures which are near to a certain limit then the metrics collapse along a special lrangeian Taurus vibration.
So we're doing something similar to what we discussed before where we were considering moving towards a singular surface with the ordinary double point singularity but now we're considering moving towards a different kind of degeneration of the complex structures which we won't um explain in detail. Uh but the statement is that when we do that and we follow the the problem is the conjecture is when we do that uh then near to that limit we will have a special lrangeian taurus vibration and the passage to the in the passage to the limit um these the metric will the sequence will collapse by collapsing the fibers of this um vibration.
So we have in the setting above we have our map from uh the scalabia manifold to some base which is the base is again half the the dimension uh and the non-s singular fibers should be special lrangeian toi in um the manifold m how about how does our map kai work in this situation or somewhat conjecturally uh if we Think of the Taurus as a think of this stuff giving a fixed description as the quotient of RM by the integer latis. Then the type of space uh parameterizing the ukidian structures or Romanian flat Romanian structures on the Taurus. It just corresponds to the uklidian structures on Rn. That's to say positive to definite quadratic forms on Rn.
So what happens in this situation or is expected to happen is that the base has a flat structure locally a flat structure and the map to type space is defined by taking the hessen of a convex function on Rn which solves the real mojare equation. The determinant of the hessen is equal to one.
So if we have such a a convex function, it's hessen at each point is a quadratic form and is just the data required to define a point in our tight model space.
And indeed one can easily check that a if we define a map in this way it does satisfy it gives a special class of solutions of this coupled equation that we wrote down for the general case.
So that's all we'll say about this um well-known situation and problem. Uh it is very important uh partly because it gives a conjectural geometric explanation of mirror symmetry in which one gets a mirror dual um calabia manifold by taking the same base and replacing the toy by dual tory. That is the general idea.
So this is all are very difficult things to prove. People have worked on it for many years. Should say in the last couple of years, great progress has been made by Yang Lee on proving the existence of these special lang vibrations in wide range of cases um outside a set of small volume. Of course, we always expect to encounter these singular fibers which are much um harder to understand exactly what happens around those.
So that's ends my first uh sort of survey part of of some chain of ideas involving Romanian geometry and specializing to this particular situation.
Now I want to go on to another um less well-known less studied perhaps um special case in which we're studying manifold seven dimensional manifolds with the structure group determined by the exceptional regroup G2.
So this group you can think of it as a subgroup of SO7 and it is could be defined in many ways.
One way is to define it as the subgroup which preserves a standard skuy symmetric crossroduct on R seven.
Well how do we write that cross product down? That can be done in again a number of ways. uh one is to take R seven and write it as the sum of the quitterians four-dimensional quitterians and a three-dimensional piece of the imaginary quitterians.
So then see to give a a sku to give a sku symmetric map from h + mh times itself back to itself we have to give a number of different components.
[clears throat] For example we have to give a component from the imaginary part of h times the imaginary part of h the imaginary part of h uh and so on. And then you can see that there are candidates for all such maps given by quitterian multiplication.
So on the imaginary part of h which is just R3 if you like this is just the ordinary crossroduct and then there are things. So the second map maybe is not second map is given by this formula in terms of multiplication. Of course, you have to fix some signs to get exactly the right cross product. You have these different pieces. To put them together, you need to choose some signs in the appropriate way. Otherwise, you get a a non-compact group that doesn't lie inside SO7.
So, that's a um [clears throat] sketch of how this cross priority is defined. And thus we get this group G2.
From the way we've defined it though, it's not really clear that this is a an interesting group. The definition appears to depend upon this decomposition of R7 into a fourdimensional and three-dimensional piece.
But in fact, there's a lot of symmetry that is not apparent immediately from the way we've defined things.
So the symmetry for example all unit vectors are the same from the point of view of this cross product G2 that's transitively on the set of unit vectors in R seven.
What algebraically what we can do we can use the standard ukidian structure on R seven and combine that with the cross product to define a three form. This is phi this is um totally anti-ymmetric in the variables x y and zed. So this is just the same formula what you're familiar with in the case of the crossroduct on R3.
So a G2 manifold we'll use this terminology for simplicity in the talk.
It's a Romanian seven manifold with a crossroduct on each tangent space which is algebraically equivalent at each point to the model above along with the the pair consisting of this cross product and the Roman metric and crucially is preserved by the parallel transport.
So this is a way of saying that the the holomy group of the manifold as it's called is contained in this exceptional group G2.
That is the fact again that uh any such a thing has reach curvature equal to zero.
And then there are various things we can do with the the differential geometry of the situation. uh the cross productduct uh just the same way as we did at the top of the screen defines a closed three form on the manifold and we have a distinguished class of submanifolds called these are fourdimensional submanifolds called coitative submanifolds uh which are characterized by the fact that the restriction of this three form should vanish.
So these are very analogous to the special langian sub manifolds we were discussing before.
Uh the deformation theory of these manifolds is also well understood of these seven manifolds at least in the compact case. If we have a compact manifold with one of these structures, then the small deformationations of the structure modulo difforphism correspond exactly to deformationations of the kmology class of the three form in the three-dimensional real kmology of the manifold.
So if you like the the modulized space of these structures has dimension in particular has dimension equal to the third betty number of the manifold.
So the direction we're going then in this section is to ask what are the possible collapse limits of sequences of G2 structures.
Then in our general picture if the base has got dimension 7 minus K the generic fiber should be a kdimensional manifold which should support metrics with zero reaching curvature.
So apart from the case uh k= 6 where we'll get these globby freeolds again uh and the case where we have flat fibers to say like as in the previous discussion the only possibility at least it's known is that um uh k is four and the fibers are little k3 surfaces thus to say kalabia manifolds of complex dimension two such as for example the surfaces of degree 4 in CDP3 we discussed before.
So let's consider that situation where we have a map a vibration above certain manifold over a three-dimensional base uh where the generic fibers are fit to the K3 manifold and we also suppose there are good reasons for doing this that the fibers are coassociative submanifolds again analogous to a special lrangeian vibrations in the previous base and we suppose that the exceptional set delta in the base is one-dimensional a link a union of embedded circles.
So we're supposing all that uh why um it's known from a construction of Kaleth from 20 years ago and uh more recent work of others in particular a recent work of Alybert that many such examples of this picture exist.
But one has not just sort of isolate examples but whole families because we can deform the structure at least a small amount uh by and that corresponds to deforming the kmology class in the three-dimensional commology.
uh than there is a preferred class in the three-dimensional coology given by the lift of the fundamental class of the base.
And the picture one expects is that as the commology class of our three form approaches that lift of the fundamental class of the base then that will force us into this situation where the fibers of this vibration collapse.
So we can sort of investigate our general picture in this situation.
The calling the tight muller space of the K3 manifold is uh completely understood through what's called teriy theorem.
though which is analogous to the classical to ready theorem which applies in the case of reman surfaces.
Uh for let's just explain that. So as he said for a colia manifold of m complex dimensions the structure group is actually su m. When m is two we have the group su2 which can be identified with the group of junior quitterians.
[snorts] So we can also think of a calabial surface of or two complex dimensions four real dimensions as having a quitterionic structure on its tangent space. It is in a sense a one-dimensional quturnionic manifold.
And then we have a triple of closed two forms expressed in the way we had before. We have the the kala form omega and then we have the real and imaginary parts of the holomorphic form. So in this case these are all two forms and they're all on the same although it doesn't appear it this way they're all the same standing. is another complex structure for example where omega 2 is the k form uh or omega 3.
So these three two forms correspond to a triple of complex structures I J K or more invariably you could say there's a whole two spheres worth of complex structures and indeed if we take such a structure then the product of X with a flat R3 we think of R3 as the imaginary quitterians does have a G2 structure we can write done using the gratonian structure on the tangent bundle of X. Essentially, we apply the recipe we used for the cross productduct to define a cross productduct on the tangent spaces of this [snorts] product.
But what should the equation be? Uh giving the slow evolution as it were of the structure on the fiber over the base.
We can understand that or explain that in terms of um [snorts] topology of the situation.
If we look at the two-dimensional karmology of our K3 manifold that is a space of dimension 22 and it has a quadratic form given by the cut product which has got signature 319 three positive dimensions directions and 19 negative ones.
[snorts] So if we have one of these calabio structures then it defines a three-dimensional subspace given by the span of these three two forms that we wrote down and this is a positive subspace with respect to this quadratic form on the in the carology.
Conversely, if you take any max positive subspace, then uh it's shown by a lot of work and contributions of many people that there is a corresponding calaba structure unique up to dipium.
Uh so actually that's slightly imprecise for that to make that happen. We need to allow also these objects with orifold singularities which corresponds to a special uh a special class of positive subspaces but we won't go into that for the purpose of this talk.
So slightly imprecisely the conclusion is that our type space in this case is the set of positive three-dimensional subspaces in R319.
Oh, locally remember what we have is a map from at least from some suitable open set in there's a missing B here a suitable open set U the content of delta into this type M space but what is the equation in this situation which should characterize this map what it is it is requirement is that this map is the Gaus map of an embedding of U into this space with its indefinite quadratic form.
So if we have such a an embedding of U into R319, then the G map assigns to each point of view the tangent space of um of a tangent space of the image of the corresponding point which is a three-dimensional subspace. So the point of this grossman uh but we want the image to be what's called a maximal sub manifold of R319 which is something we all review. So a maximal sub manifold in an indefinite space is just an analog of a minimal sub manifold in a ukidian space. to say first of all we require that the tangent spaces are all maximal positive sub subspaces with respect to the um quadratic form and then we require that the sub manifold satisfies the oil equation associated to the volume functional. We can find the volume functional in this situation in exactly the same way as in the ukidian case and therefore it produces oil equations.
So to see that explicitly if we consider the case where we have one negative dimension and we consider manifolds that can be written as graphs of a function then this is the maximal submanifold equation for a graph and it's the same as the minimal submanifold equation for graphs except that we change the sign here. In the minimal case, you'd have a plus here, which corresponds to the fact that in order to satisfy this algebraic condition on the positivity, we need the derivative of u to have length less than one.
This is a second order nonlinear PTE um a sort of quasi linear equation of LLAS type. The leading order term is roughly speaking the llus equ or variant of the llas equation of a function u.
Now the asmtotic data in some that we expect from this approach to the collap limit consists of a three manifold a link in the three manifold a flat bundle over the complement of this link with fiber this indefinite space R319 and a section of the bundle of positive three planes in E I bundle whose at each point we take the positive three planes in the fiber of E at that uh but which satisfies a partial differential equation uh which can be expressed by saying locally it's the G map of a maximal sub manifold parameterized by a section of E a section of this vector bundle well that's all away from delta what happens when we approach delta this is um perhaps the most interesting part of well the most interesting part and the part where all the difficulty in understanding things occurs.
So the the flat bundle for the kind of singularities that we want to consider which I'm not going to talk about in detail but it has the property it has non-trivial honomy going around a small loop around one of the components of L.
So when we're considering this local these local sections iota um away from delta they're given by maps or vector value functions um but we can't do that around a point of delta. Delta is now this link L.
The basic model that we require is that it looks like essentially like a surface with a branch point. So this this transverse to L in two dimensions and for simplicity taking one negative dimension rather than 19 dimensions.
We want our sub manifold to look like the graph of this two valued function the real part of Z to the 3 over2 as zed tends to zero.
So the in in classical minimal surface theory these kind of branch points are much studied in the case when we're considering two-dimensional minimal sub manifolds. Uh here it's different because we're considering things that look like that transverse to this set L but have higher dimensional singular sets.
So that brings me on just taking the time I have eight minutes left to the last part where you say this um I've tried to outline how following this line of thought studying these this particular version of this general collapsing question leads to these solutions of in that case a nonlinear elliptic PV but they have this essential value property of being two values solutions as we go near the singular set.
So that fits into um a whole class of problems and questions that have arisen in different parts of differential geometry over the past 10 years or so.
These include questions in gauge theory uh uh questions initiated by taobs and things studied by ta and also in study of calibrated submanifolds.
So I want to talk very briefly in the last five minutes about some of the some of the analytical features of these questions.
So the simplest setup is consider a Romanian manifold with the fixed metric a co-dimension 2 sub manifold and a flat real line bundle on the complement of that submanifold having being non-trivial. So having non holom minus one as we go around small loops around sigma.
So we have a llas operator then uh acting on sections of this flat bundle.
So locally away from sigma it's identical to the ordinary plus operator but uh we have interesting new phenomena due to this singularity that's sigma but let's write r for the distance to sigma.
So let's just to explain the main thing.
Let's consider this model problem of solving an inhomogeneous equation. So we take some prescribed source row which for simplicity let's say it's compactly supported away from sigma.
We want to solve the equation delta of s is row.
So if we allow if we work in a class of sections s uh which is suitably large which are roughly speaking allowed to behave like order after the 1/2 near to sigma this is uh something easy to understand you can show quite a straightforward way but is a unique solution to this equation the basic thing is that these things have derivative in L2 two. So one can for example use the ordinary Hbert SL methods to produce a solution.
And in fact I can go on that the asmtotic behavior can be understood of this thing around sigma. We're supposeding rows zero near sigma uh can be understood. They look like essentially the real part of zed to the 1/2 multiplied by a scalar factor depending on the point of sigma. So here zed is a complex normal normal coordinate local coordinate in the normal direction and t is a coordinate along sigma.
But these are not the things that we want in the applications that I'm have in mind. So particular I mean the the non the nonlinear version that we encountered in section two we're looking at a nonlinear equation but it's sort of llas type roughly speaking and we definitely don't want these solutions which behave like order r to the 1/2 those wouldn't make any sense in our problem.
But what one needs is what we might call briefly order R3 over two solutions where this coefficient alpha is identically zero. Then those have got asmtoics given in a similar way but with a power 3 over2 which is just what one needs geometrically for these applications.
So the the the feature is that typically if you fix sigma you will not find these because you solve the problem and the space you solve it in things which have order r to the 1/2. So in order to find these things you can't fix sigma you have to deform it. So the problem and the analysis is closely analogous to uh free boundary value problems where you take say some domain uh and you impo you look for say a harmonic function or something in the domain you impose both noman and dish problem boundary value problems so you don't expect to be able to solve that with a fixed boundary but you have to vary the boundary to solve it so this is the same situation schematically except that rather than a co-dimension one boundary we have this co-dimension two singular set so for example one can show um one can understand these this deformation version of this of this setup supposing we have one order r3 over two solution for with some singular set for a given row n Now we deform row not slightly then one can show that there is a unique small deformation of the singular set with a corresponding uh solution of the desired kind.
The underlying reason why you could do this is just the base elementary formula the derivative of z^ 3 / 2 is a multiple of zed to the 1/2. So roughly speaking when you deform sigma the z to the 3 over2 term when you you bring in a term involving its derivative which gives this z to the 1/2 term which you can use the counterbalance the z to the 1/2 term coming from the linear theory.
So it's not hard to understand in a sense why this should be true. But the the proof is not so easy because when you write it down you find that you encounter a problem of loss of derivatives in the natural way of setting it up.
So although the the sort of general shape of the problem is something you approach to an implicit function theorem, you have [clears throat] to use a more sophisticated kind of implicit function theorem involving the Nash Mosa theory.
So that's um one aspect of the interesting analysis which interesting kind of analysis uh geometry questions which appears when one's studying this particular situation of this general collapsing phenomena also interacts as we said with many other um questions in contemporary differential geometry and is an area which I think will see many developments in the years to come.
So that's I finished. Thank you very much for listening.
>> Okay, Professor Donaldson, thank you very much for your talk. Can you hear me?
Give me a sec please.
>> Okay. Thank you very much for your nice talk, for your interesting talk. We now are open for questions.
I don't know if someone have a question in the chat.
In the meantime, I wanted to to ask to ask something. uh for example uh for spin seven manifolds uh is there something analogous to these uh multial functions or in that setting how is how can work analog anal analogously to to to the approach of >> yes that is an interesting >> that is an interesting question a good question that one can write down a theory as it were in the spin seven case. Um but there are the the drawback is that there are not so many um potential examples at the moment. So the one can write down some interesting but slightly less familiar kinds of equations um but all within the same sort of scheme. uh but it's um [groaning] maybe there's less motivation because it's hard to find examples but it's quite likely something that will develop in the future. Um, >> okay. Nice.
>> And another question.
>> Yep.
>> Okay. I just have I have a another question if I may.
Always when you have this situation in in in which you have a a family of of or a sequence of G2 structures collapsing, you always will have a the the multival function and and they are always harmonic.
Um not exact no not not really that I I mean they satisfy this sort of nonlinear variant of the lelass equation given by this max or submanifold equation. So it's it's it's a somewhat more complicated to set up but many of the same phenomena that appear for the LLAS equal for the plus operator that I discussed appear for this more complicated nonlinear equation.
>> Oh okay okay okay.
Uh okay, Professor Donaldson maybe in the chat if someone have some question.
Let me see in the YouTube chat.
Okay, no more questions. So if I if I may a last question for you.
Erh I don't know if there is something like like a manifold or like a setting like that but considering symmetries for example in in in manifolds with homogenity one coetic one manifold you have a a family of principal orbits in some way collapsing or not not not necessarily collapsing but but tending to a singular orbital that can be like rang or minimal. I don't know if adding or or if there is an example of a setting of G2 structure but when you have more symmetries maybe an action ofity one or something like that. You just have curiosity.
>> Um let me try to think um for actual compact G2 manifolds they won't have but by the basic theory they cannot have any killing fields. I mean they're if they're genuine G2 manifolds um they're not a product say of a circle with something else. uh so symmetry doesn't help uh in an obvious way there but for many of the associated questions in particular for the in the case of say point singularities where you get um you want to describe cones on a sixdimensional manifold of a particular kind then uh symmetry techniques are very valuable they've been used by um Boscolo and Hask skins for example to construct unexpected new examples of these things exactly so kogity one techniques and produced to ODS so in in a slightly um roundabout way these symmetry techniques are um important yeah >> okay professor Andrea you can mute yourself you want make a question please >> yes hello professor Donaldson um yeah I would like to ask you something >> in the second part you describe this exceptionally groups through this cross product on R seven. So you just have you can just have this R this cross productduct on R3 and R seven right and uh do you think this algebraic characterization could also provide a useful framework for studying geometric analysis for example diffusion equations because the llian transfers to this brownian motion because I am in this area not in your area that's why I wanted to ask you this >> I didn't quite follow the question so We're looking at the cross product.
>> Yes.
And then uh if does it make sense to uh use this algebraic characterization uh to study this geometric analysis directly on these G G2 manifolds?
>> Um I'm not I'm not quite sure. It does One one can think of it as giving as of a different way of special way of representing the direct operator similar to the way that you have in three dimensions you have the curl operator. So that that is not the curl operator taking vector fields to vector fields. That is sort of special feature of three dimensions roughly speaking.
It's not something um which works in general. So there in a way this this cross product of seven dimensions uh does give something a bit like that but I'm not sure if that is exactly answering your question but that >> okay professor thank you.
Okay. So, thank you again, Professor Donaldson, for sharing with us that amazing talk and thank you for for being here. We we're very happy you you you accept our invitation to this congress.
>> Yeah. Well, thank you. I'm enjoying Thank you for asking me. Yeah. Okay.
Yeah. Bye. Bye.
Put my screen off.
Okay. So next talk and close our closing talk will be in charge of professor Maria Joseph Pacificico but it will start in about we have some troubles with with technical support but we're dealing with it.
So, thank you for your patience. Thanks.
Hello everyone. We are having some uh technical difficulties but we are solving this issue at the moment. So uh please stay here for uh the talk of professor Mar Jose Pacificico.
And also uh in uh uh 1 hour we will have our closing eman ceremony. So please stay here with us. See you soon.
Hello, Professor Pacificico.
I don't not I'm not understand because it's you have I have just this the the the program for the meeting the last policy already finished it because YouTube it suddenly stop >> yes but now you're here.
>> Thank you very much. Can we can hear you? Yes.
>> See you. Yeah, we apologize for the inconvenience, Professor Pacificico, but now you're here. So, welcome and please >> apologize.
>> Okay, now I can take my my talk because you have I have just this the the program for the meeting.
>> Yes.
>> Last finish it because YouTube it suddenly stopped.
>> Yeah.
>> Yes. But now you're here. Thank you very much. Can we can hear you? Yes.
>> We apologize for the inconvenience, Professor Pong, but now you're here. So, please come and please apologize.
>> Okay. Now I can take my my talk because you have I have just this the program from the meeting [laughter] already >> you can you can close the window of of the YouTube of the YouTube streaming please.
And we can we can stay just in here.
You can close.
No, no. We can stay just here.
>> No, no, no.
Okay.
>> Okay.
>> Okay.
I close I closed the Perfect.
>> Okay, >> now we are fine.
>> Okay, I Okay, now I can understand what's going on. Okay.
>> Okay, professor. Now we are sorry for the delay.
I'm going to present your talk.
Just a little second. I want to apologize to you and all the participant for this technical problems and for this delay in the in the program. But now it's a great pleasure for me to introduce professor Maria Jose Pacificico, one of the leading figures in dynamical systems in Brazil and Latin America. She's a full professor at the Institute of Mathematics of the Federal University of Rio de Janeiedo. Over the course of his career, she has made important contributions to the study of flows with singularities, singular hyperbolicity and lens like dynamics. Her work has has lasting influence on the development of the field. Equally important has been her dedication to the training of geom mathematicians and to streng the dynamical system community in Brazil through supervision collaboration and the organization of scientific activities. Today, Professor Pacificico will present her talk entitled an spectral de composition for singular staff close. Professor Pacificico, thank you again for being here with us. Please go ahead.
Okay, many thanks for the invitation and many thanks for the introduction and then today I will talk about the spectral the composition conjecture for star flows and uh the motivation for this is following a guidance principle in smooth dynamics which is the to understand the comp complicated dynamical system by de composing ing it into simpler invariant pieces.
And for uniformly hyperbolic system, this principle is realized by two fundamental achievements is bas spectral decomposition and Boeing's thermodynamic formalism.
One of the millstones of modern dynamics is the following theorem due to Steve Bale that which says that if you have a demorphism defined on the compact manifold which is axom then the non set can be decomposed in a disjoint union of compact invariant hyperbolic trans and local this lambda I what means a just is hyper sense that in the tangent b you have the composition That's such that the derivative the the position in the tangent bundle subspace complex sub the derivative contracts uniformly the vectors in one of the sub bundles and the derivative expands uniformly vectors in the other bundle and the no Other set is just the set that we are interested in knowing because that sets that per if you take a point in the no other set here the point the orbit or the trajectory of this this point does not return to this point and the same theorem holds for non-s singular flows that is flows such that they never vanish or any exit the flow doesn't vanish there and the boring erotic picture for this kind of flow says that the hyperbolistic has remarkable godic consequences for it in another word is that for each basic set there is a unique equilibrium state a unique measure of maximal entropy specific some explaining all this these terms that appears here. Okay. And so the question is what survives if we are dealing with flows presenting singularities.
Flows with singularities are not uniformally hyperbolic typically not and not covered by bowing theory.
So can one the main question is can one recover a buoyant type thermodynamic formalism for singular flows. Singular flows means that the flow has singularity. Okay.
And our answer is yes. the main our main erotic results it's two theorems together with Gang and Fang we prove that thereist is a C1 and danc set U contained in the set of the C1 star vector fields on the manifold with a positive topological range.
such that every flow in this set has only finitely many erotic measures of mass entropy hold set containing the in the in the set of C1 star vector fields with a positive topological entropy such that every X in the set has only finitely many equilibrium states provided that the value of the potential at the any singularity is less than the pressure of the in the whole space. Okay. Okay. And uh then let me just tell you the notions of hyperbolicity.
Given a C1 vector field on compact manifold M and the notice the singularities of X is this point. The set of points in the manifold is such that the vector field vanishes at the point. Uniform hyperbolicity means that the tangent bundle uh deco composed in a direct sum of this sub bundle. It's called stable bundle.
Plus this is the bundle tangent to to the flow. And the other sub bundle is called the unstable bundle. And here the derivative of the flow uh contracted uniformly vectors in this sub bundle.
Here the derivative expands uniformly vectors in this uh bundle here.
Moreover, this deco composition is uniform in the sense that if you have a sequence converging to some other point, the deco composition also converges to the deco composition at the limited point.
And another section is the sectional hyperbolic that is we just weaker the this notion above it. That means that you have on the at the tangent bundle you have yes which is a a uniform bundle in the sense that the d the derivative contract uniformly vectors in this sub bundle and another complimentary sub space that's we call fc y mu where what happen is that for every subspace two-dimensional subspace contained in this space here the derivative contracts volume and the sectional hyperbolist is equivalent to uniform hyperbolist when you don't have singularities that's the point and okay why to to give the answer or to prove this theorems is difficult to to achieve the result because when you have a sectional hyperbolist you have dominated splitting in the tangent bundle and on the other side if you have a multi- singular hyperbolist which is an even weaker notion of sectional hyperbolistic the tangent you don't you have this de composition in the not bundle tangent bundle the singularities for in se for sexual hyperbolic always the same index and for multi- singular hyperbolic flows they they may have different indices in the the dimension of the stable me singularity and the kind of sectional hyperbolic flows and weak for multi- singular hyperbolic flows and the structure is better understood for the section of hyperbolic case and mostly unknown for mult multi- singularity singular flow. The most singular case is much much weaker than the class hyperbolic and the motivating example for Lauren system in indeed in 63 L published his famous paper deterministic and nonperiodic flow in the atmospheric journal in which he regarded the following system of ODS as a model for thermal convection. You see that this are the these equations nowadays are known as Lawrence equation and the three equations are quite simple are polomial twoderee polomial but at that time Lis could not solve explicitly this uh this uh flow you know that indeed zist is Zero is a singularity because it vanishes the the and okay then the predictability now does the flap of a butterfly wings in Brazil set off a tornado in Israel I don't know that was the butterfly effect that Lawrence coined the term butterfly effect based on the shape of the attractor and the meaning of independency of the trajectories inhalation to the initial conditions.
The lis attractor shows that singular flows may exhibit the robust chaotic behavior and this motivate this search for an appropriate notion of hyperbolist beyond the classical one. What's the meaning? This this meaning is that even it appears so chaotic. No, but this is robust in which sense you can take this equation this vector field here. You can change a little bit this parameters.
This is the the parameters like this one. nowadays are known as the classical parameters.
But if you perturb you have the same kind of shape, you still have something similar to a butterfly. And the meaning of this and the way the the orbit or the trajectory goes around the singularity is completely brand.
You cannot say okay if it's now it's in one wing then it will go the next will go to the other one now it's completely brand and this fact happened for every pertubation perturbed the equations and that was quite amazing because how to characterize this kind of flows because certainly this flow is not uniform hyperbolic because the singularity just the any splitting that you take in the tangent bundle just the generated the the singularity disappears. Okay. So and in this direction the next step is to define what's called the flow. The addition is that the vector field has the star property if every nearby vector field has only hyperbolic critical elements.
Star flows are often regarded as the natural analog of a systems and the singular star flows extended this analogy to a systems with singularities.
You see that the the law flow is a singular star flow because the this hyper the singularity of the L flow is hyperbolic and if you perturb a little bit it is the perturb the singularity is also hyperbolic and so is the more sign, how do you say this?
This is more significant example of a star flow, singular star flow.
And uh why star flows?
Why you have the stability conjecture by palis mayo that say the structurally stability is equivalent to hyperbolicity and the stability conjecture naturally leads to start systems because a structural stability implies a star property and annoying results on these directions are foromorphism.
This was solved by Franks the me and for non singular star flows was solved by hayash g and well but for singular star flows not yet I also don't then let's unpacking the history of this kind of uh story.
The uniform hyperbolistic plus transversality implies structural stability and the stability conjecture has two steps. Step one is that structur stability implies the star condition.
Every periodic orbit of every nearby systems is hyperbolic and that was proved as I said before by the Mel Robs.
Now and the second step is that to prove that the star the converse no that a star condition implies um structural stability for diffuse. No star conditional implies uniform hyperbolist. This was proved by Leo and May and for flows when you don't have singularity by hayash and uh with singularities the same index you have sectional hyperbolist by she and the different and multi singular hyperbolist that was by bonati and the D why a system with singularities a is where is spectral the composition works thanks to the uniform hyperbolistic no singularities near a singularity sigma the hyperbolist genuinely degenerate This is a real obstruction, not just a technical one.
And the 10 box there above it tells the story of how hyperbolist was generalized to handle this while keeping the same structural holy aim that the classical hyperbolist was too hesit ative sectional hyperbolist generalizes this to like attractors same index and multi- singular hyperbolist covers different indices density and the genetic case then is each notion above replaces is uniform hyperbolist in exactly the holy of guaranteeing controlled splitting and robustness opening the door to spectral decomposition and the thermodynamic formalism which is the rest of this talk.
Then the spectral conjecture for star flows. The conjactor is every C1 starflow admit is only finitely many non-trivial CH recurrency classes which is equivalent to the conjecture that every CH recurrency class should be the homocinic class of a hyperbolic periodic or this is the natural singular counterpart of is the okay but our point of view is harder than proving the topological conjecture directly this this is as the topological conjecture for the spectral the composition for uh star flows okay and our point of view is that other than proving the topological conjecture direct directly we first establish its zerogic counterpart that is we first prove the godic version. This turns out to be accessible through thermodynamic formalism.
Then the theorem is given a continuous holder potential. Five. There is this C1 open dance set in the in the contained in the set of C1 star singular star flows with topological entropy positive such that every X in the set has only finitely many equilibrium states. Provided that the F applied it to any singularity is less strictly less than the pressure of the of the whole space.
And this lead is naturally a type for multi- singular hyperbolic classes which is the technical obstacle for uniform hyperbolic systems.
Thermodynamic formalism relies on three fundamental ingredients invariant stable and the unstable foliations expansiveness specification for multi singular hyperbolic flows.
None of this is available in general and the challenging the main challenge is recover enough hyperbolic structure to rebuild thermodynamic formalism.
Then the general strategy is you have a multi- singular hyperbolist then try to use leo geometry and then get a good orbit segments and in this good orbit segments you prove property and specification and then use what is called improved atoms and to get finite number of equilibrium states. I will try to give you an idea of this strategy. Okay.
Well, the first ingredient, the first ingredient is called the leo geometry because classical local product structure is no longer available.
Product structure means that the if the stable and unstable manifold that they intercepting in one point this point of the intersection should be near by the the [clears throat] original one and instead since this doesn't uh doesn't is available. We work with scale the neighborhoods is that compare the the the flow with the velocity of the flow on each point. You we use fake invariant fiations hyperarameterized the shadowing.
These tools allow us to recover local hyperbolic geometry even near singularities good and bad segments.
The point is that the orbit segments with the compos in two families GB and GS like this you have this is this is just one but it's like a bunch of segments of orbits and you just uh deco compose it on the sets GB is the central one and GB like the B is contains the wood orb segments and selected the true hyperbolic behavior and control the recurrence.
On this collection we prove bowing pro and specification and GS contains the remaining orbit segments.
The key point is to show that they carry a strictly smaller pressure.
excuse me and therefore do not contribute to equilibrium states.
We are trying to use here the Boeing strategy. Boeing strategy since in the beginning he used that bowing property that I will explain is specification responsiveness. These three things implies existency of equilibrium states. We are just trying to follow Boing strategy but it's not enough. We have to use what's called the cle hag improve the clement hag thmpson [clears throat] strategy or critio what which was exactly this the to deco compose the the orbit segments in this kind of families the good ones and the other ones that you can that they carry strictly smaller pressure and therefore do not contribute to equilibrium states and the good orbit segments. The collection is constructed from points with infinitely many slow recurrent place times.
This point is exhibit a persistent hyperbolistic control recurrency away from singularities uniform local geometry and this is precisely the setting where geometry can be fully exploited this please time is times okay you don't have uniform hyperbolist but please prove that you you you can find infinitely many points or times such that the orbit through this point behaves like uh if it were hyperbolic and it's low recovery because this is not enough to prove you have to define an another type of please time that we call the slow recurrent please time is just to deal with the segments that pass nearby a singularity.
And then to establish the bowing property, we first identify good orbit segments through infinite slow recurrent time which provide persistence persistent hyperbolic behavior along the art. You have this is represented is a a ball a bowling ball that you follow the [clears throat] the the orbit of points in this in this neighborhood. they are nearby till some iterated.
And [clears throat] we uh just prove this that the the informal proposition to this point is that if xt is the good orbit segment and y belongs to it boing ball then y is scaled the shadow by the orbit of x. This mean this means that the distance between among the between the [clears throat] fs of x and fs of the orbits point by point iterate is comparable with the velocity of the flow at the point for every s in this segment of orbit.
hensive closeness in bowling [clears throat] balls automatically enters the scaled hyperbolic geometry of Leo.
Okay.
How do we win? The usual please times are not are not enough.
Once the orbit comes closer to a singularity, we need a new kind of place time and the slow recurrency hyperbolic time for a good orbit segment XT. This represents the segment and not the only point and for any s in this interval. The segment that when you follow you will evolve by time can only spend a small portion of its time inside the neighborhood of the singularity.
This is the singularity and this is represent a neighborhood where you lose hyperbolist because here the flow vanishes and then you are losing hyperbolist nearby here and then what you do we are able were able to find this kind of slow please time that the or this kind of when the amount of time that it spends inside this neighborhood is small compared with the time that it spends outside W till the next entrance in this neighborhood. You have bad segments but the other segments have enough hyperbolicity to cover what is loed in this neighborhood.
And with this just can you prove the B property. Okay. And the read is the specification property.
Specification property is like this. You have one segment of orbit. Here you have an orbit. And what you get? You get that the the orbit of a nearby point is shadow but in one piece here and then it goes outside but then it comes nearby again and then go outside and then come returning to be nearby the orbit of the point.
The starting point is the easiest the existency of infinite slow recurrent please times which the detected points with persistent hyperbolic behavior.
Let me test.
And uh the informal proposition is that there exists a large compact set of good points detectable by invariant measures whose local invariant manifolds intersect transversely those of a fixed hyperbolic periodic at a uniform scale.
This transverse intersections provide the mechanism needed to glue orbit segments and he cover specification.
You see that all this what we are if you putting everything together you have that bowing property implies bounded distortion specification you can orbit growing and together they allow us to apply uh this critarium the clean hagats to improve the critarium End of orbit time a finite number of equilibrium is what just in a few few words what is this because clitarium that you could follow bing strategy to get equilibrium status that was obtained for theomorphisms in the beginning and the cleaning Hagat this for not only for diffor improve uh bowing strategy but the cling unfortunately does not fits for us because of the singularity. Then we had to work a little bit more just to adapt this criteria to uh to our setting. And this what is this in a few words means that you just work in such a way that you can follow the moving initial strategy to get equilibrium state. In another words, you have to prove responsiveness, specification and bowing property and you adapted the in the setting where you are working. Is this this is the point and from one class to all star flows.
Okay. Each isolated multi- singular hyperbolic class admits a unique equilibrium state. This was proved by Kier and Buzzy.
And combining this with the goic spectral decomposition, we have only finitely many equilibrium states for C1 open and dense star flows.
in particular finitely many measures of maximal entropy.
And back to the spectrum of the composition conjecture, the topological conjecture remains often.
Our result establish it erotic counterpart. Eerodic de composition is proved.
But this brings us one step closer to the original spectral the composition uh conjecture and indeed the godic result already yields a partial topological consequence that I will state now and has a strong topological consequence the spectral the god spectral de composition is a the coral area that this says that for a C1 open on the set of star flows for every constant positive constant there resist only finitely many chain recovery classes with topological entropy bigger than [clears throat] C. Moreover, each is such a class is a homocinic class of hyperbolic periodic support at the most one local measure of maximal entropy entropy.
You have this [clears throat] adotomy that with the same cos and that there is a Brazil set of one star flows such that for every X in the set and every non-trivial CH recurrency class C of X We have that if the topological entropy in this class is bigger than zero then this contains some periodic point P and is an isolated homoc clinicic class.
And if the topological entropy of the flow in this class is zero, then C is sectional hyperbolic either for X or minus X and contains no periodical orbits. And in this case, every erodic invariant measure with support contend must satisfies that mu is the dak measure at the singularity on each singularity on some singularity in the [clears throat] class.
Okay. So what is one remaining obstacle is that our result reduce the topological spectral the composition conjecture to a single cast that for C1 open and dens flows every non-trivial chain recovery class has positive topological entropy because of this dichotomy.
No.
And if true immediately implies the topological spectrum, the composition conject is this what we would like to prove not now but in the next years I don't know in the in the future.
And then my co authors on this is and the reference for this are this uh works here that we start quite recently from two the year 2000 till now and uh okay and this is I I will stop here. Thank you very much for your attention.
Thank you, Professor Pacificico. Very nice talk. H we have time for some questions. H in the chat. We have a question. Is there an analog the composition result for non-compact manifolds?
I lost the Oh my god. Okay.
>> Okay. Let me repeat the the question.
Did I >> uh the first question is is there an anal analog?
>> You know, I don't know. I am >> what? Okay.
Uh I stop the comfort. Okay.
>> I will stop. Okay. Now I have Okay. Now I am here. What?
>> Okay. Professor, thank you for your very nice presentation again. H we have some questions in the chat. So I am going to the first question.
H is there and analyze the composition results for non compact manifolds.
Sorry, what is the question? You know, I'm not seeing my my slide. [laughter] >> No.
>> Uh, let me see.
>> I don't know what happened.
Let me >> You're sharing your your full screen >> is here or no? No.
>> Yes, I now we are seeing but it is not in full screen. No, it's behind that window. Yes. Okay.
>> Okay.
And then what was the question? I don't know.
Well, unfortunately I I the all this is quite technical. If I go through the proof, I would never finish it because it's quite complicated. I try just to pass the idea that to work. What we are following since the beginning is just the boring principle to prove in the since the beginning that you have some responsiveness uh specification property and then you can prove um existence of equilibri states. That was the main idea behind all the the the the problems here. How to adapt this and then once Clement Hagen Thompson adapt for some setting that they were interesting and then we adapted this setting just to recover speciveness Boeing property and specification in our setting.
It's more the final message is this that we have to work just to recover completely. Now the idea the buoyance idea to the existency of equilibrium states okay >> okay we have some some other questions professor warp b I don't know this could one could one hope to prove full spectral deco composition by showing that every zero entropy multi- singular hyperbolic class reduces to a finite that configuration of singularities.
>> No, unfortunately you have to prove that exactly that in this setting you it cannot have entropy here a recurrent class cannot have uh entropy here. It's just that what is this sorry thistomy that this case doesn't exist in another way that if you have a a C1 starflow then and uh with and you have a non-trivial CH recurren class then the the topological entropy has to be positive. This is what one has to prove to to get to achieve the topological uh uh spectral the composition. No, what we we have is just the godic one in the sense in the in the meaning that we have a finite many equilibrium states.
No recurrent classes with finite many equilibrium states. What's the theorem?
The theorem is I I am a little bit because what I talk I talk most of the time not about the multi- singular hyperbolic I talk mostly about the sectional hyperbolic [clears throat] hyperbolic flows. Why the approach here is much more uh complicated in the sense that you don't have any geometry.
You don't you may not have any singularity. You don't have tangent bundle to to deal with. you don't know how to to start you know then it's much more suffered but we succeeded to work on this and uh okay you have much more work to to set everything and again the the difficulty is how to adapt and generalize Ed the Boeing approach to get the [clears throat] finite number of equilibrium states with this is for this set [clears throat] here multi- singular hyperbolic flows much more complicated.
Okay, we have one one last question.
Under what additional assumptions would be the number of measures with maximal entropy be locally constant under C1 perturbations?
Uh this I I I I don't know what I know is that the what we know locally constant by now I I don't know how to answer this correctly.
Sorry.
>> Okay. Perfect. Oh, thank you. And if I may may make a a question, I would like to to ask is is there any bound on the number of of the number of erodic measures of maximal entropy in terms for example of the number of singularities or or something like that?
>> No, we didn't. It's the I don't know. I don't know. It's just a finite number, but I don't know if it's it's related to the number of singularities or not of the flow. By now, I don't know how to answer this.
>> Okay, Professor Pacific, we have some some messages. Julio says, "Thank you so much for the talk." Professor Wenbo says, "Thank you." And I say thank you too because that interesting talk is a good way to to close this this conference. And now I invite you all to to stay to the closing ceremony. We are going to announce our IMA prize 2026 and I'm I'm going to >> stay sorry I stay on on the zoom or I go through the >> on the zoom professor please.
>> Okay. Okay.
>> On on the zoom over here please.
>> Okay. So the one I'm going to pass the word for you and you are going to share this closing session please [clears throat] are you there?
Okay, sorry. We have a little delay again.
Okay, thank you very much. Thank you to all the participants that that has been following the scientific program.
Um just to confirm we have with us um professor Tatiana Toro, right? just um for practical things. Let's see if if Professor Toro can unmute herself.
>> I'm here.
>> Very good. Um good.
So, Professor Jointo, you can try to unmute yourself to see if uh everything works well.
Professor Tooff, can you hear me?
>> Yes, I can hear you.
>> Thank you, Professor.
Uh, could you try to activate your video to see if your camera is working well?
>> Yes, of course. Uh, yes.
>> Okay, very good. So, we have with us Professor Dinto. Okay. Professor Karina Navaro is here. Professor Brian Graales is here.
>> Hello. Hello.
>> Oh, hello.
>> Hello.
>> Very good. Also, Pedro, right?
>> Okay.
>> Okay. Thank you very much. It's really my great pleasure to give you to give to all of you the welcome to the man prize ceremony and to the closing ceremony of our conference. Um we really have been enjoying this week of the many many conferences uh of a very nice scientific program of very inspiring talks right but then I think we are just uh to some minutes to know who is the who will be the the winner of the man prize which actually is the most important prize that offer our the scientific society.
So [clears throat] uh in order to have uh our closing ceremony and also the ceremony of the IKMAN prize, it's really my great honor to inform to the participants that we have with us not only authorities, authorities of Latin America, international community of mathematicians from Latin America, but also we have with us um professor Tatiana Toro the vice president of the international mathematical union and professor Jakinto president of the international society for analysis his applications and computation as our invite invite guest. So thank you very much to professor Toro and to professor to for being here for making time in in their very busy agenda for for this ceremony and um the following will be the plan for the for the ceremony. First we are going to hear some words of professor Tatiana Toro vice president of the EMU.
Later some words by professor Jakintov, president of ISAC. And then we are going to have uh we are going to hear from Karina Navaro Gonzalez, president of Iikman and from Brian Graales, vice president of Iikman. Uh the name of the winner of the Iikman prize. Right. So after that we are going to have some words about uh some words by Andrea Ortado our business scientific director in Iikman and about information about the inerson program of the man conference that actually will take place in August by Pero Fernandez Espinosa who actually is our new scientific director.
So thank you very much to all the participants that are following us and I would like to invite professor Tatiana too vice president of the international mathematical union. Please go ahead professor to >> thank you. So first of all good evening and go good afternoon to all of you. It is an honor to be addressing the audience of the 2026 international community of mathematicians from Latin America Ikeman Congress. I hope that you all have had a very productive and stimulating week. Ikeman at the time the international conference multiddisciplinary aspects in mathematics and its applications was created in 2020 and hosted at the universal colombia. The conference brought together mathematicians across diverse fields fostering collaboration and sharing groundbreaking research. By 2022, the conference had grown significantly, drawing an expansive community of mathematicians from Latin America and beyond. And from this success was born the international community of mathematicians from Latin America, Ikeman Latin America.
This is a thriving community dedicated to advancing mathematics in the region.
Ikeland works very closely with La Mathematica Latin America kar to strengthen mathematical research, foster collaboration and advanced training by bringing together established researchers, emerging graduates emerging scholars and graduate students from across Latin America.
Today, AGMAN Latin America stands as a hub for mathematical innovation and collaboration, proudly showcasing the talent and potential of Latin America's mathematical community. I will also like to highlight the establishment in 2026 of IKM Africa to promote mathematical research, collaboration, and visibility across the African country.
One of the greatest challenges facing students and mathematicians in Latin America is isolation. IGMA helps overcome that challenge by creating a space where researchers can exchange ideas, learn about the latest development and welcome part of a and become part of a vibrant mathematical community.
As a student at the University Colombia in Bgotaa many years ago, I will have cherished the opportunities that AGMAN now offers to the next generation.
Agm promotes the work of researchers across Latin America by providing a platform where their ideas can be shared, their achievements recognized, and their contributions to mathematics highlighted. In doing so, it strengthens the visibility of mathematical research in the region and foster a greater sense of community and collaboration.
I would like to also express my admiration and gratitude to the IGM executive committee for their vision, dedication, and tireless effort in building this remarkable community. By strengthening mathematics across the global south, they're not only creating opportunities for individual researchers, but also enriching the entire mathematical enterprise. A stronger and more connected global mathematical community benefits all of us. Thank you very much. And I will say I'd like to um congratulate the yetto be known winner of the Aman prize. Thank you.
Thank you very much, Professor Tatiana Toro, for your very kind words.
So uh it's my great pleasure then to invite professor Drainto president of ISAC the international society for analysis its applications and computations our invite guest for this lossing ceremony.
>> Okay thank you very much. Uh can I share um pile here?
>> Yes, professor, please.
>> So [sighs and gasps] um um so um can you see this all of you?
>> Yes, professor. We can see your presentation.
>> Okay. Oh, okay. Sorry. Uh now I see that it's should be in the beginning. I just want to put it here.
Um so uh sorry I I think I made a mistake when I took a note in my calendar. I thought it should be uh actually for some reason I noticed the Tuesday in one place and I thought it was the last thing. So sorry for this.
So um in any case um thank you very much for having the opportunity to uh be a part of this closing ceremony of this very um important uh meeting and I'm so happy to see that uh how to say mathematics and analysis in Latin America is becoming so strong and with young talent mathematicians uh and um and also how to say a very go in in the in the organization of a lot. So I think this is a important u part of the future of mathematics. It will be in Latin America. This is for sure uh true and I'm also sad that I couldn't participate in person. Okay. uh on uh this. So now I can see that uh you cannot see the whole thing here. Just a moment. I will do it like this. Now you can see it uh better here. Okay.
Uh so um u because of this I could not uh completely update the file I wanted to have. Okay. But um uh so so I will uh talk a little bit about uh Isaac and uh of course as I said um it's nice to see how strong uh Latin America becomes but also I'm so happy to see that we can have this very good collaborations uh and that we can somehow have um have would um I mean as much as we are able to support these activities from Isaac part uh uh uh is the for at least Isaac is the other part because uh ICMA becomes a really a key um collaborator.
one can almost say also that big um cross fertilization between our communities and so on.
So uh a little bit about ISAC uh so um uh this is the it stands for the international society for analysis its application and computations and here you can see some um some important things that appears with Isac. So a very important um part of ISAC is of course all the different ISAC congresses that are gathering lot of people uh together.
Uh actually one of the really main ideas with Isac of course it should promote analysis its applications and computations the subject of Isaac but also somehow it was uh very early I mean really from the beginning uh it was also that um one should try to really connect um uh people especially from different part of of the world. Also, how to say promoting u the exchange between how to say um wealthy countries and less wealthy countries how to say so that there was somehow um this was from the beginning.
It should not be how to say a rich man club. It should more be an attempt to putting bridges between and also how to say encourage uh um exchanges between different u part of the world and um this is still a cornerstone of Isaac.
Okay. Beside of course uh well the subjects uh you can see here that so far we have um had 15 uh congresses and uh the first one was in Delaware in 1997 and the last one it was in Astan Kazakhstan.
It is actually the first time. It was uh they were very proud of having this event uh because it was the the last one uh because it was the first they said um big uh scientific event in central Asia.
And the next time I hope that I will have the pleasure to see all of you uh with um in Shaman in China.
But also you can see here which is really important also I mean this is extremely important with Shaman in China but also extremely important is that 2029 you will see it will be in Colombia.
Uh so but really I hope we will meet already in Shaman. Okay. But of course if we are not meeting in Xiam we will for sure meet 2029 in Colombia. Okay.
Can you promise me that professor Kadona?
>> Yes sure what about we have very good things committed with that.
>> And what about uh professor Gonzalez?
Can you also pro promise that?
>> Yes of course I promise you.
>> Okay. Very very good. Okay. I'm very happy to hear this. Okay. And uh here you can also see about the uh the in the ISAC congresses there are several sessions usually around 15 but it can also go up.
I I suspect for example in China there will might be more than 20 actually and so on. And there were also a lot of sessions in Kazakhstan in the last congress.
um how this is a little bit how is um ISAC uh um organized. Uh so um we have within ISAC uh special so-called special interest group uh abbreviated SIG and um here are some of them uh but but uh these are not the last ones I think we have give a little bit more uh so um we have complex analysis etc clifford and quinionic analysis is and uh and so on and so on. So there are a lot of uh different kind of uh special interest group. Our last one was uh in a fractional calculus and uh we were act it seems that we are able to gather into Isac a big part of the editorial board of the most uh how to say well I would say well they say themselves that it should be the best journal in fractional uh calculus uh a springer journal okay so that's a little bit how it is okay and um and so on and so on. Uh and of course uh we enjoy very much when one can have it in such way that um you can have some activities within Iskac if you have special mathematical society. Uh but uh also we um enjoy uh very much as it works now with MAM uh that you have really something very huge and lot of activities outside and so on. So this is somehow one should see this as a platform. Okay. So I'm very happy for this uh very successful conference that you have organized now here. Okay.
So that's a little bit uh how it is.
Okay.
Since 2025 uh the we have this pre the officers of Isaac is here. But of course there is then the officers of this society is nothing compared to especially not the president if they have not how to say a good um Isac board. But of course this is still not that important as the ISAC members.
Okay. And I'm very happy to that uh well we have a lot of ISAC members with ICMA.
So we can then have this kind of cross fertilizations.
So that's a little bit about that and so on. And also we have books here in Isac Associated Journals and um uh we also have ISAC news that distributes to the members two times per year and so on and so on. So that's a little bit um and at each ISAC congress also in similar way as here in ICMA we give so-called Isac award to a young outstanding researcher and also we have well some uh these are some recent awardes um this is not an updated I wanted to update this because we have some very nice award awards also recently but uh um so the and so on. So we give it the last one was given 2025 in AA now uh so um and so on um and um and so on. You can also see some of these people have to recognize for example Michael Rashansky is I think you have encountered him. What do you say professor Kadona he's he's not so inactive yes that's true and also some others somehow you can also see Terence Tao for example so and so on so um if you are not an ISAC member yet you are very welcome to become one so you just uh contact one of the officers and hopefully one can then this can also So hopefully be helpful for ICMM AM. We I believe that we can both make us stronger together how to say.
So um thank you very much for for having the talk and really I wish you how to say well I would say that uh it was so good conference and also I'm feeling confident that your next conferences and also especially when you organize the next ISA congress will be very successful. So thank you very much for your attention.
This is uh just thank you for attention.
Okay.
>> Thank you. Thank you very much professor Jakinto for your >> very nice presentation.
Just uh a comment um in the next man Africa actually professor Jakinto is one of our plenary speakers.
I really invite all of you to follow professor to plenary in October.
>> So okay.
>> Thank you very much.
>> Okay. So then uh now it's time to know then uh who will be the winner of the man prize and for this special moment of our scientific program I invite Karina Gonzalez president of man and professor Brian Graales vice president of man who will give us the name of the man prize winner.
Okay. Hello everybody.
So we will start with the with my speech.
So ladies and gentlemen, on behalf of the international community of mathematicians from Latin America, I am honored to present the Eigiman Prize 2026 to Professor Maria Jose Pacificico.
I would like to begin by thanking her for accepting this distinction and for join us on this occasion.
The man prize recognizes scientific achievement.
It also recognizes work that has contribute to the development of mathematics in Latin America.
Professor Pacificico's career gives us a strong reason for honoring her in both respects.
In honoring professor Pacificico, the man prize continues its tradition of recognizing outstanding Latin American mathematicians following previous recipients that Toro today here with us and Carlos Kenny.
Maria Jose Pacificico was born in Guariba in the state of S. Paulo. She completed her degree in mathematics at S. Paulo in sorry S. Paulo State University in Hocon SO State University in Hoklar in 1973.
She then continue her studies at the Institute for Pure and Applied Mathematics known as IMPA where she received her master degree in 1976 and her doctorate in 1980 under the supervision of Wellington de Melo. After completing her doctorate, she held a faculty position at Flumin Federal University in 1982. She joined the federal university of Hio Janeum where she has spent most of her academic career. She became a full professor in 2006 and has remains closely involved in research, graduate education and university administration.
Her publication record was already notable in the early 1980s.
By 1984 her work had appeared in invention mathematically the journal of of differential equations and her god theory and dynamical system. In 1986 she published in topology and again in inventions mathematica.
These early papers began a research record sustained for more than four decades. Professor Pacificico is internationally recognized for her contributions to the dynamical systems. Her main areas of work include flows with singularities, lawren type attractors, partial hyperbolicity and entropy.
Her results have become an important part of the literature on dynamical systems and she is widely recognized for her major contributions to the development of singular hyperbolicity.
The formation of students and young researchers is another important important part of her work. Her updates academic records list 16 completed doctoral thesis, 14 master's dissertations and 31 post-doctoral researcher under her supervision or joint supervision.
This number represents many years of reading, discussion and individual attention.
A number of this researchers later became her co-authors and continued their own academic careers.
these aspects if her or work deserves the same attention as her publications.
A mathematical community depends on researchers who are willing to train others and to help them develop independent work. Professor Pacificico's activity has an advisor has extended her contribution well beyond her own articles and books. She has also taken on demanding institutional responsibilities at the federal university of Hio Janeu.
She coordinated the graduate program in mathematics from 2012 to 2017 and again from 22 to 2024.
At the national level, she served during three separate periods of the advisor committee for mathematics and statistic of Brazil national council from scientific and technological development.
Her service has also had an international dimension. She had worked on committees of the of the world academic of sciences including panels responsible of the selection of ours and new members. She has also contributed to regional programs of scientific cooperation in mathematics.
Her scientific standing has been recognized by major institution in Brazil and abroad. She received her first research productivity fellowship from Brazil's national council for scientific and technological development in 1981.
She currently holds level 1A the highest category in that fellowship system. She had repetitively received the distinction scientist awarded by the research foundation of the states of Rio Janeiro. She was elect a full member of the Brazilian Academy of Sciences in two uh 2005 and became a member of the war academic of science in two in 2007.
She was also awarded the rank on commander of Brazilian n national order of scientific merit in 2024. The Brazilian Mathematical Society, the Institute for Pure and Applied Mathematics and Brazil Ministry of Science, Technology and Innovation are her her day her the prize Ellas Mathematica in the category outstanding scientist in mathematics in Brazil. This hour recognized an established scientific career and her role as a woman in a field in which positions of research leadership have long been occupied mainly by men. This part of her history should be stated directly.
Professor Pacificico established herself through the quality and consistence of her her work. Sorry. She later occupied position from which she could participate in decision affecting mathematics in Brazil in Burwat.
The Igman prize 2026 recognizes the quality of professor Pacific research.
It recognizes her sustained work across more than four decades. It recognizes the student she has trained and the responsibility she has assumed.
It also recognizes her place in the history of women in mathematics in Brazil and Latin America. Professor P Maria Jose Pacific, thank you for accepting this award. Thank you for your work, for your contribution to the education of new mathematicians and for your service to our scientific community. It's my honor to present you with the Eigman Prize 2026, Professor Pacificico. Congratulations.
Thank you. Can Can you hear me?
>> Yes, I can hear you. Well, >> thank you very Okay. Now I think the connection is not the best.
>> I sincerely >> Okay.
>> I sincerely Okay.
I sincerely thank the Award committee for selecting me for this distinction.
I also wish to share my this recognition with my many co-authors and collaborators indeed throughout my career. They have been essential in shaping my mathematical journal and this award is in many ways a recognition of our collective work.
I have greatly enjoyed this conference.
unfortunately could not attend all the still in progress at my university and this is the busiest time of the year for us. Even so, I found the conference truly inspiring. I brought together it brought together outstanding researchers from a wide brand of mathematical variance creating an exceptional scientific atmosphere. I also was delighted to see such a good balance between women and men among the invited speakers.
As someone working in dynamical system, I was especially pleased that our field was represent as well and I believe that meetings like this place is on play an important role in strengthening mathematics development in Latin America increasing its international visibility.
and fostering collaborations that will have a uh blessing impact. No.
And then thank you once again for this great honor and great thank you to all the for professor Cardona and all the others involved in the committed here. Thank you very much.
I'm really honored to to receive this uh award. Thank you.
>> We have some comments on YouTube.
Everybody is congratulating Professor Pacificico for for the prize.
Very good feedback from our community.
So um thank you very much to Karina for for the speech and congratulations Professor Maros Pacificico for this prize. Uh it's really an honor for us to have you here and thank you for accepting the prize of course. So let's continue with the closing ceremony. I would like to invite uh Andrea Ortado vice scientific director of Latin America.
>> Hello >> to share some words with us. Please Andrea, go ahead.
>> Uh hello everyone. Good afternoon. Uh before we conclude this uh wonderful conference, I would like to share a few personal words.
Being part of the IGMA Latin America community has been one of the most rewarding experiences of my academic journey. Over the past three years, I have had the privilege of participating in different initiatives from the Umalaikman Latin America seminar to the women in mathematics projects and the man young researchers program. Through these experiences, I have met remarkable mathematicians, collaborators, mentors, and above all, wonderful friends.
A few months ago, I had the great honor of being appointed by scientific director of Iikman Latin America.
Receiving this invitation was both humbling and deeply meaningful to me. To be entrusted with this responsibility while I am still pursuing my PhD is an honor that I do not take lightly.
I am sincerely grateful to the executive committee to professor Karina Gonzalez, Professor Brian Graales, Professor Dwan Cardona, um Pedro Fernando Fernandez and everyone who believed in me and gave me this opportunity.
I completed my bachelor degree in mathematics in Colombia and in 2017 I left my country to pursue my master's degree abroad.
Since then, every years has brought new challenges. Building a scientific career in a foreign country is not always easy.
It means adapting to new cultures, new languages, and new academic environments while being far from your family and everything that once felt familiar.
Looking back today, I feel incredibly grateful.
This appointment uh reminds me that dedication, perseverance, and genuine passion for mathematics can open doors that once seem impossible.
Throughout these years, I have learned that challenges do not define us. Our response to them does. Wherever I have been, I have tried to face every obstacle with resilience, hard work, and the determination to keep learning and contributing to our mathematical community.
Those are values I will continue to carry with me wherever my career takes me.
I would like to thank all the speakers, participants, organizers, and everyone who work behind the scenes to make Latin America 2026 such an success. Your dedication and enthusiasm are what make uh this community so special.
If there is one message I would like to leave with our students and young researchers is this. Do not be afraid to dream beyond your borders.
Your path may not be easy, but with perseverance, integrity, and hard work, opportunities will come.
I am standing here today as proof that they can. Finally, I encourage everyone, especially our students and young researchers, to stay connected with Eigman, participate, collaborate, ask questions, and support one another.
Together we are building not only a stronger mathematical community but also a community where people can grow, belong and inspire one another. Thank you all for making this journey so meaningful.
Um thank you very much everyone.
>> Thank you very much Andrea uh for your very inspiring words.
Then I would like to invite uh Topo Fernando Fernandez Espinosa, our current scientific director who actually then is the director of Latin America in Colombia has a research group there uh that actually is affiliated to the minister of science and technology of Colombia. So please Pedro.
>> Thank you Dan. Thank you everybody. Uh can you hear me properly?
Yes.
>> Okay. I I I just want to start by saying that I am very happy to be here. I am very very happy to be part of this very wonderful team of of friends. I think and uh I also want to say that words can express how happy I am and how honored I am uh to be part of of this community.
And also I think words cannot express how happy and honored we are to have organized this meeting and have this uh wonderful week of mathematician of mathematics sorry and uh share with this very beautiful community not only in Latin America uh also around the world.
Uh I would like to also thanks again uh to our speaker for uh their time for their this position with us. Also I would like to say thank you for all the for the participants of our conference. Without them uh I think nothing has sense and you know now this is the end of one of the stage of this Sigman 2026 conference. H this uh the today is the the mark or marks the conclusion of of this uh this part of IGMA but uh this is not the the end of our conference in August. I would like to say that we have an second stage of IGMAN 2026 conference uh will will be in person and which will take place in Dutama Bojaka Colombia which is now my place and I would like to invite all of you to participate in person in that uh moment of our conference we will have a plinary talk some courses and also some contributing being uh contribute talks mainly by students undergraduate master and doctoral student. So if you want to join us you are uh really welcome. If you want uh h give a talk just please feel free to contact us and we are very happy to receive hearing in this beautiful town in Bojaka and uh I think this that's it. I am very happy today and uh I also as uh Karina and also as Andrea mentioned I'm very happy to be part of the be part of this beautiful team. I consider all of you my friends. So I I am I am now very very happy to be here.
So thank you very much and see you in August here in Bojaka. Um thank you >> thank you very much Pedro for um the invitation to follow in the next in person program of the of our conference in in Vitama in Bojaka. So professor to if you want to visit um Colombia you are welcome to to visit Bojaka to visit Bojaka >> okay >> for the inerson programs. Thank you very much. [laughter] I I would like to but but we will see there is some distances etc. So it's not the closest place for me and >> otherwise you have second opportunity in NSAC 2029.
>> Yeah that's true that's as well but thank you very much for the kind invitation.
>> Thank you very much. So with this then we finish our scientific program of the virtual conference and thank you very much to the participants that uh have joining have have joined the sessions.
So thank you very much with this then we finish our conference for this week.
So see you in the inerson program. Thank you very much.
>> Bye bye bye. Bye bye. Byebye. Thank you.
Many thank you.
>> Professor Cardona write in the chat for the the meeting in August. The name of it because I could not understand you. You talk Spanish accent.
>> [laughter] >> Do you understand? Yes, it is.
>> Yes.
>> Yes. We are going to have a program you Yeah. What is the write in the in the chat because I cannot understand when you talking in Spanish then don't I don't know.
Okay, thank you. [laughter] >> Okay, >> thank you. Maybe we can finish streaming maybe.
Related Videos
Definition:Bounded variation and if f is monotonic on [a,b] then f is Bounded variation on [a,b]
wingsofmathematicsbytanush2507
4K views•2019-09-05
Prof Chris Holmes | Bayesian fitting and evaluation of complex models arising in...
uclfacultyofpopulationheal9290
564 views•2019-07-03
Patrick Landreman: A Crash Course in Applied Linear Algebra | PyData New York 2019
PyDataTV
9K views•2019-11-30
Approximating the Standard Deviation from Data of a Histogram
donnasmith8529
15K views•2019-09-26
HSC Maths Standard 2 | "At Least One" Probability Rule
ATARNotesHSC
697 views•2019-05-20
Spectral Sequences Live! 17: The Grothendieck spectral sequence
k-theory8604
395 views•2025-11-10
Structural Equation Modeling for Beginners
QuantFish
1K views•2025-09-30
Exploring Practical Applications of Linear and NonLinear Models In Business Research Dr.Jeelan Basha
MallikarjunaDKaggal
258 views•2025-05-26
Trending
we're almost finished the house (ep.125)
JennaPhipps
347K views•2026-07-22
We Finally Know Where Saturn’s Rings Came From
astrumspace
79K views•2026-07-22
BIG BET: Cathie Wood goes ALL IN on Elon Musk
FoxBusiness
89K views•2026-07-22
MIC DROP: Smithsonian Director Called Out For Woke Propaganda
TheAmalaEkpunobi
37K views•2026-07-23