Research on the theory of elastic surfaces - Laura Monk

Ajouté : 2026-05-11

[00:00:07][music] >> Thank you very much and thank you for the invitation. It's a great pleasure to be talking about Sophie Germain's research today on this special day. Uh so I'm going to try and present her works that gave her this prize at the academy and explain the figures that we just saw uh on the vibrating plates.

[00:00:35]So uh as we discussed uh there's this uh German physicist and musician called Ernst Chladni who uh found these figures. So he's not the necessarily the first one, but he's the one who really investigated them that you can obtain. So you can actually obtain very intricate figures uh by using the bow in different places uh on the on a plate.

[00:01:01]So here are some examples of such figures. So he was actually very talented performer. So he spent two years in Paris and during this occasion he went to a lot of different places and he was very outspoken person and a pleasure to listen to and he would demonstrate all of these very intricate patterns. And at this point no one had any idea as to how to model them.

[00:01:24]Um one of the high peaks of his uh visit to Paris was two different occasions. He did go to the French Academy of Sciences and show these figures to a lot of prominent scientists.

[00:01:36]Uh and he was invited by Napoleon to show them in the Tuileries Palace.

[00:01:42]Upon this occasion Napoleon was absolutely fascinated and he offered him a grant to translate his book into French so that his research could be more well known uh to the French uh speaker.

[00:01:56]And he also offered to set up pri- uh money to he offered also offered money to set a prize uh to give a mathematical theory of the vibration of an elastic surface and to compare the theory to experimental results.

[00:02:13]Following that uh the scientists at the academy met and actually tried a little bit to solve the problem themselves. They didn't just straight on open the prize. They were like, "Well, is this actually an interesting problem? Can we actually solve it?" And they decided it was actually very difficult. And they opened the prize. So this uh Grand Prix of the French Academy of Science was a regular occurrence. There was every few There were very few every few years. There was very set rules. So in particular the prize was always 3,000 uh francs.

[00:02:48]And um so the brief is pretty much exactly what Napoleon suggested as a brief.

[00:02:57]And on the first occasion uh all the uh it was considered that there was no satisfactory solution to this problem. That is not very rare. I looked at all the um list of problems. It did happen a few times.

[00:03:13]Uh and in that case it would just be reopened until we find a satisfactory solution.

[00:03:19]So it got extended once.

[00:03:21]Uh Sophie Germain uh won a honorable mention, but her work was not deemed sufficient to give the full prize. But then two years later she was offered the full prize.

[00:03:32]And the question was closed.

[00:03:34]Uh upon all these occasions her entry was the only entry. She's the only person who thought, "I'm going to tackle this problem and solve it." Uh which shows how intimidating the problem was at this point.

[00:03:47]So um I want to walk you through uh why this problem was difficult and the novelty of her solution and the way people were thinking at the time and the ideas uh that were important to see here.

[00:04:05]And in order to do that I need to start by talking about the case of one dimension. So here a plate is an object of two dimensions.

[00:04:13]Uh instead of talking about the plate for now I'm going to talk about the beam. So that's really the object people were interested in.

[00:04:20]Uh so it's what we call beam theory and it's really as the word as the name suggests. I have a beam and I want to study it. In particular I wonder how can it vibrate? And it's commonly considered that the first appearance in history of this problem is in Da Vinci's notes where he wonders how a beam can bend.

[00:04:40]So that is a very old.

[00:04:43]Um it's Nowadays we use still uh answers to this problem which arose in the uh 18th century.

[00:04:59]So Bernoulli thought about this problem and what he thought was that so in physics you want to understand what forces uh an object uh uh uh endures. And so here if I have a curve, here I have a gray curve, you can think of it as a beam.

[00:05:20]Um at every point I have a circle which is kind of the best circle that approximates the curve. That is what we call the osculating circle.

[00:05:31]>> [snorts] >> And its radius is going to be um a measurement of how flat the curve is is at this point. So on this curve I have some points which are not very flat. So the circles are quite small.

[00:05:45]And some points which are quite flat and where the circles are very big.

[00:05:50]So the inverse of this radius, one over the radius, is going to measure how flat or not flat the curve is at this point.

[00:05:59]So in particular the the the K, the curvature, is small when the curve is flat and quite big when it's not very flat. And what Bernoulli thought was that if I want to uh if I have a beam and I want to deform it, the amount of force I need to put in is the curvature at this point.

[00:06:20]And that was a very interesting hypothesis. And building on this uh Euler found an equation that describes how a a beam vibrates.

[00:06:32]So I won't bore you too much with mathematical notations uh cuz I don't want to assume you know about differential equations and stuff like that. Uh but this is the equation that satisfies a beam that's vibrating. And so in our mathematical language here Z is the height of a point on the beam. I'm going to give you pictures in just a second. Z is the height. And this formula relates uh the evolution in time of the height of the point to the way the uh shape varies in space. So these two quantities D squared Z over D T squared and D fourth Z over D X squared are respectively how the shape varies in time and in space. And this equation tells you that these two things communicate. And there's this number here which depends on the physical properties of the beam like its elasticity, its mass and things like that.

[00:07:29]So actually you can solve these equations quite easily in simple cases.

[00:07:34]And there's special solutions which I'm going to show you in just a minute which can be expressed in terms of a product of a sine wave in space and a sine wave in time.

[00:07:46]What is very striking and we'll see in a minute is that because of this equation the the frequencies in space and time are related by a formula which depends on the physical property of the beam. So let's see that.

[00:08:01]Hopefully that works.

[00:08:04]Uh oops.

[00:08:07]Okey-dokey.

[00:08:09]So this is a free beam. As a crazy mathematician I'm going to imagine an infinite beam that has no ends because this is a very natural thing to do. We'll put end points to it very soon.

[00:08:23]Uh so this shape is the shape that the beam takes. And if so this is a paused beam.

[00:08:31]And now if I want to let time evolve it's it's going to vibrate.

[00:08:38]And so if I look at one specific point in the beam, like my marked point here, it's just going to go up and down.

[00:08:46]And so there's two frequencies here.

[00:08:49]There's the frequencies in the space direction and there's the speed at which the point goes up and down. And these are the two frequencies which are related to each other um by this formula depending on the physical properties of the beam.

[00:09:03]Now I'll stop being a a silly mathematician and I'll actually Oh sorry. So as a silly mathematician I can just pick any frequency I want. So I can put a wave calmer wave which is going to move very slowly.

[00:09:21]Or I can put a very excited wave which is going to jump and jump and down very fast.

[00:09:28]Now uh this is only part of the answer of the problem, this equation. Uh and because here I have so many so many solutions what happens in real life when you look at a beam is that you need to think about boundary conditions. So for instance I could imagine I take a beam which is fixed at both its end points.

[00:09:50]So like I have two buildings. I put a beam that's fixed at the both ends.

[00:09:54]What's going to happen?

[00:09:56]Well, in this situation, this point here uh it's such a jumping so fast I'm not sure I'll catch it.

[00:10:03]Um I like All right, this point here is not going to if I look if I put it here, because the beam has been fixed, it shouldn't be allowed to jump up and down like this.

[00:10:18]So, this is what we call a boundary condition. If you look at the beam fixed at box both its end points, this end point here will not be allowed to move up and down. And if you put this solution this constraint uh that is going to restrict the possible frequencies you can have.

[00:10:33]Uh so, here in this specific situation, because of how I made every every choice, it means that the space frequency will have to be an integer.

[00:10:42]If it's an integer in this specific situation, this point will be fixed to be zero and will not be allowed to move.

[00:10:50]So, now I have new shapes for my solutions.

[00:10:53]They look like this. They're like cute little waves that do stop at the end points.

[00:10:57]And I have a bunch of them. So, the first one would be like this and vibrating very slowly.

[00:11:05]The second one a bit faster.

[00:11:08]Et cetera.

[00:11:10]So, this is actually what you hear when you take a guitar and uh vibrate its string, you hear all these harmonics.

[00:11:17]And the fact that there's these specific frequencies that appear to be integer in my example, but it would be a number related to the physical properties and the length of the beam.

[00:11:25]Uh this is because you have constraints you those boundary conditions and now you only hear these specific sounds or you only see these specific uh vibrations appear. We saw that in the demonstration where in most places you couldn't hear anything, but that sometimes there was a sound coming.

[00:11:45]The sound is what we see in this plot.

[00:11:49]Okay, so this is the situation one dimension already a very interesting problem.

[00:11:54]Um Now, let's uh talk about uh well, finish talking about this case.

[00:12:01]So, if we put specific boundary conditions, we get resonances, which are specific numbers which uh for which there's a specific solution that arises.

[00:12:12]And uh this theory has been developed in very pragmatic matters. Sometimes these days, as mathematicians, we wonder what's the applications. At this point, there was no question of the applications. Uh because actually some solutions of this equation, because of the four here, some solutions of these equations are not just little oscillating waves, but actually exponential.

[00:12:35]And uh this is why sometimes bridges break if you hit the resonance. So, there's been historical cases which were really scaring people at this time where you had a perfectly good-looking bridge and people would walk on it exactly at the right frequency and the brick bridge just collapses. And this is why, because this equation has some exponential solutions which will just if you hit this exact frequency, bam, everything breaks down.

[00:13:02]So, these are very concrete uh questions which are very important. Uh these have been used for constructing the Eiffel Tower and Ferris wheels and many other applications.

[00:13:14]Okay, but now uh so, this is very well understood probably 50 100 years before all of this. Uh why is it so hard to go from one to two dimensions?

[00:13:26]Um actually Napoleon asked this question to uh Chladni during his presentation. And Chladni was very smart physicist and he had done a whole book on the problem.

[00:13:37]It's not like he just was playing the violin. He actually had thought a lot about this problem. And he told Napoleon that one is not yet able to apply a calculation to areas curved in more than one direction.

[00:13:50]So, what were really bothering people at the time is that we understand how a string or beam, a one-dimensional object, is curved, but for two dimension, we don't know what curvature is.

[00:14:04]Some people say, but people put it a bit uh with caution in uh sources I've looked at, that Lagrange would have would have said upon setting the problem that it requires a new kind of analysis and that might have scared people off, because if someone asked for a minute at him thought it was really hard and you would have to have a crazy new idea, that is a little bit scary.

[00:14:26]Especially you had only two years to solve this problem, right? It's not like a a very long time.

[00:14:32]And another big problem is that people and more specifically Laplace at the time had developed a very strong idea that all of these problems should actually be seen in terms of the interactions of molecules. So, Laplace at this moment in time was super big fan of Newton's law of gravity and the idea that different bodies would attract and each each other according to certain laws. And he really really firmly believed that all problems through capillary attraction, these problems, the beam problem, all of these should be explained in terms of these objects are made of molecules and we will describe how they interact and this is how we we will solve the problem.

[00:15:21]This is not a good viewpoint. Everything makes very complicated this way. Uh it's just really hard and these all these effects he was focused on uh they they actually um are so small at these scales that they're completely irrelevant. But at this point, he was very influential and he really strongly believed that. So, people only tackled this problem with this viewpoint of thinking of this plate as a bunch of molecules and how all these molecules interact. And this is very complicated.

[00:15:52]Okay, so now I'm going to talk about why uh the curvature for a surface is different from than from a a line.

[00:16:00]Um so, the thing that was probably a problem for people at the time is that if you have a surface, so here you have my little surface which is shown uh in white and on the mesh.

[00:16:15]And I have a point here.

[00:16:17]I can put a plane a normal to the surface and the intersection of the surface with this plane will be a curve.

[00:16:27]And here I have an osculating circle.

[00:16:30]So, this is all very well, but then when I rotate the plane, the circle will change.

[00:16:35]And that means there's infinitely many different um osculating circles at this given point.

[00:16:43]So, this is a bit more scary.

[00:16:47]Uh it's not completely not understood though at the time, because Euler proved in 1760 that there's two special planes um which for which the curvature is maximum and minimum.

[00:17:02]So, I plotted them here. So, you can see here the blue one is minimum because the circle is very big. So, it's quite flat.

[00:17:10]So, the curvature is small.

[00:17:12]Uh and the red one is where the curvature is maximum. It's the place where the surface is most bended.

[00:17:18]So, and these two planes are always perpendicular to each other. So, these are called the principal curvatures. So, now you do have mostly two numbers to describe the curvature, which is really the case, because he also provided a formula.

[00:17:33]Now, if I know K1 and K2, the two principal curvatures, and I take another plane and I give an angle between those initial planes and my new plane, there's a formula giving the curvature for this plane to the principal curvatures. So, all the curvatures actually only depend on these two numbers K1 and K2.

[00:17:54]So, this is actually quite a lot of information.

[00:17:57]And this is where things were when Sophie Germain uh started thinking about this question.

[00:18:05]So, what Sophie Germain did was look at Euler's work for the beam problem. And she I I think um we saw in the first talk that she actually didn't want to enter the competition initially, uh but actually it came to her that uh there was a very natural adaptation of Bernoulli's instinct of looking at the curvature, which was not to look at K1 and K2, the principal curvatures, but what she calls what is now called the mean curvature, which is the average of the two numbers. So, now this is one number, the mean curvature, and she said she suggest we should just do like Bernoulli and Euler, but with the mean curvature instead of the curvature.

[00:18:46]This is the quantity that is relevant here. If I have a plate and I want to deform it, how hard I have to push and pull is exactly the mean curvature is her assumption. And she was actually so shocked that no one thought about it and so satisfied with her idea that she actually felt compelled to submit a solution, because she was like, this is just so great and uh she had very little time to write down her solution. It was very rushed, but she was so impressed by it, she really wanted to share it.

[00:19:21]So, this mean curvature is actually always sorry.

[00:19:25]Always obtained for when you take the two planes to now be uh 45° shifted from the principal curvatures. So, I have my two planes of principal curvature. I turn by 45°. At this point, the two circles will be exactly the same size and the size is related to the uh mean curvature.

[00:19:45]It's actually true for any two planes.

[00:19:46]So, I take any two planes, uh I take the average of both the curvatures, it's going to give the mean curvature. So, all these things she proved uh in later work.

[00:19:57]So, this is a very beautiful generalization of the 1D case.

[00:20:02]And uh she uh details a lot of observations on this new object that she introduced in her asset uh which was published posthumously.

[00:20:12]Uh and notably, she sell says, "Geometers should decide whether it should be adopted." And I can tell you that this is something you teach in uh at at third or fourth year uh curriculum in mathematics. It's or even before, probably. It's just a very used notion to this day that remains uh a very interesting object.

[00:20:35]Okay. So, now how do we go from here?

[00:20:38]Here, this is where the problem happens because Sophie Germain uh did not have the luxury of uh going to formal educations, and uh she was working quite alone. And these are very difficult problems, and unfortunately, she made a mistake when she very hurriedly tried to send her her answer. And this was pointed out uh by Lagrange, uh who So, remember, Lagrange actually thought about this problem. He was amongst the people who decided to set the problem, and they thought about it, and they did not solve it. But now that he had this good idea of looking at the curvature, he said, "Well, actually, if this is what how it works, this should be the equation." So, he wrote down what the the equation should be because this is the kind of things he was very used to doing. This is his field of expertise.

[00:21:28]So, once he had this beautiful idea of using the mean curvature, he was directly able to find the equation and to prove that Sophie Germain's equation was wrong. It did not come from her hypothesis.

[00:21:40]Um and then he says to her, "Uh you just need to use my one page 148 of this very complicated book, and that that's will work." So, this is a not a very helpful So, it's a very big very obscure textbook of very modern methods that only Lagrange understood fully. This is a bit steep to just be like, "Okay, just use my book."

[00:22:05]Uh and unfortunately, um So, this is the correct equation. To this day, this is how we model a beam uh a a plate that vibrates, and it's known as the as the biharmonic plate equation.

[00:22:20]And unfortunately, Sophie Germain never managed to go from her hypothesis to this equation.

[00:22:26]And uh there was no traces of Lagrange finding the equation except from this letter by Legendre, uh and it was never written anywhere.

[00:22:37]So, what she did instead was to take for granted going from her brilliant hypothesis to this great equation proven by by Lagrange, and then she provided loads of experimental evidence. She had people make plat- plates made of glass, uh and she was like out of from her own money uh vibrating it and measuring all the different resonances that she would see.

[00:23:06]And uh So, you can see in one of her manuscripts her listing all the different sounds that she can make from the plate.

[00:23:17]Um So, >> [clears throat] >> Okay. Uh modern plate theory, so the way we now think of this problem is a bit more general than this equation proven by Sophie Germain and Lagrange. Uh it was settled between 1850 and 1880s uh by uh Kirchhoff and Love.

[00:23:41]And I found this very nice to see that in the very big article that Kirchhoff made to start this whole question, he straight on says, "This problem has been initially solved by Sophie Germain." And then he goes on a very thorough explanation of what her contributions and works are. So, his work actually proves that her method only works in a very uh specific limit and needs to be adapted to cover more general cases. So, now we understand a lot more. So, indeed, he kind of faults her as- assumption and shows that it needs to be adapted, but he really attributes those results to her and discusses them at length.

[00:24:23]Okay.

[00:24:24]Um So, this equation is enough to now explain those figures that we see.

[00:24:32]So, similarly to uh the one-dimensional picture I showed you before, you have exactly the same thing happening in two dimensions.

[00:24:44]Uh so, this is a square plate. Uh unfortunately, out of lack of time, I did not put a fixed point in the middle, so we will not see exactly the same patterns. But here, it's a plate which has free edges, so the boundary condition here is that the edges uh vibrate up and down, but freely.

[00:25:02]And in this situation, the same thing hap- happens as in one dimension, which is that you have these oscillating patterns. Uh so, there's a basic shape of the plate, and each individual point goes up and down.

[00:25:18]Um and because of the boundary conditions, because we look at the plate and not the infinite piece of the plane, you have very specific values, resonances, which appear, and they give different shapes. So, here I can change which mode I look at, and depending on the value, the shape of this function that appears will change.

[00:25:41]And I'll see diff- I'll hear different sounds and see different shapes appear.

[00:25:48]Now, what about the salt? I did not talk about the salt yet. Um if I put salt on this plate, so this obviously it's a very exaggerated. We didn't see the plate goes This is the oscillations are tiny.

[00:26:01]But you can think of putting salt on this.

[00:26:05]Um so, remember each individual point goes up and down.

[00:26:09]Um if you have a piece of salt on something that goes up and down, it's going to be kicked off and fly away.

[00:26:15]Except if you look at the point where the thing does not move. So, if you think about this example, it's a bit easier to see. If I put my point here, and I'm salt, I'm going to fly away.

[00:26:31]Whereas, if I put my point here at a node, uh actually, I don't move really. So, I will not fly away. And if I land here, I'll always stay here.

[00:26:41]So, in one dimension, it's a bit silly to put salt on a beam, but on a plate, this is what's going to happen. You have nodal Oops.

[00:26:51]lines, uh which are the places where Sorry.

[00:26:57]I lost my This is a good eyesight exercise for me.

[00:27:02]Um so, you have nodal lines, which are the places which do not move, and this is where the salt will remain because any other place is vibrating up and down.

[00:27:13]So, in her works, uh Sophie actually represented a lot of these uh figures that she can obtain by vibrating a plate.

[00:27:23]She This is actually the one-dimensional image as I was showing it to you, and this is the two-dimensional situation.

[00:27:33]Okay. So, now there's a bit of a hairy subject, which is the subject of Poisson. Actually, uh wondered a lot about whether to talk about it because it actually takes a lot of place in any presentation of this thing, but I actually think it's a bit sad to give space to men who uh pollute and do not uh contribute in a meaningful way. Uh so, I'll go on this very briefly.

[00:27:55]But basically, the situation is that people think that Laplace, who was part of the academy, wanted his student, who he felt a very uh uh fatherly relationship to, uh to get into the academy. He wanted to advance his career, and he was looking for a problem to put that he could solve. Uh and so, the problem was a bit like set for Poisson to solve. Except he did not manage to solve it within the allocated time.

[00:28:22]To be admitted to the academy, someone has to die. So, it's a bit of a waiting game. Uh and then someone did. So, Poisson did get accepted to the academy. Uh and at the second iteration, he was a judge. Uh so, he could not participate.

[00:28:35]He was a judge, so he could read Sophie Germain's work and undermine it. And he was a judge, so he could uh show up and come to the academy and say, "Oh, I want to read my new works in the middle of the competition, which solved the problem that is open to competition for external people using her solution." So, that is not great practices. Somehow, no one got up and said like, "This is ridiculous.

[00:29:00]Don't do this."

[00:29:01]Um so, he, in the middle of the second round of competition, read a memoir which solves this problem using the molecular mentality. This is really hard. It's a horrible computation. No one wants to read that. It's So happens that somehow, with a lot of very horrible, complicated steps, through a big mystery, he arrives at the equation proven by um Lagrange and Sophie Germain.

[00:29:30]Uh this is a bit miraculous. No idea why.

[00:29:33]Um but one sure thing is that he had access to her memoir, her research, and did not um attribute anything to her.

[00:29:42]Just independently found the same equation.

[00:29:46]Um That was not very for her because he was actually a very well-respected mathematician. She actually respected her his work very much. So, it's a bit uh upsetting when you're working on something and someone else who you respect says you're wrong, not even you're wrong, but this is actually what should be done and ignores your work and uh it's not very pleasant, I think.

[00:30:09]Uh and she actually talks a lot about this in her memoir uh Recherches sur la théorie des surfaces élastiques, of which the uh the talk is named.

[00:30:18]Um she does not name him, but talks about him as a very well-respected very well-known name.

[00:30:25]Um so, none of the contributions of Poisson to this specific question stayed in history. They were all disproven quite radically.

[00:30:34]Uh it was really not the good problem for him. And but it it is actually someone who contributed amazing things to other fields of mathematics. Uh he's a very well-known name. Just this wasn't the right uh thing for him.

[00:30:49]And there was a decade-long controversy between Germain, Navier, and Poisson as to who did what and what was the right thing. It was quite ugly, actually.

[00:30:59]Okay, so I want to conclude this talk by a few little things that came to my mind while I was preparing it.

[00:31:05]Uh and the first one that was quite striking to me is the question of name recognition.

[00:31:10]So, throughout this research, I did not encounter even one name I did not already know.

[00:31:17]Uh every single name was things that I have already written by the someone theorem or using the someone theory. Um except Sophie Germain.

[00:31:28]It is very sad, but when I was asked to give this talk, I was very puzzled as I did not know that Sophie Germain had contributed anything to my field.

[00:31:39]Um this is quite alarming because I have given several presentations including the Chladni figures. Like, I have shown videos of Chladni figures and somehow managed to do that without knowing Sophie Germain uh contributed to solving this problem.

[00:31:54]Um this cannot be a coincidence if I know every name except the one woman. But uh um um So, uh another very concrete example of that is that there is actually two notions of curvature.

[00:32:13]So, I told you there's two numbers, K1 and K2, the principal curvatures, and you can find one number, the average of K1 and K2, the mean curvature, and that's an interesting quantity. There's another one which is the product. That's called the Gaussian curvature because it was invented by Gauss at the same exact time. So, the question is why is the uh mean curvature called the mean curvature and the other curvature the Gaussian curvature?

[00:32:40]Uh there's an interesting article suggesting to call the other curvature the Germain curvature. I think I'll do that from now on.

[00:32:48]Um Actually, Sophie Germain was very very happy to see that her dear friend Gauss was speaking about similar things. She wrote in a letter about this to him saying that I cannot tell you how astonished and at the same time how satisfied I was in learning that a renowned mathematician almost simultaneously had the idea of an analogy that seems to me so rational that I never understood how no one had thought of it sooner.

[00:33:17]So, this is really speaks to her enthusiasm.

[00:33:21]She was just very happy that they found the same thing or similar things at similar times.

[00:33:27]And it's very interesting because those two curvatures, the Gaussian curvature and the Germain curvature, really measured completely different things. So, you you look at the formula, you're like, well, there's K1 + K2 over 2 K1 K2.

[00:33:41]Well, it's two things you can do with K1 and K2.

[00:33:45]What happens is that sorry, I forgot to bring a piece of paper, so I have a piece of article by Sophie Germain as the only piece of paper I have.

[00:33:53]Um So, the difference between those two notions of curvature is that they don't actually talk about the same thing.

[00:34:01]The Gaussian curvature only talks about what happens on the surface itself.

[00:34:09]Um so, you have to think that you live on the surface and you just live on the surface and you're completely unaware of the fact that the surface might have been put in R3. So, this piece of paper has been put in R3, where we are now.

[00:34:23]Um one notion of curvature doesn't care about this, the Gaussian curvature does not care, whilst the Germain curvature is actually talking exactly about this.

[00:34:34]So, this piece of paper has Gaussian curvature zero. That's it. I don't need to tell you where it is or how I put it.

[00:34:42]But I can actually deform it and I can make it into a cylinder if I want.

[00:34:46]And this also has Gaussian curvature zero.

[00:34:49]You can see that because if you put a point here, you have the one direction where the piece of paper is flat and one where it's curved.

[00:34:58]On the direction where it's flat, the curvature is zero, so the product will be zero. So, even if I make this shape, a little ant living on the surface will not know the difference, but uh so the curvature will still be zero. However, the Germain curvature here is different because now I have a direction where the curvature is zero and one way the surface is curved. So, the Germain curvature is actually half of the curvature in this direction.

[00:35:23]And this is very important because in the problem of the plates, the problem of the plates is all about how my plate is deformed. So, if I looked at the Gaussian curvature, nothing is happening. It's really the Germain curvature which was relevant to this problem.

[00:35:41]All right.

[00:35:43]So, um another good example of the question of name attribution is that the biharmonic wave plate equation could have been called the Germain-Lagrange equation as is done in uh a biography of Sophie Sophie Germain.

[00:35:59]Um so, I think there was a few opportunities to easily fix this problem. Um on the Wikipedia page of curvature, uh Sophie Germain is not even attributed to the mean curvature, so I'll change that after. I haven't come around to it.

[00:36:14]She's attributed on the mean curvature page, but not on the general curvature page. So, I'll I'll add that when I come back home.

[00:36:21]Um okay.

[00:36:23]So, a little uh last-minute thought also, I think it's important to think about the importance of mistakes in mathematics. If we don't feel comfortable to make mistakes, we actually don't do much. See, she was the only one to contribute to this competition.

[00:36:39]That is very brave. Uh no one else dared to participate and probably because they were afraid they would say something silly or make mistakes. Uh in this story, many mistakes were made. The silly molecular thing by Laplace was nonsense and no one is pretending otherwise. Everything Poisson contributed was really bad, but she's the only one who we keep insisting that she made mistakes, she derived the wrong equation, this thing wasn't very formal.

[00:37:04]Actually, I think it was very brave to show these things even though she didn't have the tools to prove them fully.

[00:37:10]So, I really liked her writing, so I read quite a bit of this research on the theory of elastic surfaces that she wrote.

[00:37:20]And um in the introduction, she really spoke a lot about these feelings she had about the whole Poisson thing.

[00:37:28]And uh it is interesting the maturity that comes and the different way of thinking.

[00:37:36]Uh so, basically, she says that there were two different different hypotheses, so hers and Poisson's. Uh and that one has a famous name attached to it and therefore, it's normal to distrust hers.

[00:37:49]And she has made every effort to renounce it. She wanted to think Poisson was right. And now she's going to explain in this book how she needs external judgment because she keeps thinking about this, but she just thinks it's just a good idea and a good way to go about it. So, please uh please tell me where I'm wrong, basically.

[00:38:11]Um she apologizes for being a bit pushy for taking assertive tone, but she just says she doesn't want to be doubting in every sentence, but please please do uh um she seeks from him a critical examination. And she will no doubt be forgiven for not concealing any of the advantages that I believe I recognize in my hypothesis. So, I'm I'm sorry, but this seems looks like a great hypothesis. Please tell me where I'm wrong. And I think we lack a lot of this in science and in mathematics. We're very scared of being wrong. And this is not how good science is made. And I I found it very refreshing to read someone talk about these conflict that you can have. Uh the idea of showing to each other our works and asking for feedback is really the way things are supposed to work.

[00:39:03]Okay, so just to wrap things up, 250 years later, uh spectral geometry, so the the uh study of these kind of figures that appear on these plates is a very big field of mathematics. I'm part of it. Um and it's related to many many modern applications. So, here for instance, this is what what's called a cardioid, and these are exactly the same as those uh salt figures that would appear for very high energy of this cardioid, and people look at this to think about quantum chaos and electrons.

[00:39:38]Um these are the um resonances, so those special lambdas for some important hyperbolic surfaces that can arise in number theory, notably.

[00:39:51]Uh and they form very intricate uh crazy-looking patterns, uh which are very interesting. Thank you for your attention.

[00:39:59]>> [applause]