Understanding the Oracle - Govind Menon

Hinzugefügt: 2026-05-14

[00:00:15]It's great pleasure to welcome you to tonight's friends talk and to introduce our speaker professor Govin Mannon Eric Alan Tuckfellow in the in school of mathematics at the institute.

[00:00:29]Govin is a professor of applied mathematics at Brown University. His path into mathematics has been quite unique and broad. He trained first as um in mechanical engineering at IIT and then in theoretical and applied mechanics at Cornell and later received his PhD in applied mathematics from Brown.

[00:00:53]Since then his work has moved across mathematical physics, disorder system, algorithm turbulence and most recently the mathematical foundation of the artificial intelligence.

[00:01:07]The breath that breath feels especially fitting for tonight's topic understanding the oracle.

[00:01:16]The title reminds us that the relationship between the mathematical thought and machine intelligence reached back to the birth of the computer in the 1940s. A history with deep ties with I is yet recent advance in AI have brought this longstanding question into a focus.

[00:01:37]What is what does it mean to have a have a mathematical ideas? What is intuition?

[00:01:44]What is creativity?

[00:01:47]And what does it truly mean to understand?

[00:01:50]These are not merely a technical questions. They are human, philosophical and mathematical all at once. as Golvin is particularly well suited to guide us through them. His current work explores how ideas from geometry, learning, optimization, and mathematical culture might help us to think more clearly about artificial intelligence not just as a tool but as something that challenge how we understand our own thinking.

[00:02:23]One of the things I feel especially compelling about Goloving's work is that he does not approach mathematics as a closed form of knowledge. He's attentive to how idea is formed, shaped and transformed. at I is he has also been involved in the mathematical forklaw project an effort to examine the emergence and evolution of mathematics idea and to consider how the field itself might change in the years to come. So tonight as we think about Oracle algorithm and intelligence we are also invited to think about ourselves how we recognize a good idea. how we trust our mathematical insights and how human understanding may differ from or perhaps resemble the outcome from the machines. So please join me in warm welcoming Govin. Thank you.

[00:03:27]So uh thanks very much uh Giin. So, uh, there's, uh, I'm going to tell you one of my favorite jokes, which is that in math education, people never want what they actually paid for. So, anyway, I thought I'd try to give you, uh, a sense of, uh, some of the interesting things happening, uh, in my life. uh and I would like to kind of begin by uh rooting my talk kind of very uh firmly uh in the traditions um of the school of mathematics uh at the IS and um uh my uh kind of joy this year has been to kind of work with Akay Vancesh and a group of uh other uh scholars um in kind of uh rethinking the humanistic uh foundations of mathematics thinking about the relationship uh between humanism and science.

[00:04:16]uh and our inspiration in some ways has been to revitalize the spirit of Herman Vale uh one of the founding members of the school of mathematics and uh I thought I would begin with this uh quotation that uh veil has in his book uh that's where the title of the talk comes from um I'm not going to read it out in the original Greek but as you can see uh veil's translation uh is about the oracle uh that neither reveals nor hides uh but gives tokens so when uh veil mentioned is uh by tokens he meant symbolic systems uh not the tokens that we have in LLMs today but uh it felt like the right pun to start the talk so uh the tokens that I'm talking about are things like this uh these are symbols uh these symbols capture meaning uh for interpreters um so Isidor Rabi once said that uh scientists especially in the nuclear period are high priests of a particular age. Uh so these symbols are uh symbols that connect us uh to nature and the nature in this case uh is uh that these symbols are uh these tokens in in Bale's language are uh something called nonlinear partial differential equations and they are the mathematical language uh for field theories in physics. So the first of these equations uh over here oh I guess that isn't quite shining uh is is is uh the equations are a collection of equations introduced by Oiler uh Swiss mathematician in the 1700s and what he was doing was trying to kind of apply Newton's law to the world around us and he was thinking about fluids. Uh the other equation down below uh is an equation which was perhaps not written in exactly this form but it's an equation that I've been thinking about for about 10 years now.

[00:06:07]Uh and that's uh related to the work of John Nash uh in the book his work in the 1950s. So I think uh you know Nash is someone who's very closely associated with the institute and many of you are probably familiar with his work in game theory. uh his it's he always had this uh tension um and uh he always felt like he needed to be a real mathematician. He never saw his work in game theory as real mathematics. So I'll try to explain to you uh what his real mathematics uh was about. Um and uh it's it's a kind of fascinating story.

[00:06:43]So in fact uh I want to show you you know I want to begin with the tokens to kind of begin to show you how these tokens uh carry meaning and how they related very closely uh both to the work of kind of members from the founding period of the institute and uh continue to be of importance today. And so the first equation over there uh uh is related to a fundamental phenomena uh of turbulence in fluids. It is in some ways the fundamental unsolved problem in classical physics and it moved into mathematics or kind of came to prominence in mathematics in the 1940s uh in relation to various projects uh related to or various problems uh related to the Manhattan project and so it was really forman's work so there's a kind of 1948 or 1949 uh paper by forman called recent theories and turbulence uh and the paper is still absolutely marvelous What's down below uh is uh an image of one of the solutions uh conceived by Nash. Uh that is actually one of the very few images we have uh of the solutions to that equation. Uh because these solutions are extremely counterintuitive.

[00:07:54]Uh but the problem itself again goes back to work of uh faculty in the school of mathematics. uh these are Herman Vale and Hasslow Whitney and they were thinking about the foundations of geometry relating Einstein's work in physics uh back to mathematics in the period from 1920 to 1940 and on the right hand side uh is an arrow uh so that's uh I want to kind of uh one of the kind of big breakthrough results uh in this field was by Camilo Dellis a current faculty member in the school of mathematics and I have DLES plus because you know it was Dellis plus co-authors.

[00:08:31]Uh but anyone who knows Camilo knows it's kind of prod you know he's got this kind of prodigious energy uh and so uh I put in a plus because it's just become this industry and so uh roughly speaking the kind of spirit of this talk is to kind of show you how systems of symbols or how different bodies of mathematics are related have a certain unity uh and how this unity now transcends these areas of mathematics and moves into other let's say human worlds and uh the the kind of astonishing aspect of this work of Camo uh and his team was to kind of recognize a completely unexpected connection between these two different field equations. So the first equation uh the first uh sort of picture up there is a physical experiment. Uh the second picture is of a numerical experiment and what's going on in this numerical experiment is I use the word ideal crumpling of a sphere. And so in kind of popular language, what this means is that as you can see, that's a sphere that looks all wrinkled. Um, but if we judged a sphere by, you know, just imagining something like a great circle on the earth and, you know, you've got an ant crawling uh on the sphere, uh, the point really of that crumpling is that the ant does not know that it's being crumpled. So there are different notions of measuring length and you know and uh the issue over here is we could imagine a two-dimensional ant measuring uh length on that on that sphere or we could imagine a three-dimensional grad student measuring length between two points on the outside and these measurements would be exactly the same.

[00:10:08]And so what Nash's theorem says is that you could take the entire earth and you could crush it down put in your pocket and the ant would not know the difference. So it's a very very counterintuitive result. Uh to date we don't have any numerical computations of this result and that's actually part of the problem and that's the the way I began to think about this problem. So uh my first stent at the institute was about 10 years ago and uh as you can see there's there's a little bridge over here and so what Camo did was to take Nash's methods and apply it to construct kind of completely counterintuitive solutions to the equations of fluid mechanics. I had been working on statistical theories of turbulence and uh I wanted to walk across this bridge uh the other way and the question that I began to ask is you know you have to have a secret hammer to go at this and my idea was that in turbulence we think about statistics we think about probability theory it's a different field of mathematics and I was like huh nobody's really tried to use probability theory um to to look at this work um and so I decided to take a stab at Nash uh you know I didn't know if he any good. U so I haven't actually solved this problem but it's been quite a ride. So uh what's the issue? So in in some ways the idea of probability uh is is related to you know the joke of uh of the drunken sailor who kind of comes out of the bar and then takes one step this way, one step that way and you know before you know it has has wandered all around. But the mathematical foundations uh of this area uh were uh one set of mathematical foundations coming out of physics uh was in uh one of the kind of breakthrough papers of Einste Einstein in his uh year of mira miracles when he uh he he kind of uh for the first time um uh provided a clean and simple experiment that would settle whether uh atoms existed. So there had been a fierce debate on the existence of atoms uh up until the early 1900s and Einstein by kind of working uh with the theory of random walks uh mathematicians called it Brownian motion uh settled the debate on atoms. So it's not just that he came up with the photoelectric effect special theory of relatry he also came up with the proof of atoms you quite a guy. So uh so somehow the study of turbulence uh the study in physics is an intellectual descendant of this problem. Uh the real issue when you try to take these ideas of randomness and move them over uh to the problems of of uh of geometry is that you want the entire sphere to fluctuate together. So that ends up being a kind of super hard problem. Um and even after 10 years uh of rather hard work uh we don't quite have the results that we want. uh though we do have good stories to tell you. So uh one of the interesting things about mathematics is the way you know you can kind of relate it's an abstraction. Uh there are abstractions and there are contextualizations.

[00:13:06]And so here uh I've begun uh with the contextualization of mathematical physics. So it's it's our use of mathematics to describe uh the world around us. But in fact, one of the first uses of probability theory is uh to describe human experience. Um so I I I think we would agree, you know, complate guy. Uh that poetry uh describes the human experience. So I I've lived in New England for about 30 years. Um I'm still not quite sure if I like it. You know, it's a kind of complicated place. And so I thought I would pick for you two subversive poems which have very much uh the character of New England. Uh so on the left uh is uh a poem of uh Emily Dickinson uh from uh sorry it's the Franklin edition not Johnson and Franklin uh but it's a it it kind of captures uh Dickinson uh in her kind of raw element and uh so you see uh on the left hand side clearly a kind of very feminine aspect uh a kind of intuitive uh uh intuitive kind of transcendent uh experience but still kind of very much uh rooted in in in the earth of New England. And on the right hand side uh we have um uh you know one of the early poems uh from from from Robert Leel uh from 1946. Um and I think you know the the poem speaks for itself. So over here both poems are kind of subversive uh in their own way and with lol you see a Boston Brahman kind of you know really really wrestling uh with his conscience with his kind of sense uh of inheritance.

[00:14:50]So um it turns out that uh one of the first kind of um one of the fundamental uh models of probability theory, one of the things that we use uh repeatedly uh is is is the idea of marov chains. And what Marov the Russian mathematician was doing was trying to apply um mathematics to understand poetry. Um so rather than show you what Marov would do, you know how brutal could you be, right? like you take this stuff, it's got so much feeling and you take this feeling and you just take it out and feed it into the algorithm. So that's what I did. So I took these two poems and I mated them, you know. So we made uh I made a kind of bastard child of Emily Dickinson and Robert Lol. And you know this child's name is Ember Lovenen. Um and as you can see it seems to have some feeling. So you should always read the fine print.

[00:15:47]So Ember Loansen uh is uh an alias for my favorite grad student or perhaps my favorite teacher uh Google Gemini.

[00:15:56]So you know uh I I show you this to kind of show you the ease uh with which kind of mathematical ideas one can move with mathematical ideas between different fields. Uh but human beings are are complicated and Emily Dickinson I think was really complicated and in the first version uh of Dickinson's poems uh in 1890 um you know uh Todd and Higginson really tried to pin her down. So Dickinson's poems should be read uh aloud, you know, so she's got pauses. Uh she uh doesn't use uh she's unruly, you know, really this unruly spirit. And then they tried to tame her uh by kind of nailing down pieces of it, right? So here you go. Um with friendliness becomes if anybody's fear, you know, the redness of the rose to the red rose upon the hill doesn't quite work. But anyway, this was the first published version.

[00:16:54]Um, and it turns out that Robert LOL's poerm has another published version. So, I thought I'd do, you know, another experiment. And there you go. Here you have two bastard children, uh, on May 4th, uh, and May 5th, respectively. So, I decided to call the first one Ember Lewinsson and the second one Ramly Dole.

[00:17:11]But you see how easy it is to kind of play uh, with randomness. And at the same time you see uh the need to kind of use randomness because even the written word uh when one is describing human experience is actually uh rather fluid.

[00:17:27]So I want to show you this uh to give you a sense um you know I I recognize that most of you are not professional mathematicians and so I I this is to give you a sense of kind of how could you know how do mathematicians actually think like why is there a relationship uh between humanism uh and and mathemat and mathematics? Why are poetry and mathematics uh related? But over here you what you see is that you know uh both poetry and mathematics capture in in a certain way emotional feelings uh and uh they are kind of efficient codings of the human experience. Uh that's mathematical terminology but I think you know what I mean. And so I kind of want to show you that in both of these pro in both of these uh uh in both poetry and mathematics there's a very strong aesthetic element where you're constantly feeling probing uh to get through a real the real kind of ideas down and uh with this aspect of creativity there's always a kind of imperfection so somehow you're you're you know Dickinson in particular was never quite satisfied with what she did even when she wrote her poems there were kind of many words within many kind variance and so uh what I thought was that you know I would show you that metaphor as a way to kind of show you how one thinks about uh original sources in mathematics. So these are the two papers of Nash uh that I've been I've spent about 10 years studying very very carefully. Uh I've spent most of my time on that paper from 19 1954 uh which is the paper that captures um uh the image that I first showed you.

[00:19:02]uh these papers uh really define um in many ways uh one of Nash's primary mathemat primary legacies uh in mathematics. They are kind of theorems of extraordinary power u because of their versatility.

[00:19:16]Um in some ways uh they are also rather tragically the last real work he did uh shortly after uh this paper uh you know uh mental disease uh took him uh and he never recovered uh you know I mean there was you know there was kind of 30 years there was a Nobel Prize all of that but the the real fertility of his mind uh was in some ways lost and so uh you know for me there's there's a you know I really do see Nash as my as my spiritual teacher uh my guru and so you know for me there's this there's very much a spirit of kind of looking at this and kind of understanding everything uh about his life kind of sensing the meaning uh through through the work and it's a little bit like Dickinson what you do is you take the poem you take pieces off it and you play around with it so here are some pieces so that's one piece from 1954 that's one piece uh from 1956 and that's the other piece that's uh with tough work you really have to put everything And so that's that's when I was born and I needed everything in my life to actually you know uh look at his work and then you kind of replicate and you mix it in in a turbulent mental environment. So you know uh AI came along there were two stints at is there was a pandemic political chaos teenagers so happened you know so 10 years went by and at the end this 10 after the end of these 10 years uh you know there were there was a kind of respectable uh bunch of things that came out. So first of all there is uh real mathematical work so real mathematical ideas in the sense that stuff that's replicable that can be used uh by anyone else. uh in particular it brought me great pleasure to work with Dominic Anowan who's Camilo's uh student uh Camilo Dalis in in in the school of mathematics. So Dominic at this point is the only human being who knows how to kind of go go on up and down the bridge. I know how to go one way on the bridge. Camilo knows how to go the other way down on the bridge but Dominic knows how to go back and forth. So you know it's really uh the right guy. uh I began to spend a lot of time so you know uh during the pandemic I served as the chair of the faculty uh at Brown University uh because I could see that academia was changing and so for me it struck me as a as a time to it was a once in a-lifetime opportunity uh I thought uh to see a university go through a crisis the crisis has not ended so you know many of my other colleagues have still been involved but I saw it as this moment you know it was a pandemic education was going online I I saw it as a moment when let's say a transition from theater to cinema. Uh and so I I spent a lot of time uh thinking about institutions and I just want to kind of come back to this tptic of ideas institution and individuals. I'll come back to that at the end of the talk. But uh one of the most delightful things uh the story of people not just mathematics the ideas but people who do it uh is a project uh that Akshai Bankadesh uh on the faculty of the school of mathematics at the IS and the broader IS community uh looked at. So uh you know I feel really proud uh to represent the work uh of this group. Uh I see Alan here and a couple of other people. So what's the what's the project? So the project in some ways is to reimagine uh the creative process in mathematics and its relationship uh to the world around us and um in some ways you could think about the project as being almost faith-based. So kind of as opposed to kind of uh uh you know putting yourself into the world of the algorithm uh to put yourself into the world of the human to put yourself into kind of the intimate spaces like Dickinson. And so our our the structure of our meeting is that um we meet we met once a week uh you know in both the fall and the spring terms. Uh we discussed original sources so really to see the evolution of mathematics uh over centuries. Uh so you know it it's a and I set the agenda more or less for the fall term. uh but in the spring term uh we brought on board uh sessions organized by uh many of the other people with us and in many in in every single case uh we considered uh context along with mathematical ideas. So mathematical ideas are all about abstraction but we also considered like in in many cases uh extensive amounts of context like stories about people or actually ancillary writings of various forms uh to appreciate the humanity of individuals. So it's really an act of imagination rather than an act of technique. So you know going back to uh the famous uh statements of Einstein. So we began uh with the work uh of forman and m Coloan pits uh really discovering really discussing the birth of neural networks which is the starting point in some ways of the modern theory of computing. Uh we went into the the evolution of ma of geometry in the 1800s which led to the crisis in foundation.

[00:24:12]So uh for you know for those of you who have a kind of mathematical training that that's kind of who's who. We looked carefully uh at the evolution of the character of physical law. Uh you know as I said Einstein settled the question of atomism but uh there were strong opponents uh mark in particular so this mark of the mark number quite a character. Uh so we read you know Lenin hated Mark. So you know that gives you a sense uh of the range of mark. Uh so he was he was excellent at polyomics but he was a sort of brilliant scholar. Uh ended up on the losing side of many debates but he still fought the fight.

[00:24:50]So we felt it was very important to kind of read Mark uh in the original and then we discussed a bunch of papers on philosophy and the practice of mathematics. So if you look at this side of the table it's actually very much uh the mathematics of the sort that I showed you. Mathematics that fits into the world of Herman Vale. uh mathematics that involves uh natural law. So it was not an accident. That's what I understood best. Uh that was the kind of common language uh the common space that Aku and I had. So that that was largely what fall was about. Uh and at the end of the fall semester we got together and we asked like when we started this thing we weren't sure if we would make it past two seminars honestly you know. Uh so but people were still with us and at the end of the semester we were like okay what do you guys want to see? uh do you want to like this all began with this like okay AI is coming it's doomsday what are we going to do um but uh you know we're like okay you know are you are you are you guys willing to go further into the past and so uh there was actually a lot of enthusiasm that so I I I'm going to show you a little bit of Newton uh to give you a feeling for Newton and there was a bunch of other things and we thought a lot about mathematics as a cultural system so over here uh we really tapped into many of the other faculty uh and members at the institute. So Alma Steinart who's a historian of mathematics organized one session uh Karen Olenbeck who is uh you know one of the world's leading mathematicians and a pioneer uh and a role model uh spoke uh to us about kind of uh the battles that she had to fight uh for equality of women and mathematics or at least for progress in women in mathematics and David Nerberg uh the ma you know the director of the institute uh joined us for a session on mathematics and religion uh and I'm going to dig into this uh because it gets it's pretty it's it it's good.

[00:26:42]Okay. So uh the thing that I really learned about Newton so I've been thinking about Newton for about 35 years since I began mechanical engineering and unfortunately I learned that I don't know anything. So uh the fascinating thing to me about uh about reading Newton in the original is to see the mutation of ideas so that uh stuff gets totally turned inside out. So the modern treatment of Newton's laws uh or the modern treatment of Kepler's laws. So Kepler's laws are the laws of planetary motion. It says that the area swept out by the par by the planetary orbit is proportional to time. So the way it goes is as follows. So you use calculus. So you assume Newton's law of gravitation.

[00:27:24]Uh you write out Newton's second law as a differential equation using vector calculus. So this is kind of freshman year stuff uh from college. and then you integrate it up and out comes Kepler's law.

[00:27:36]Uh what Newton does is the complete opposite. So what Newton does is he starts out with Kepler's law does not use calculus uh uses like a sense of limits which goes back to the ancient Greeks and uh he uses conic section. So again methods going back to the ancient Greeks and out pops the law of gravitation.

[00:27:58]Uh it is incredibly hard to read Newton at first and then you begin to feel his cadence and then it's just unbelievable.

[00:28:06]It's really just unbelievable. It's like this he can he can do stuff that you don't think anyone can do. So these are just two images uh but I just want to give you uh the feeling of how to do it.

[00:28:18]So in fact uh it's it's not just so why read original sources? Well, of course you refresh your mind. your your challenge your uh you know you realize uh the grandeur of the great scheme of things but in fact it's a lot more than that it actually takes me back to the problem that I began with in fact one of the ways of of seeing uh Newton's derivation uh is actually very closely related to how I think about AI it's kind of how you think about models and how we think about explanations of data so in a sense what Newton does is to uh what we do today in our textbooks is to assume a theory use certain mechanical methods and to obtain consistency with the data. What Newton does is in some sense strangely much more modern where one starts out with a data. So this is what a machine learning model would do and out pops a theory. So uh one of the kind of striking aspects of Camilo Dalis's work on turbulence was this creation of unphysical solutions. And for a long time I felt that somehow what was required was a theory which would kind of say that this is not unphysical but to relate uh the problem to one of measurement that's a slightly technical question but in some ways uh I felt reinspired and reinvigorated to see in some ways that spirit all the way back in Newton.

[00:29:38]So uh Newton uh is a wonderful writer.

[00:29:42]It's really it's really something to read Newton. And I thought I'd just give you a couple of paragraphs so you kind of get a feeling uh for how Newton writes. So you know it's uh this is from 16 87. So you know there's of course a little bit of the awkwardness in the language. Uh but aside from that it's actually uh rather easy to read. So the first uh this sentence over here is the very first sentence of the preface to the Prancipia.

[00:30:10]And uh the thing that you see in the sentence is that Newton is always looking back to the ancients. So when you think back, you know, for us uh we think about Newton as being way back in the past, but Newton is looking further back to Papus of Alexandria. And one of the things he says is that, you know, he's laying aside substantial forms and occult qualities. So this is a dig at Kepler. So when you read Kepler, which we also did, it's all wrong. like it's just like the guy is crazy. He's got these musical theories. His book is called like you know harmonies of the spheres but somewhere in this kind of junk is a little bit like which is this piece that you pick out which is Newton's laws and Newton's having none of it. So he says I have in this treatise cultivated mathematics so far as it regards philosophy which is what we would call physics today. So you know that's that's a sentence for example from chapter one uh of Newton.

[00:31:07]In in in the next chapter what he does is he lays out the concepts very very carefully and the concept he thinks about with extreme care is time. He's like what does time mean? How do I think about it? And so he goes through this process where he says you know only I must observe that the vulgar conceive these quantities under no other notions but from the relation they bear to sensible objects. So it's a little bit like you know going back and reading like Ben Franklin or something like that but you see that what he's really doing is you think about time. So he says absolute true and mathematical time flows of its own nature and he sort of dismb he dis disambiguates uh how uh one speaks about these concepts in popular language from how we think about it how he thinks about it.

[00:31:54]And finally he's playful. So there are parts this is the part just before he obtains uh the centripal law he you know he's just saying like you know why don't we try this so it's kind of very very engaging and so what's really really fascinating to me about Newton is where this is coming from uh and so there's another book there are several other books that Newton wrote and here's another book uh he wrote in 1733 uh this is a book uh he dies in 1727 it's published 6 years after his life. So there are many other physics books but this one this was the one he really cared about.

[00:32:31]So this is New Newton's work on the prophecies of Daniel. So this is Newton's work. He was a very very uh careful Bible scholar. He was a heretic.

[00:32:40]Uh he did not believe in the trinity. So he could never really be a professor at Trinity College. And he wrote this book after uh the book was published only after his death because the penalty uh for heresy uh was death. So this is a spiritual engine. This is where the stuff comes from. And uh at first sight you can't believe it. You're like wait like rational stuff, irrational stuff like what's going on? Like this is Newton we're talking about right? So he goes to this great extent but say he wants nothing to do with the occult but there it is you know. So he spends all his life doing this. And to kind of give you an applesto apples comparison I thought I would pick up you know pieces from it. So let's see. So the first one is looking back at the ancients. The second one is kind of careful working of definitions and the third is kind of the playfulness.

[00:33:30]So here's the first one. So here he's thinking about time. So this is just like from chapter six of the prophecies.

[00:33:37]So here he goes through a kind of careful description of the history of the Huns. So you know the kings of the Huns were these various people. So it's like brothers of Monzuk. So it's like reading f fantasy literature but it's Newton. You know he's checked every single detail over here.

[00:33:53]And then you begin to see his system. So for Newton, the word of the lord and the word of nature is exactly the same. And so what he does when he think about the prophecies, he says that the language is taken from the analogy between the world natural which he understands from the principia and the empire or kingdom considered as world politics. So he's kind of thinking about the system. So he's going to look at the prophecies of Daniel and he's going to code the prophecies of Daniel and convert it into real events. Right? So uh somewhere a little bit so he says the conflration of the earth or turning a country into a lake of fire for the consumption of a kingdom by war. So he's translating so that phrase so in a prophecy when you read about a configuration of the earth what it means is that a kingdom is going to go to war. The being in a furnace means that for being in slavery under another nation.

[00:34:44]uh the ascending up of the smoke of any burning thing forever and ever for the continuation of a conquered people under the misery of perpetual subjection and slavery. The scorching heat of the sun for vexation was persecutions and troubles inflicted by the king. So he's very very careful and deeply obsessed with his freedom of the spirit. And then as you go down it gets even so now this is actually in the introduction for the book. It says the authority of emperors, kings and princes is human. The authority of council, senates, bishops and pre prespits is human. The authorities of the prophet is divine. Um he does not believe in the idolatory. So he does not believe in the trinity. And for him the world is going to hell uh in the ad in the year 1800 uh 18 uh 800 when shallman becomes the emperor. So for him that's the kind of beginning of the end of the world. And then he goes on and you know uh I'll leave you to read the italics in the last sentence and the italics are Newton's. So these are his words. The man is an absolute zealot and one of Newton's predictions is that the world will end in 2060.

[00:35:51]Okay. So this is based on the prophecies of Daniel. He does a careful mapping using this stuff and that's what he says. So I know that his theory in the prinipia works. I'm not sure if this theory works, but like stuff ain't looking so good right now. So, I'm edging my bets over here. So, you know, I I had lunch with Jihen and I was like, this stuff is in my head. I don't know how to get it out my head, right? Like on Newton, I was like, what's going on?

[00:36:19]And so, Jun tells me like, okay, you know, you have to look at religion. You have to tell us about Ramanujan. Um, so I don't know. I can't look into Ramanjun's mind, but I'm going to give you some stuff. I'm going to tell you how I think. I I really do think about religion a lot and I want to show you to this I'm going to show you like you know here like let's do the kind of raw version of the is like you know the stuff you didn't want to know about crazy mathematicians but let me show you you know how I think uh about about religion and how one thinks like you know how the stuff related in my own head uh and I think that's very important uh because you know the issue in some ways over here is that we we think about uh a sort of distinction between let's say a left brain and a right brain and some ways this creative process seems to emerge from a contradiction uh within ourselves within our own souls. So, um, I want to bring a little bit of Bollywood. Uh, so I think a lot about chance and, uh, that guy on the left, uh, is the emperor of Bollywood, you know. So, I heard there's some movie about Nash and like Russell Crowe got actors, but like that guy is the emperor of Bollywood. Okay, so that's Abatab Bachan uh, and he was my childhood hero and as you can see, he's the great gambler. So, this is like a Bollywood movie from the 1970s. what's on the right uh is uh actually a Persian uh painting and this called the book of dice. So it's a central event uh in the Mahabharata. Okay. So you know that that's the religion in which I grew up.

[00:37:50]So the Mahabharata is uh an Indo-Uropean epic. Uh it's got many narrative layers.

[00:37:56]Uh it's got many intersecting stories and it was it's dated to about 800 uh BCE. Uh so it's about 3,000 years old.

[00:38:04]Uh it's uh about uh 10 times the size uh I think of the Iliad uh and the Odyssey combined. Uh it's got 100,000 verses in 18 parts and basically describes a saga of dynastic succession and the central event uh of of the Mahabharata uh is a frateral war between two cousin uh between two two families. So there's kind of two families of first cousins and they basically slaughter uh one mother. So that's that's the it's all twisted to power in in various ways. So it's it's a story that goes back uh 3,000 years. It's infinitely mutable.

[00:38:40]There are several uh regional variations. So India has like you know 30 official languages and many of these languages u there are versions of the Mahabharat especially in in in the south of India. Uh so there are linguistic variants and there are many folk renderings.

[00:38:57]So the folk renderings that I grew up were that comic books the best way of indoctrinating the young. So these are comic books uh the names are you know so it's amarata which literally translates to immortal stories. So this is stuff that I grew up with and the stuff that I grew up with actually each of these images captures an event a kind of important event within the Mahabharata.

[00:39:23]So that's the Lord Krishna as a child, a kind of lovable rogue. Uh you know, he loves to eat like milk and dairy and stuff like that. U you like a boy from Wisconsin.

[00:39:35]This is Karna uh the kind of you know the the sad hero. He should have been he's actually the firstborn of the Pondas. He should have been the king but through sort of uh he's an illegitimate son. He's actually the son of the the son. uh and so this this picture over here refers to a particular event in the war when his knowledge fails him at uh the time that he needs it the most. So he's actually fighting his own brother and he gets killed. His brother is Arjuna the great warrior and this is Abhimmanyu uh the son of Arjuna who dies uh in a particular event in the war. And this event over here is Drapppathi. So uh this is kind of polyandry is back in the day right like everything goes. So uh Draadi has five husbands uh all of whom are the pandibas. But in this episode of here what happens is in this game of dice her husband uh who is very very truthful but has a weakness for gambling in fact gambles her away uh to his evil cousins and here her evil cousins are actually raping her in public. So it's it's it's got uh a great deal of violence uh the Mahabharata. So in fact the classical written form is not in these comic books. So this is a comic book about Drona. So Dona is the great teacher in the Mahaba tradition.

[00:40:52]And the story over here is actually a very poignant story. It's about the kind of uh the very dedicated teacher who you know like all academics has a salary which is necessary but not sufficient.

[00:41:03]And so the story over here is that his son has never tasted milk. And the children around they kind of you know tease him and mix some kind of rice flour and water and kind of feed it the sun. And the son's very happy that he's drinking milk until he tells his father and the father recognizes in his humiliation that he needs to go out there and make some money. And so he goes uh and you know goes to his childhood's friend's place is humiliated again. So this kind of sagas of revenge but this is how this stuff is actually recorded. So these are pal manuscripts and they go back uh many centuries. Uh this is of course is the the version you know as the society was changing as people were forgetting their heritage. a fairly conservative um uh publisher kind of decided that you know you had to do these these comic books.

[00:41:51]So uh the central event in the Mahabharata is the Bhagwat Gita. So at least that's one of the conventional ways of thinking about it. And what happens in the Gita is that it's the event just before the war. And so Arjuna uh the great warrior does not want to kill his teachers, his cousins, everyone on the other side. And he's got his charioter Krishna who was this lovable rogue but now Krishna reveals to him his true nature. He's the lord of the universe. And so the Gita is actually uh the moral code of the warrior within the cosmic scheme. And so it's often identified with the Mahabharata itself though textual analysis. So there are schools of thought that say that Gita was a later edition in the 2 century Christian era uh not something which came from the original and in fact the Mahabharata can be read uh without an emphasis on the Gita. So here are uh pictures uh from uh uh of kind of sculptures that represent not the kind of comic book version but the variants of the bahabarata or the the variants which are all related to this particular event. So it's this particular event. So what happens is the war is about to begin and uh Arjuna does not want to fight and then Krishna kind of gives him this reveals himself as the lord of the universe and and gives him this didactic message on duty and how you have to fulfill your duty and you have to fight the war and uh in some ways he says that this is your destiny.

[00:43:22]This is the code and you have to follow through. So as it turns out this episode is very very closely tied uh to the IAS uh through certain very famous quotes.

[00:43:32]And so here uh I give you the Sanskrit version on the lefty sahasra yadiha.

[00:43:45]So uh this is one of the verses uh so this one forms for example the text uh of Oppenheimer's one of the biographies of Oppenheimer and this is that if the radiance of a thousand sons were to burst forth at once in the sky that would be like the splendor of the mighty one. So that that quote is actually corresponding uh to the sculpture uh the the vishwar rupa uh of of uh and then of course uh there is uh another quote which follows uh which is speaks uh the actual translation is I am time the destroyer of worlds uh but Oppenheimer uh translated to now I'm become death the destroyer of worlds. So you see uh the Mahabharata is very important uh you know for of course the Hindu conception of the world but it's got this grandeur uh a kind of universal grandeur and uh Oppenheimer was greatly attracted to it uh in the 1920s in his studies of Sanskrit but you can see uh as I told you that there are kind of different readings of the Mahabharata and to give you a flavor of the moral complexity uh of the Mahabharata let me give you a contrasting episode and a contrasting reading you know to be uh to bring out my inner Emily Dickinson and to be a little bit subversive. So, uh it turns out that uh you know the war takes place and so you Mahabharata has 18 books and only about the first nine of those books really make it into the children version because after the ninth book it really becomes a book for grown-ups. Uh in particular the 10th book uh is a book called the massacre at night and it is the distinction between um just war and unjust war. So the way uh the the war goes is that you know the shatrias the warriors they each start each morning and they fight one another and each day of the war you know there's a particular episode so one of the kind of striking episodes so this is an episode which actually you know u uh is the episode of Abimanu um you know it's it's this it's it's it's this kid he's this kind of fantastic warrior and they're like okay there is this great formation only you and your father know how to break into it. Your father's not here. Can you do it? We will protect you. So his uncles go with the son Abiimmanu. Uh but he gets trapped and they can't get him and Abi Manu Arjun's favorite son is killed.

[00:46:08]Uh and so you know there are episodes like this. Every single day the war is fought according to ritual. What happens in the 10th uh book is actually quite different. So at this point the karabas have lost and runa's sona has been killed but he's killed in very unjust manner and his son is one of the last people standing. So there are three warriors standing and what these three warriors standing decide to do is that they decide to break into the panda camp at night and attack them when no one is thinking. So to violate the rules of law, to violate the rules of duty and uh in fact it turns out that there is another god of the Indian trinity who actually possesses uh Ashwatama uh Duna's son as he goes in and this god uh is actually in in some ways the dark god uh it's it's Shiva but Shiva appears in uh the feminine version of Khali. And so these are images uh so both of these images so this one's not from the Met but the Vishwarupa uh and the Chamunda over here are are both in the Met. So these are you know this this 10th century sculpture and so let me give you a version of the unjust war and so what happens is that it's an extremely bloody chapter. So uh so Kali uh appears so Kali is related to time. So somehow this concept of time appears in the religious imagery and you as you can see uh there's this kind of um you know uh not quite sure and maybe this is not quite right for an after dinner but it turns out that it's ashwatama enters uh the camp the pondas are not there but he kills uh everyone one by one in a very very ritualistic manner. So Disha Dumna uh the man who kills his his his teacher uh he he's possessed he kind of enters into the tent and uh he he kind of jumps on Dishadumna and he tramples him uh and he tramples him on his gentiles till he dies. So this is in the text going back uh you know 3,000 years. Uh similarly uh he kills each of the sons of the pandas one by one. He kills uh a particular uh transgender warrior shikandin by slicing him in half. So it's really this extremely uh violent piece and so it reflects I think you know another reading of the epic a kind of older reading which is about uh the acts of violence and unjust war. So in fact these traditions are actually living traditions uh and they're actually related to mathematics because in the n in the 15th century there was a school of mathematics in Kerala uh where I'm from and uh within the school it's a kind of school of the oracle. So somehow there's a kind of divine oracle and the oracle uh in in the school uh is called village para which means a living vessel for light. So over here there's a kind of syncritic tradition and this in the syncritic tradition uh the authority of the oracle stems not from possession uh and it's kind of very different from so I I I put this stuff out there to to kind of say that this is actually my tradition. So this is actually our temple uh where I grew up or not where I grew up but where my grandmother grew up uh and this is a picture uh from that temple and these are actually living living kind of religious feelings uh in a particular way.

[00:49:34]I was in Hiroshima uh last week uh and I want to show you kind of another uh Sanskrit prayer just to kind of give you uh the sense. So this is a Sanskrit prayer which is on uh the peace bell uh in Hiroshima and this is from the 3rd century or uh one uh 100 Christian era.

[00:49:53]It's one of the first Sanskrit prayers in Buddhism. Buddhism was written for the common people. It was written pra uh not Sanskrit which is like Latin. And so over here you see uh this poem uh which uh is one of the kind of founding uh books of Mahayana Buddhism. And so over here you see the kind of counterpoint uh to uh a sort of uh I don't know uh it it was moving to me you know there was the kind of openheimer quotation from from the Bhagwat Gita and on the other hand uh you have over here uh the quotation uh of the lord of kind of vast light uh and so you know I think uh the spirit of this is quite clear. So you see in these poems, Dickinson, you know, Lurel, uh, uh, the the poems over here, this kind of interplay of light and dark and somehow the sense of kind of this tensions within, uh, within within our own spirits. And, uh, last week uh, you know, when I was in Japan, uh, I went out to, uh, uh, Pete Hart kind of very kindly introduced me to, uh, uh, people who study the origin of life, uh, in Tokyo. And I found you know this this beautiful picture uh on the wall by a Japanese artist uh Kaio Namura who has an artistic modern artistic conception uh of the origin of life. So she's kind of drawing a picture of the panser panspermic theory of kind of life spreading. Uh and that picture actually reminded me very much of uh an indigenous picture uh of indigenous art uh you know in in in kind of respect for Ashka's uh Australian heritage. And this is from an art project in 1971 by a school teacher who showed up in you know kind of totally benited uh area of uh Australia uh in the center of the country called Papunia. And what happened over there was that he went there as a school teacher. He began to teach children how to use the art. And what happened is that the elders in the community actually began and kind of began for the first time to kind of take their traditional knowledge of the trimmings and uh and and make moodles associated and this actually evolved into a very famous art movement uh the papuna toa art collective. So now the stuff is super expansive but at that time it was really this kind of transfer of knowledge. So I want to go back to where I began uh about the oracle. So I began with symbols, symbols that people like me understand, symbols that have meaning uh for people like me. One of the fascinating things about AI today is I think these systems of meaning are now scalable. Any many many people uh have them. So if I think about how ideas emerge, they are you know there are if I think about the mind of Newton, there's this kind of complex of ideas within um and ideas live within individuals who are kind of housed within institutions.

[00:52:47]But today mathematical ideas uh are in some ways much much bigger than the individuals because uh there are you know the ideas in some ways live in computers the computers. So I play with Gemini all the time. Gemini has read every math book ever. And as I work with Gemini I'm training Gemini in my ideas.

[00:53:08]So not just the kind of formal stuff but also the informal stuff. I talk to Gemini about aborinal art. I talk to Gemini about poetry. I talked to Gemini about mathematics. So Gemini knows everything about me. My grad students only really about know about my math. So I'm transferring my knowledge to the computer.

[00:53:27]Um so I think the ideas live outside of the instit of the individuals and some ways our institutions are actually in in pretty dangerous territory. Uh so I know the numbers at Brown the positions are just not very good. Um and so I think this is where mathematics is going in the near future. And when I got into faculty governance, uh there was this paper I read uh going into 2022. Uh it's paper called reinvigorating uh economic governance. I didn't know at the time that Kevin Worsh was going to become the the head of the Fed, but it's a great paper. It's it's Hoover. It's from the Hoover Institute. So it's pretty conservative, but I think that's exactly why uh I felt it was very important to read it carefully. And I think this you know uh uh the reason I put this out there is I think it's very very important uh to think carefully about our institutions to think carefully uh about how our institutions can continue uh to be spaces for um the mixing of complicated ideas and I thought I would just end with kind of pictures. So I like visiting religious sites across the world. So these are boys who took me across the river in Barnesy. That's a little girl in Jerusalem. Um that one I love. That's just from like one month ago. And these are my kids uh both in Mexico. So that's the story. Thanks.

[00:54:43]Thanks for your time and thanks for your support of the Institute.

[00:54:51]>> Thank you Govin so much for sharing that with us. Uh we do have time for a few questions.

[00:54:59]>> Yeah.

[00:55:02]>> In conversation. Last week at the president's talk, I met John Hopfield >> who was the whole idea of Nobel Prize, the the brain and computers. Yeah. So, where do you see math taking us? People say that AI is really simple math.

[00:55:17]>> How do you see AI from a math perspective?

[00:55:20]>> So, uh there there are kind of three different levels. So, I I think about both the math of AI and AI format. So, uh in relation to John's work, um you know, and relation to kind of work that we did in our seminar. So one of the questions is why do neural networks actually work? So there are neural networks which are artificial models for artificial models created by Melo and Pitts. Um in so you know there's a there's kind of weird math question about why does this stuff actually work much better than we think uh it'll work.

[00:55:51]So there's still kind of territory over there that we don't understand but a theory is emerging. At the same time, there's been a uh an engineering spirit uh within u uh the use of neural network methods which is just like let's just build this stuff and see where it takes us and that's led to reasoning systems and LLMs. So that's the kind of stuff that I'm using when I'm using uh LM. So it's pretty clear that reasoning works even though we don't think it should work but it works extremely well uh on these machines. So just in terms of the practice of mathematics and the understanding uh there are vast open spaces uh for us to explore. So I'm very very bullish on mathematics uh in the sense that mathematics is vastly decentralized.

[00:56:39]There's it's extremely well capitalized.

[00:56:42]Uh it's cheap to develop schools of mathematics. So it's not just that like you know the United States has been prominent in mathematics for about 70 years but today you know China has invested a great deal in mathematics, India has invested a great deal in mathematics. There are two fields medalist from Iran. There's a field medalist from Korea. So it's very very diverse. It's uh it's the most scalable of all intellectual endeavors in the sense that it's relatively cheap to train mathematicians. So Romania. So, so in in that sense I'm very bullish on mathematics because what mathematicians need are tools uh they need students and they need problems they really need things to study so all of these frontiers have been opening up so there's in addition to the stuff that we do traditionally there are new frontiers between mathematics and linguistics there are frontiers between mathematics and biology so that stuff I think about a lot um there are dangerous I think ethical questions that need to be considered. So I don't think the privacy of human thought is in violate, right?

[00:57:44]So so there's there's stuff related to also not just thinking about um uh neural networks or kind of like human and machine intelligence but think about biological intelligence more broadly. So all animals so you know all mammals or all mobile animals kind of navigate space. That's actually a math problem.

[00:58:03]So your GPS is solving that problem but birds are solving those problems. So the sense of you know like human centric mathematics may expand. So for me I find it incredibly promising uh to to study. The concern of course is a little bit like openheimer that's why I showed it to you. So there's truth and beauty uh the the founding kind of motto of the institute and there's let's say truth and nonviolence which is the founding motto of you know my the country of my birth India. beauty can sometimes be extremely violent. So scientists may follow ideas because of their intrinsic beauty. So physicists uh were guided by really just by beauty. So it's really from Einstein's work uh that the atom bomb comes out. You know of course there are kind of systems there are political circumstances. So that that's in some ways why I went back to Newton. The the tension you know between uh the choices one makes as an individual what impels us to do certain things that I don't think is something that can be governed in any sort of easy way. So what we were looking for in some ways you know a we have kind of different motivations but in some ways uh I think where we agree is what we're looking for is an educational chain reaction. So that's why we wanted to make a simple project uh a way of kind of educating young people in the spirit of the discipline so that as the discipline grows that the ethical core uh the spirit of sharing the spirit of commonality the spirit of discussion and disscent the acceptance for outsiders that the spirit would stay the same. So it's it's it's it's you know sorry for being a little bit verbose but it's very very complicated territory uh because of the the kind of various pieces uh in play u AI skues young mathematics skues young so you know one of the reasons I'm on the other side of 50 and there's a sense in which I can't work you know you just age out of certain sectors of mathematics think about it as sport you just can't do certain things anymore and so there's very much a sense of uh for me a sense of stewardship, a sense of preservation of knowledge, but also a sense of kind of passing it on. So I think about that, you know, in some ways the reason I show you the pictures of my kids is I think about that as an immigrant. There are pieces you hold, there are pieces you let go. I think that's the story of the land. I think that's the story of the discipline. So >> yeah, >> um the slide >> Peter can I'm just going to give you the mic so people can hear your question.

[01:00:41]>> I'll speak loud.

[01:00:43]>> This one? Yeah. Yes. So institutions held within individuals uh and ideas hold those. If you had to guess where we will be in 2030 years in that sequence what will it look like?

[01:00:57]>> So I I have to begin with confession. So actually when I start out with this I what I you know over there I have in a human lifestyle over here I want to have in a civilization time frame because I really want to show you in some ways uh the long life of ideas far outside individuals. So that's why you know I I really began with the contrast between mathematics and religion. So mathematics uh in the concept of proof. So one a Russian mathematician Vladimir Aldnort calls mathematics the religion of Uklid.

[01:01:24]There's a certain emotional resonance uh that mathematicians feel and there's a sense in which you know you really like the ideas have stayed the same uh in very minor variations over 2,000 to 3,000 years uh despite civilizations languages coming and going. So that's why I felt that the ideas are transcended and kind of like uh on the longer scale it changes um on a on a 20 to 30 year horizon. So let me give you uh my understanding of uh the academic landscape. So first of all the the the private liberal arts uh colleges uh that's not sustainable. So you know if you look at the statistics you you know you look at the data you really see uh these colleges going down. So Hampshire College for example just went went bankrupt uh and had to close their doors. But in fact there are many kind of lesserknown places which have been going down. So uh the model is not sustainable because uh of uh an absence of real basic issues of governance. So um there's an absence of transparency uh at all levels, right? Like nobody knows what it costs, nobody knows what it takes to get in. Uh nobody knows what you're actually learning. Nobody knows if the credential is accurate. So this has to be cleaned up, right? So um the uh unfortunate truth is that while we need reform, what we've actually got is a kind of a revolution. So you know the the system needed to be changed, but it needed it like you can't bludgeon. So we are kind of in a a dynamic of kind of extortion from the outside to rectify uh the situation.

[01:03:00]uh we are seeing um you know kind of uh trustees are now exercising much more oversight on what's going on. Uh but on the other hand uh one of the things that can go wrong in academia is that you can make short-term decisions with long-term consequences. So you've got lifetime tenure and so if you decide to invest in an area so there like you know a lot of this interdisiplinary stuff in university it's junk. So, you know, somebody comes along and kind of says like, "Oh, we're going to do this great thing." You know, a new president comes along. It's always it's like, you know, like this movie of um it's uh there's a Sandra Bulock movie. It's like I think it's called Miss Congeniality, right?

[01:03:37]Like, you know, you go to the the Missorld competition and everybody says, "What should you do? World peace. World peace. World peace." And like, you know, it's a new president comes along, they're like, "What are you going to do?" Oh, I'm going to clear you know, we're going to we're going to fix hunger. We're going to fix climate. So, it's just some blah blah blah. But like, you know, you don't want to do that. I'm a mathematician. I won't do mathematics following mathematics as opposed to like you know some bureaucrat telling me what to do. So there's been a massive tension of this sort which has been brewing. You know faculty are unhappy with the administrators administrators don't know it. So the internal governance has to be cleaned up but it can't be kind of you can't end up with a situation where you're kind of locking down on on descent.

[01:04:16]So uh nevertheless the elite universities if you look at them there's no there's no kind of decline in uh the admissions rates. So still have you know admissions rates are seem to just keep going down. So it's very very difficult to build new universities. So uh I I would say uh if if I were to be optimistic I would say that somehow uh the institutions are actually resilient.

[01:04:39]So we're still standing despite many attacks. Um and in that sense uh I think you know institutions like Princeton or Brown they've been around for 250 years uh in all you know there's a sense in which they will be uh around. I think however uh when I think about mathematics what really really concerns me at this moment in time is that it's very expensive to grow faculty. So the way faculty grow is that you have to kind of obtain an endowment get new lines. So it's very difficult to grow faculty. On the other hand, this is exactly the moment when you need more people. So what's happening right now is that most of the kind of uh research is actually taking place in the labs including fundamental research. So if you think back to the 1950s, you know, which was a very fertile area for mathematics, there was Bell Labs, but there was also kind of a broad base. So somehow if um if we can get our act uh you know maybe because of international pressures things of that nature uh it's it's possible that we'll go back to some sort of broad-based uh educational system but you know at some level it also reflects uh the social dynamics. What does society actually want? How do we actually value learning? So I personally am actually very deeply invested in the idea of uh just thinking carefully about how to scale education. So I think the Brown model. So I I am a firm believer in the liberal arts education. So I I really do believe in the idea of humanism and science being together. I do think all of our fundamental challenges are of this nature. So take a look at kind of gender politics. It's really about like there are lines on norms decided by religion but then it's like science becomes this alternate but you need to recognize that science and humanism are not different. So I do believe in the education I do believe in kind of long-term frameworks. Um I think it's going to come you know it has to be kind of multipolar reform and it may also just be that at this point especially with mathematics I do think it's a tinder box. Uh four new corporations have been created in mathematics in the last year. Uh right now these corporations are focused on kind of they're really there's something called the math formalization program going on.

[01:06:56]So it's really kind of taking all of mathematics and kind of converting in some ways to computer checkable facts.

[01:07:02]In some ways you could think about this pro pro project as being the analog of the human genome project. So I I think there are many many opportunities for you know understanding how mathematics can be used. uh but there's a bottleneck of personnel uh bottleneck of institutions like mine where people are really doing uh these things. So uh I I think this is my perspective on on this piece of the academic landscape. It may also be that you need to kind of break up the universities. So you know there's no reason why Harvard College and Harvard Business School and Harvard Medical School and Harvard so you've got this very very unwieldy conglomerate. In some ways this is basically like you think about the university as a holding corporation uh which is holding talent.

[01:07:45]So think about a creative agency which has got a bunch of you need you know these are kind of spiky people and you need like guardrails around them but there are different models. So there's also a sense in which I do feel um I don't want to bet against the entrepreneurial spirit of people in this country and I don't want to bet against the extraordinary talent pool of young people we have. So we've got an extraordinary talent pool of young people like 10,000 posttos in the Boston area. We've got new areas of inquiry. Uh there's an abundance of capital. Uh and there's actually an abundance of space.

[01:08:17]So maybe these these like you know somebody could go out and buy Hampshire College come up with a new mission. You have to have clarity in your mission.

[01:08:25]You have to have good governance and you need to kind of uh move with that. So I I think it's actually an extraordinary space for opportunity that you know that's really one of the reasons uh you know I feel like it's it's good to have a candid conversation uh which brings in brings in people. Um math education in this country is somehow catastrophe. Uh that's that's been a problem. You can do math at low cost you know like at many countries you figure it out. So at some point like you can't keep buying people from the outside. So that too is an area where stuff can be fixed. that was an area where people are thinking. Um so many many kind of spaces for opportunity but it's also necessary uh to give uh way on some primary conceptions uh on what a university is and what it should be.

[01:09:15]>> Thanks for the question.

[01:09:17]>> Think we have time for one more question.

[01:09:24]>> Well that's never happened.

[01:09:27]Govin, thank you so much >> for this presentation.