Phymaths podcast # 78 || Dr. Daniel Litt

Oh yeah, nice to be here. How's it going?

Yeah. So math, I don't know, I guess I've liked it for a long time. I guess I mostly got into it as an undergraduate in college, though I had pretty broad interests in general. I don't think it was a foregone conclusion that I would be a mathematician. Like I thought about writing fiction or doing other things like that, or doing physics. Algebraic geometry specifically, I think, was just sort of random, you know, was contingent. When I got to graduate school, you know, I think I decided I liked math. I went to graduate school. I briefly considered doing functional analysis or like, you know, sort of more kind of classical number theory. I do do some number theory now, but of kind of an algebraic geometric nature. But yeah, I think I just enjoyed chatting with the algebraic geometers at my graduate program. And then since then I've, you know, moved in various other directions and now do quite different things than I did then. But yeah, I think it's basically contingent.

Yeah. So, so what is algebraic geometry about? You have a system of polynomial equations, and you can look at the space of solutions to those. So, I don't know, like if you have the equation y = x^2, the real solutions are a parabola, the complex solutions are a sphere minus a point. And you know, if you have more complicated equations, you get much more complicated spaces. So those are spaces which you can just study as abstract shapes, like for sort of forgetting that they had this algebraic nature. And then this field topology, which is devoted to the study of shapes. And in particular, we have lots of good invariants of shapes. So homology kind of counts holes in the space. For example, the fundamental group is about loops in the space. So it's an algebraic object whose elements are loops in the space up to deformation with a fixed base point. And then there's an operation, you can compose them. You can go around one loop and then another. Okay, so if you have some random space, you have these invariants, you can study them. That's what the subject of algebraic topology is about. So trying to understand something about shapes. But it turns out that in algebraic geometry, these invariants, so the fundamental group and so on, have a lot of extra structure that kind of comes from the fact that the space is defined by algebraic means, by polynomials. So I think one of the slogans I found very motivating is that the behavior of the fundamental group, or maybe representations of the fundamental group, so sort of auxiliary spaces one can construct out of it, should in some way reflect not just the shape of this object but somehow the algebra behind it. And I don't know, for example, if you have a system of polynomial equations, you don't have to just look at complex solutions, you can look at rational solutions, solutions mod p, and that kind of thing. And so it turns out these invariants kind of reflect some of the arithmetic behavior of the polynomials defining the shape too. And so that's at least my motivation, is to try to kind of understand, I guess, kind of connections between arithmetic and geometry. So like between numbers and shapes.

Great. Okay. So, let me kind of back up a bit and introduce the question. So, right. So, I described just now some sort of extra structure on topological and algebraic varieties. So, like there's constraints on the shapes that can be defined by polynomial equations. So, just as an example, if you get a smooth compact complex manifold, it's the set of solutions to a system of polynomial equations. It always has an even first Betti number. There's a sort of even number of independent one-dimensional loops in the space. So that's some kind of constraint on topology. But you can also ask what are the constraints on topology, not just of shapes, but of maps between them. So for example, maybe the simplest situation is maybe you have a family of systems of polynomials. So systems of polynomials where the coefficients are themselves parameterized in some way. So you get a family of algebraic varieties, of sets of solutions for these things, and you can ask what are the constraints on the topology of this map. Okay, and so one way of asking that question is to try to kind of extract some algebraic invariants of the map. So for example a representation, and so that's a local system on the base of this family of varieties. And you can ask which ones kind of come from this algebraic construction, come from polynomials. So yeah, so we have a lot of knowledge about like sort of the special properties those local systems, those representations have. So I don't know, some examples are they underlie a Z-variation of Hodge structure, if you know what that means, their étale realizations are arithmetic, so they kind of interact in some way with the Galois theory of the field over which your polynomials are defined, and so on. And then we have a lot of conjectures that in fact all these sort of necessary conditions that we know about are in fact sufficient. So those conjectures are like wildly open in general, but one can try to understand something about them and verify predictions they make and that kind of thing.

Sure. Yeah. So, so there's a conjecture of Simpson that a complex local system underlying a Z-variation of Hodge structure is of geometric origin, comes from a family of algebraic varieties.

But you should think of that as some kind of non-abelian version of the Hodge conjecture.

Yeah. So, okay. So, here's a conjecture I really like. It's related to something called the Grothendieck-Katz p-curvature conjecture, which is maybe a bit technical to get into on this podcast. Though I appreciate that you seem to be happy to talk about technical things. Yeah. So, here's just a very basic question. So, you have an algebraic differential equation. So, I don't know, an example is like f prime of z = a * f / z or something like that. So, that has a solution z to the a. And so, you want to know when can you explicitly write down solutions? So one way of formalizing that is when are the algebraic functions, when do the solutions satisfy not just a differential equation but a polynomial equation with polynomial coefficients. So a necessary condition, and this was observed by Eisenstein in the 1850s, but is that if you take the Taylor expansion of one of these solutions, only finitely many prime numbers appear in the denominators of the Taylor coefficients. So that's just a fact about all algebraic functions, that when you take their Taylor expansions, they have this arithmetic property. And it's kind of natural to ask, is this necessary condition sufficient? It's sort of hard to think of what other conditions might be necessary, but it turns out it's not sufficient. So there's lots and lots of examples of solutions to differential equations where if you take their Taylor expansions at a particular point, then they kind of only have finitely many primes appearing in the denominators of the Taylor coefficients, but the solutions themselves are transcendental. But if you just study these examples, it turns out you were always taking the Taylor expansion at a singular point of your differential equation. So I think the natural conjecture is that if you have, let's say, an algebraic differential equation and a solution whose Taylor expansion at a non-singular point has only finitely many primes appearing in the denominator, then it's algebraic. So that at least we don't know a counterexample to. And then my collaborator Josh Lamb and I have proven a number of cases of it. So I think there's a kind of strong evidence this conjecture is correct. At least there is some evidence. I think, you know, we don't know this conjecture is true for like second-order linear ODE though, so it's a very basic sort of gap in our knowledge about, you know, it does seem from the evidence that there's some kind of fundamental connection between differential equations and number theory. But yeah, I think we're just starting to understand it.

That's right. Yes. Yeah. So, so I mean maybe just as an example, if you take, yeah, like e to the z, the Taylor coefficients are 1 over n factorial. So every prime number appears as a divisor of n factorial for sufficiently large n, right? But if you take, I don't know, square root of 1 + z, then you'll only see powers of two in the denominators.

Right. Exactly. Yeah. So there's some interesting cancellation that happens. So then the numerator also is divisible by a lot of primes when you have an algebraic function, and so it cancels out most of those primes appearing in the denominator.

Yeah, so I don't know if it literally came from that. So that is, I think, the sort of maybe what people were thinking about when they first started asking about which solutions to differential equations are algebraic. So the sort of first situation where this question is interesting is hypergeometric functions. So those are solutions to second-order differential equations with three regular singular points, basically, and you can just really explicitly write down the Taylor series of these solutions. And then there's some essentially finite list, which, as you say, was found by Schwarz in the 1870s, of parameters for which these solutions are actually algebraic. And yeah, so I think there's sort of an ahistorical... so Schwarz kind of found this list using just pure complex analysis. There was no arithmetic, but there's sort of an ahistorical way you might have approached it where you just look at these Taylor expansions and you check when do only finitely many primes appear in the coefficients. And Landau observed in the early 20th century that you could also use this to recover Schwarz's list. But yeah, so I think that isn't literally how people understood the algebraicity of certain hypergeometric functions back in the day, but it is, yeah, I mean, it's a way they might have. And it's kind of, there's kind of a mystery here. So I mentioned Eisenstein was the one who observed that algebraic functions have this property that only finitely many primes appear in their Taylor coefficients, in the denominators of their Taylor coefficients. And in the paper where he found this, he kind of writes something like, 'I found some amazing applications of this result to the theory of solutions to differential equations whose Taylor expansions are simple to write down but, you know, which don't have simple finite expressions in general.' And then he died a few months after publishing this paper, and so we don't know what those applications were, but I think it's quite possible that he had some ideas about hypergeometric functions.

That's right.

Yeah. So that conjecture is basically this previous conjecture except you ask that the Taylor expansions of all solutions at all points have some condition on the primes appearing in the denominators of the Taylor expansion. It has a nice different formulation. So you have a linear ODE, and the other way to state this Grothendieck-Katz conjecture is that all the solutions are algebraic functions if and only if it has enough algebraic solutions mod p for almost all p. So enough here means like the space of such has dimension equal to the order of the ODE.

Yeah, I think so. I mean, okay, so first of all, these conjectures are like super open, so, you know, we don't know they're true, so it's a little hard to talk about.

Not really, no. There's different ways of formulating it, but at least the nonlinear case includes the linear case as a special case, and linear ODEs are in particular also nonlinear ODEs, so they're...

Yeah, it's, well, any foliation.

The same question as the local systems arising from geometry.

Yeah. So, okay, we now know that they all do. I think E6 was actually the last open case. So that was my interest in it. Yeah. So the origin of this question is, so Serre asked a slightly simpler question, which is he asked, so if you have an algebraic variety, the set of solutions to a system of polynomial equations, there's a way of associating to that a group called the motivic Galois group, and we know that whatever that is, it's something called a semisimple algebraic group, and those have been classified. So there's four infinite families, A_n, B_n, C_n, and D_n, and then five exceptional ones: G2, F4, E6, E7, E8. And so it's sort of easy to get the infinite families. And so the interesting case was to do the five exceptional cases. And so those were all done by other people over the last 30 years. So E6 was the last case, which was due to a large group. So Boxer, Calogari, Emerton, Matipus, Pera, Padrius, I'm probably missing someone. And so the question I was wondering about, and I can say why, is basically just can you make families of these things? So, and this is a stronger question, of course, if you can make a family, you can make one. And it boils down to just like, can you find a family of varieties where the monodromy action of pi_1 of the base on the cohomology is Zariski dense in one of these exceptional groups. Okay, so you could just like study the existing constructions for exceptional groups, and if you do so, it was not so hard to see that you could get families for all exceptional groups except for E6. So the existing construction for E6 only gave some sporadic sort of countable collection of examples. So we need to look somewhere else. And cubic threefolds were kind of a natural place to look just because we knew they had some connection to E6 already. So my collaborator Thomas Kramer had observed that there's some kind of connection between the space of lines on a cubic threefold and E6, and we were able to exploit that to find these E6 local systems. Actually, one really fun thing is we've also figured out how to use it to make F4 local systems since that paper came out, but that's not written up yet.

Yeah, families exist. Yeah.

Oh no, I mean, so I think most of the existing examples are over a curve, so a one-dimensional family, but we expect, I think one expects to be able to make lots and lots of interesting examples. We just, you know, don't have a good machine for doing so. So you're sort of searching for a needle in a haystack. I think it's, you know, it's not, this is not really a precise question, but like one question that motivates me a little bit is like, just can you find a really beautiful example. So this E6 construction from cubic threefolds I think is quite beautiful. There's a lot of interesting geometry there. I think most of the other constructions have some sort of more, you know, maybe they're motivated by geometric language or something. So there's some very deep and interesting math, but one really can't get, I don't really know how to get my hands on the geometry. So I kind of still would like to find more cool examples.

Yeah. So this is Joël Yun's work. I think he was able to find, I think, E7, E8, and F4 local systems of geometric origin using some... I mean, I don't think it literally used the Langlands program, but, you know, the ideas were motivated by ideas from the Langlands program.

That's right. Yeah. So G2 I think was the first one to be found, arguably in some sense by Katz, but I think it was worked out carefully by Deligne, and then Deligne, Pilloud, and Sawin.

And there was related work by Gross and Savin much earlier that maybe... Yeah, in many of these cases also there's sort of like, for example for E6 and E7, we knew examples of local systems like the E6 and E7 local systems that are conjecturally of geometric origin, but we don't know how to prove that.

No, no. I mean, so there's a classification of semisimple groups, the simple ones, or, you know, there's four infinite families, A_n, B_n, C_n, and D_n, those are all easy to do, so you can make families of curves where those appear. And then, yeah, so now the exceptional ones are finished too. And then once you have all this, so those are the simple algebraic groups, and then once you have the simple ones, you can take the products to get everything.

Yeah. So in some sense this problem is solved now, but I think it's, you know, I wouldn't say all the examples are themselves so well understood, and I think there's value in finding more examples of where the geometry is kind of more transparently connected to the exceptional nature of the group, for example.

Yeah, sure.

Well, I think it, you know, it may... At least I would say I was definitely a little late to the party, but I think by February, I think I sort of understood where things were going. So, yeah, I don't think if you were paying attention that it felt like a one-off thing necessarily, but yeah, now it's quite obvious that it's impossible.

I don't think I had a... I mean, yeah, so like Navier-Stokes, I guess, is the only one we know has been solved. That I don't think I would have... In fact, yeah, I think you can find some tweet where I said like, okay, I wouldn't be surprised if it gets solved in 2027 or something like that. But Navier-Stokes, there was already a lot of work, you know, AI-assisted work. I think I would have been surprised if you told me that it was like largely autonomous, like there was not meaningful human involvement. So that I would have found quite surprising. But mostly as a matter of timescale, not like that had happened at all. Like I think if you would have asked me like, do you think Navier-Stokes will be solved by 2035 or something, I would have been like, I don't know anything about fluid dynamics, but I would have been like, sure, why not.

I mean, I still think one has to write. Like writing is like how you think, and, you know, but yeah, primarily writing will be something one does for oneself, I think. Just in part because like it's very easy to generate text now. So I expect like more math text to be generated in the next five years than in the previous thousand years or something like that seems pretty plausible to me. And so generating more text with the hope that someone else might read it is kind of, you know, now it's a challenging thing to expect to happen, right? Like I think you should write for yourself. But yeah, I mean, you know, we're generating a lot of mathematical text, for that text to have value, like people have to be able to understand it and interact with it in some way. And so I think, you know, for example, if you want to get a math PhD, it seems to me the value of that is like developing your own expertise. And then, okay, what will you be doing with that expertise? Maybe writing papers or something. I don't exactly know what the work products of mathematics will look like in the next five or 10 years. It seems that seems like something that's kind of up in the air. But it, you know, it's also like clearly kind of a golden age for mathematics in that we now have like the AI systems can sometimes give us answers to very hard questions, and it's like on us to understand those answers. So yeah, I mean, I don't think I have a good answer for like what our work products should be, like what, you know, what outputs should we be sharing with other people, but like if you love math and like want to learn math, you certainly have to develop personal expertise, and then you also have to interact with math. And I mean, it does seem to me like that the only way to do that in a sustained way right now is still getting, you know, math education, like majoring in math or getting a PhD in math. So yeah, but yeah, and of course the future of the field is very much up in the air, and I don't have answers about what it will end up looking like.

Yeah. Yeah. I mean, my expectation is that defenses will have to become a lot more rigorous, right? Like, so the historical way of evaluating a PhD student is like you primarily evaluate them based on the text of the thesis, and then the defense is kind of performative. I think that like obviously if we want to certify that our math PhDs know some math, that is not acceptable anymore, just because, you know, you can generate a sort of decent PhD thesis, or at least mathematically decent PhD thesis, by doing that right now. And then, you know, it's sort of important that the author of the PhD thesis has read the PhD thesis, for example, and understands it. And, you know, I think my view is that a minimal requirement for a PhD still has to be some kind of original contribution to the scholarly literature, and, you know, that means that some intellectual labor has to be done by the author, by the PhD candidate. But yeah, I think in terms of evaluating people, not just in math but in everything, like written text will not be reliable as a way of trying to understand what people know and what they can do. And so I think we'll, you know, for example, hiring postdocs, I think letters of recommendation will have a lot more value than publications. I think we'll probably do a lot more interviews. Yeah, so that's a big change to the structure of the profession.

Yeah, I don't really know what the future of the journal system is going to be, right? Like already it's kind of not clear to me what the purpose of it is. Like I think de facto a large part of the purpose of the journal system is like evaluating results and evaluating one another. Like, you know, historically if you have a paper in a fancy journal, that helps you get a job. So it's like a proxy. So now it's like not a proxy for, you know, your personal qualities. It's a proxy for like what model you're using or something like that. And then also, I mean, right, what I said is that I expect like more mathematical texts to be produced in the next five years than in the previous thousand years, and that's just like not possible for humans to read all of that. There's just not enough mathematicians. So yeah, certainly the nature of the journal system and peer review has to change a lot. I don't know what direction it will take. But, you know, my hope is that we find ways to, you know, still have humans writing for other humans and humans reading what other humans are writing. But, you know, that seems to me like it will require some pretty substantial changes to the profession, and I don't know. Yeah, like I certainly, yeah, human-only peer review does not seem possible to me, just as a, whether one likes it or not, just as a practical matter. I think a lot of text is going to have to be evaluated by AI at a first pass, just because of pure quantity.