Daniel Litt: The arithmetic of differential equations (NTWS)

It's really nice to be here. Uh let me reiterate Alina's request that you ask lots of questions. Uh I always find Zoom seminars a little bit depressing if I'm just talking to the screen. So yeah, please please ask lots of questions. Um uh everything I'm going to say is joint work with Josh Lamb. And let me just remark uh some of this uh some of this work has appeared in print uh in a paper from this January, but some of it is work in progress. And I I want to mention that because I'm very worried that recent developments will lead people to talk less about work in progress and increase secrecy in our profession. So I'm very consciously trying to talk about work in progress. And what I'm thinking about I I encourage you all to do this. I think it's important that we keep talking to each other about mathematics even in the face of of technological change. Uh okay. Uh so I want to uh talk about the arithmetic of differential equations. Uh so let me just start by introducing what I think is the the maybe base basic question uh that that motivates the subject. So um in the 1870s uh Schwarz classified algebraic solutions to hypergeometric differential. So hypergeometric functions that are algebraic. And I think this uh caused folks to ask just in general like can we tell like when does an ordinary differential equation let's say with algebraic coefficients or rational coefficients have algebraic solutions. So in other words you write down some differential equation. uh you want to know um when when does it have a uh getter here uh when when does it have a solution which is in the algebraic closure of the field of rational functions. So when does it satisfy a polinomial with polinomial coefficients? Uh so my uh like very like if I had to explain this question to a a grade schooler I would just say like the basic question is someone writes down a differential equation can you explicitly write down a solution or here explicitly means as an algebraic function. Uh so let's just like uh maybe it's not clear to you why I would be talking about this question in number three seminar. Uh so let's uh let's work out an example. Uh so here's like a very uh basic differential equation. Uh so the derivative of f is equal to a * f / z where a is a complex number. Uh okay. Uh so uh I I assume many people in the audience have taken a calculus class before. So what are the what are the solutions to this differential equity? I have all that. Does anyone want to type it into chat or something? Oh, there's someone has something. Oh, yes. Okay, great. Julian Rosen says constant time C to the N. That's right. So, constant time C to the A. Um, okay. So, obviously this is algebraic if the constant is zero. Um but uh for C non zero uh this is algebraic if and only if the structure constant a is a rational number and that's kind of easy to see. So if a is a rational number you take the solution to the power given by the denominator that's a polomial. Uh so so that gives you a polinomial satisfied by this solution. uh if a is not a rational number uh it's it's not hard to see that this function isn't finitely branching and algebraic functions have finally many branches so it can't be okay um and what I just want to point out is that this is an arithmetic condition on the structure constants of the of the original differential equation so this suggests that this question which seems like maybe it's like a question in theory of just like I don't know anal analysis or something is actually a question of number. >> Okay. Um great and uh I think there's now kind of a a standard uh conjectural answer to this question of fuks which is the growth and decad curvature conjecture. Um so let me kind of give it in a a simple form. So let's say a is an r by r matrix of rational functions with rational coefficients. And then we want to study the solution ddz minus a times some vector of polorphic functions is zero. So in other words the derivative of f is equal to a * s. Uh and so the growth casc curvature conjecture tries to ask the following question or answer the following question. When are all solutions to this differential equation algebraic functions? Meaning all the all the entries of a solution vector f are algebraic. Um and the conjecture is this. So all solutions are algebraic if and only if some arithmetic condition is satisfied. So you take this differential operator to the p power. You reduce it mod p and you want it to be identically zero for almost all um and uh this is called the p curvature conjecture. Uh the reason why is that this operator ddz minus a to the p is called the puce mod p is called the p curvature of the differential equation. Um so uh there's one there's many formulations of this conjecture. One can formulate it this in the generality of like flat vector bundles on a uh variety defined over a finally generated Z algebra. I'll discuss some version of that formulation later but it's equivalent to this one that I wrote down. So it really this really is some very elementary statement about linear ordinary differential. Um great. Uh are there any questions? Oh yeah, maybe let me just remark before I continue. Uh so why does it make sense to reduce this mod p? So this is a matrix with rational functions with rational coefficients. Uh only finitely many primes appear in the denominators of those coefficients. So for most P it makes sense for just one p. Uh it looks like there's a question from Mushi Goldbury. You should feel free to just unmute yourself and ask the question. Uh yeah, I was just wondering um solutions in what space of functions in the conjecture. >> Yeah. So you can you can understand this as holorphic solutions if you'd like local holorphic solutions or just power series. So a a differential equation like this always has a basis of local holorphic solutions and I would just like those holorphic functions to also be functions. But if you want you can also just take a formal tailor expansion of a solution. So it's it's kind of um it's a fun little induction exercise to argue that uh away from the poles of these entries you can always write down a basis for power series solutions and then I would like those formal power series to be algebraic over uh cube CC. >> Okay, thank you. >> Great. Um yeah, are there are there any other questions about this engine? There is one more question from Say um in the chat. Uh say would you like to unmute and ask the question is what would the algebraicity of a power series mean? Great. So I just mean if I have a an algebraic function I can take a tailor expansion at a at a regular point. Um and I mean it is one of those um another way of saying that is that there's a map uh from rational functions to laurance series and inclusion and a power series is algebraic if it is algebraic over the rational functions which are a sub field of the field of Laurent series satisfies the your power series satisfies a polomial whose coefficients are polinomials. Um, great. And so you can see that this is some kind of arithmetic issue. Uh, where does it come from? I think there are a bunch of ways to think about it, but one way to think about it is that if you just write down uh formal solutions to a differential equation, you get expressions like adz minus a to the n over n factorial of theory. Uh so what this condition means is that so if this is divisible by p it means that when you look at the p tailor coefficient that's some expression involving this divided by p factorial it's saying that doesn't have a p in the denominator. So uh that that's that's one way of understanding this condition. Okay. Uh let's let's return to our example and try to work it out. Uh so so we were studying this example uh ddz minus a over z f of z equals z. Let's see what happens when we take this differential to the operator to the p power and reduce model equal. So if you apply that to a monomial Z to the n, you get the follow. Get n - a * n - a minus one up to n - a - p + one uh * z to the n minus p. This is an exercise. Um so this uh condition that this is zero is saying that all of these expressions are zero. So in other words a is equal to n or n minus one or n minus 2 or n minus 3 etc when you reduce mod. Um so uh this is zero if and only if uh a mod p let's let's just say pretend a was an algebraic number for now it's in fp. Um okay so the b curvature conjecture in this case says that a uh being in fp when you reduce mod p for almost all p implies that a is a rational number. Um and so this happens for almost all P if and only if A is through in Q by the chart of density theorem. So this is a fun exercise uh if you haven't seen it before. So, so this very special case of the B curvature conjecture um just for this differential equation ddc minus a over z f is zero uh it I don't know a way of proving it without tpar density so I think that suggests to you the potential okay other questions about this example before I move on >> uh c can I so so I mean so here yeah this is true if and only if phase in Q in in the same conjecture looks like you're working over Q adjoin Z anyway is is this supposed to be like a Q are some other >> Yeah. Yeah. So if you put a Q I there was a little abuse of notation. So what I wrote in the conjecture I put Q here. This is equivalent to the conjecture over a finite generated C algebra by some restriction of scalers trick. Uh but yeah in this example you should think A is in cub. That's that's a very good point. Yeah. So you can make sense of this with with with QR coefficients too or even any finitely generated >> uh yeah. Um okay other questions okay uh so so we do not have a lot of evidence for this conjecture uh let me kind of briefly go over what's known so the the forward uh implication here uh that if all the solutions are algebraic then this holds is very easy um so if you uh go home after the talk and try you'll succeed in a few minutes uh the backwards direction is basically totally open so I I gave you one example where we know it's true um But yeah, so so the the hard part is to show that if this thing is zero for almost all P, then all the solutions are going to break. Uh we do have some evidence. So I think the strongest evidence we have is the following theorem of cats um from 1972. And he showed that the conjecture is true for a class of differential equations called Bicards equations. Um so uh what I would like to explain in this talk is a generalization of this result of cats. So uh but before going into it maybe I just want to talk a little bit about the history of the of the um conjecture. Uh so my understanding is that um Katz was at IHS and he was studying this P curvature operator. So this what happens if you take the P power of a differential operator and do it mod P and he gave a talk on it at IHS and growth and asks as a question in the seminar is you know the growth cat curvature conjecture true growth and never wrote anything about it but the conjecture appeared in in work of cats uh attributed to growth after this and so this is one of the first papers cats wrote on the savage act where he he proved the conjecture for kuk's equations uh so let me kind of briefly give you a definition of a picard folks equation and then I'll tell you a little bit more about the other evidence we have for the conjecture. Um so the situation is as follows. So you have a a smooth propamorphism of complex varieties. Um so x to s um and a pard equation is a differential equation satisfied by a period function. So what do I mean by that? I mean I take some cycle K cycle gamma S on one of the fibers of this map. Um so that gives me a cycle on all the nearby fibers because if I have a smooth propomorphism all the nearby fibers are have a cycle I can just form it in a natural way and then I also take a differential form on the total space index and then I get a function by integrating this differential form restricted to a fiber along my cycle on that fiber. So for example, if you've seen an elliptic integral, that's an example where you integrate the invariant differential on elliptic curve, a moving elliptic curve against one of the loops on the curve. So this this notion of period function is a massive generalization of an elliptic integral and it's a general theorem uh that these functions satisfy differential equations. Uh what's not obvious, it's due to cats and is that these differential equations themselves are always algebraic. Um so if you know about the theory of gman connections uh one can understand this as follows. So you take the flat bundle given by taking the local system on s whose fi is the coy of a fiber of pi. So that's r i pi lower square of c. You tensor it with o that gives you a vector bundle on s and that has what's called the gman connection which is given by the identity here and the exterior derivative here. So if you're familiar with the correspondence between differential equations and flat bundles, this is the flat bundle version of the story. Uh if not, just think this is some kind of fancy version of a differential equation. Um maybe just one word about why these functions satisfy differential equations. So it boils down to the fact that coalology of great varieties is finite dimensional. So if you start differentiating this function you can move the diff differential inside the integral and then eventually you get some inference up to a fun uh great okay so this is the class of functions for of differential equations for which we know the peak curvature conjecture uh with I think one exception so uh there's also deep work of Chernobki Chernobski Andre and Bosch uh which prove the peak curvature conjecture for which proves the peakure conjecture for differential equations with solvable. So loosely speaking, you should think of that that's the case where this matrix A here is like uh so I think that's like more or less a complete list of cases in which the temperature is known. There's a few other sporadic cases, but they're you should think they're all quite close to this. Um okay. Uh are there any questions about this? Um so uh the peer conjecture is some kind of answer to this question of oops like when are the solutions to a differential equation algebraic but I I personally don't find it totally satisfying because like not all differential equations are linear. Um so what I would like to talk about is a version of this story for nonlinear differential equations. Um, and there's there are several conjectures about this. Uh, but let me give you uh I think the best known one. So, this was annunciated uh in 1999 in an unpublished preprint of Echodol, Shepherd, Baron, and Taylor. Um, and then I think it first appeared in print in in beautiful work Bosch. Uh, and here's the conjecture. So, you take a smooth seam over a finitely generated C algebra inside of the complex numbers. um you take a foliation on that. So that's just a subbundle of the tangent bundle which is closed under the le bracket. So this is a quite general notion of a nonlinear differential equation. If you have one of these things, what does it do at each point? It gives you a subspace of the tangent space. You can try to integrate those subspaces to get a a complex analytic sub manifold. Uh so that's called a leaf of leaf of the leaf of the foliation. And the conjecture is that all leaves of this foliation of the complex numbers are algebraic if and only if some arithmetic condition holds. So when you reduce this folation fod p it's closed under the following operation namely taking a vector field to its p power. Um so uh this might be an unfamiliar operation. The the upshot here is that if you have a vector field on a variety in characteristic p well that's a way of differentiating functions. It's a differential operator. You can apply it p times And it turns out that in characteristic P that's again a vector. Um so this is sort of boils down to the same fact as like the freshman stream the binomial coefficients with the P in the top are divisible by P. Um okay. Uh so this is some this condition is some version of this previous condition like our differential P is 0 to one. Uh so again this conjecture uh very little is known about it. uh before talking about it I want to talk a little bit about just like probably the notion of a foliation might not be super familiar so I just want to draw some pictures um which might might help figure out what I'm talking about so here's an example so let's let my x be a2 and I want to explain why a foliation is kind of a differential equation so I'm going to make out of this differential equation frime of u is some function of u and f of u uh I'm going to make a foliation so here's what I'll do so I want to solve this differential equation. Here's a vector field. I can construct out of it ddu plus this function g of uv * ddv. Um you might graph that that vector field. It might look something like this. So this is called if you've taken a calculus class, this is called a phase diagram of the differential equation. Um and I'll take my foliation to be the span of this vector field. And now how is this related to solutions of differential my differential equation? If I flow along this vector field, I get some curves like this. And these curves are the graph of of a solution. So these are the the these are these curves give you the the kind of graph of the ordered pairs before you. Okay. So this is maybe maybe some way of thinking about why is she some version of a differential equation. It's a good exercise. I mean here I did this for a non possibly nonlinear restorator differential equation. And it's a good exercise to work out the formula for a higher problem. Um, so maybe I'll give another example. Uh, so for those of you who know what a flat vector bundle is, the total space of a flat vector bundle carries a natural whose leaves are the flat sections. Um, and uh, if you if you specialize this conduct gesture of echodal, shepherd, baron, taylor to that case, you recover the peer. So this is a strict generalization of the conjecture we earlier saw for linear. Okay. Uh so I'm about to state the main theorem. Uh and then I'll spend the rest of the talk pretty much explaining what it means. Uh so before I do that, are there any questions about this? Uh maybe before I I uh I take maybe let me just remark on what's known. And so I think um so the the known cases of this that are in a genuinely nonlinear situation are essentially all due to Bosch. Um and uh one I think one should think of them as kind of related to this case of the the solvable case of the Pure conjecture like this case where A is up triangular. Um there are situations where you have some group in the picture maybe. So for example if you take a a foliation on an algebra group which is invariant less invariant your translation than than Bosch and I think Antoine Shamberl this um okay so so I want to explain a result towards this conjecture um which uh uh I think of as some analog result of cast equations that I explained earlier um and then I want to explain a little bit about how this is related to the classical curvature conjecture So here's the setup. I start with a smooth proper of algebraic varieties where S is smooth over some finitely generated C algebra R um which lives inside of the complex numbers and then I'm going to form the following object. So it's the modulized space M dram x over s comma r. Uh so this is the space of flat connections on x over s of rank r. This is a stack but you can ignore that for this talk. A point of this is a flat connection on a fiber of this. So if you think this is a space whose points parameterized different um so uh this is some kind of analog of dramology. It's a nonabelian version and it carries a nonabelian version of connection which is called the isum monotonation. Uh if you don't know what that is don't worry that's the point of the talk is to explain that. uh but the theorem which is joint work with Josh Lamb is that this conjecture this conjecture I could all shepherd Baron Taylor and Bosch is true in this case so there's some foliation on this space and if it's closed under powers then all leaves foliation are great um so I think of this as some kind of nonabulan version cats are patients that I annunciated okay so most of the talk I want to just spend explaining what this is uh before I do so are there any questions Okay. Uh maybe let me make a remark before moving on which is so katz's proof of his theorem about picard fuk's equations uh really exploits a lot of the structure of the coalology of algebraic right induces con theory uses the you know integral structure on singularology uses various structures on algebraic drumology mod and the various compatibilities between so I think of it as some instance of the theory of motives um based uh So in large part this theorem is an excuse to develop an analogous structures. But I think that there's um some kind of evidence for some maybe not quite rigorous uh nonabilian theory of motives which is about uh the structure of this object related objects. And so it turns out that all the structures cats exploits in its theorems. So for example the hajindex theorem seem to have analog setting and that that's I kind of at the end of the talk I want to explain a little bit about that. Uh okay so what is isomonic? So I told you that a point of this is a flat connection. So uh a foliation should be a way of describing certain families of flat connections which should be the leaves. Um so in order to explain this I have to say a little bit about the ruman hilbert correspondence. Uh so what's that? So if I have a complex manifold y is an equivalence of categories between holorphic vector bundles on e with a flat connection and complex local systems on y. Um so here you should think if you're not familiar with the bet what a flat bet bundle is you should think this is like a differential equation and this is like a representation of the fundamental so let me just write down a formula for this equivalence of categories in in in both languages. So here if I have a flat bundle E NABA I can just send it to the kernel of this connection NABA. Synabolism E to E tends to omega 1 kernel is some ch and it's a theorem that that is always a local system of rank isomeorphic to E rank the same as E and here if I have a complex local system B I can send it to the vector bundle B tensored with structure sheet with the connection identity tensor D the exterior. So we actually saw this already when I formula simulations. Okay. Uh let me just briefly say in words how to understand this in classical differential equations. So if I have if someone gives you like a linear ordinary differential equation on an open subset of the aine line. So just some open in C. Uh you can always write down a local basis of polymorphic solutions. Um so now from that I can I claim I construct can construct a representation of the fundamental group of my open subset of the f of C. What do I do? Well given a loop I analytically continue my basis of solutions around the loop that gives me a new basis of solutions. So in other words I found a change of basis matrix. Um and what is the monodrome representation? The monodrome representation is the representation which associates to a loop the change of basis matrix I just achieved by analytic continuation. So uh so here this association is given by analytic continuation of solutions by differential equation. Okay. So this is all we need to understand the isom monitoration. Um so I will describe it in terms of its leaves. Uh so in other words I'm telling you what the solutions to the differential equation are. If someone tells you the solutions to the differential equation you know what the differential equation is right? So in this case if I give you uh complex manifolds which kind of fill out uh complex sub manifolds filling out some manifold they're tangent spaces give me the foliation. Uh so here uh the isomodon mutilation on this mod space of flat connections is to defined by the following property. So its leaves are families of flat connections on the fibers of x toss whose monogrammy representation is locally independent of s. Uh so let me just unwind that a little bit. So on each fiber of this map X to S I have a flat vector bundle because X to S is a smooth compromorphism all the fibers are homotopia equivalent. So they all have the same fundamental group. So it makes sense I can take the monogrammy representation of my flat vector bundle. So this representation obtained by analytic continuation. Um that's a that's representation of a group which is independent of the fiber I'm looking at. So I'm just asking that that representation the isomeorphism class of that representation is constant. Okay. Uh so uh before giving you some kind of background on this and examples, I will give examples, I promise. Let me just give you a warning to make you even more uncomfortable. Um so this space is not smooth and it's not a steep, it's a stack. Um so uh sorry uh here s little s is a point of s. So I'm I'm question so I'm thinking of a point of mamm as a point of s point of capital s and then a flat bundle on the fiber over that point. So so Annette I hope that answers your question. Um okay so so the warning is that this is not literally a foliation. So strictly speaking it does not literally fit into this framework of the whole shepherd baron conjecture. The reason is that this is neither smooth nor a scheme. It's a singular stack. Um and the way of making sense of this is that this is a a crystal of stacks over S and one can formulate this shepherd baron Taylor conjecture in that generality but I just encourage you not to worry about it but Josh and I have a paper pur nonmology which you can look at if you' like the precise statement. Um okay. Uh so uh the other small warning is that the description I gave this foliation is very analytic. So it relied on the Remon Hilbert correspondence. So it is not at all object obvious that this the foliation I'm defining is an algebraic object. But uh so Simpson maybe plus a little bit of work shows that this foliation is algebraic and in fact descends to whatever finally generated Z algebra what starts looks like. Okay. uh so okay so that sounds kind of uh unpleasant unaware let me just give you some examples so many classical differential equations are is mononogrammy differential equations so maybe the most famous example is uh this so the panel basics equation is an example of a differential equation which has an interpretation as an isometeration so if you'd like you can think of the result I annunciated as being about the panel basics equation specifically so what I explained implies that the peak curvature conjecture is true for this this equation for any values of alpha, beta, gamma, delta. Um the uh maybe let me just unwind a uh more general example which I think it's the one I found most useful when I was trying to learn the subject which is the schleser system. So I told you here that this uh dam is a space of differential equations. So let's like talk about the simplest differential equations. So let's take a bunch of complex numbers lambda 1 through lambda n and I want to study differential equations with mild singularities regular singularities on p1 minus these end points. Okay, so what do these look like? So these look like the following. I take ddz plus some matrix of complex numbers over z minus lambda i times my vector functions. regular singularities just means that um you hear these we only pin poles um and then the condition that this is here I said the lambda i are complex numbers so I want this differential equation to be non-s singular at infinity and that's just the condition that the sum of these matrices ai it's zero okay so I want to explain what is monotropy means in this example so it answers the following question so if I start moving the lambda i how do I vary the AI to keep the monogrammy of this uh representation on a change. So I told you analytic continuation gives some representation associated to this differential equation and if I start moving the lambda I but I don't change the ais that representation is going to change. So there there's some way of changing the ais while fixing the lambda i's to keep the representation constant. That makes sense because pi one of the space is independent of lengths as long as they're distinct. Um okay so so here's the answer um as you vary the lambda j's the ais have to satisfy this differential so commutator ai so the derivative of ai with respect to lambda j is the computator divided by the difference if i is not j and to figure out how ai should change if you move lambda i uh you use well you just differentiate this expression so so uh uh this line tells me but what all but one term of the sum is and then that lets us solve any tr so this is some crazy differential equation and again but again it's an totally classical thing and you can think of the result I annunciated above as saying this echodol sheep baron Taylor conjecture is true for this differential equation um and that some statement like you know if uh if I take if I look at this differential equation and I want to know if all the solutions on some uh for some family of lambdas that are algebraic then I have to you know do some operation and hold me the powers of these expressions and reduce them on p for infinitely many and actually writing it down would be complicated something uh okay uh so so uh let me just recall the theorem uh now that we supposedly know what the words mean uh so here's the theorem I have a smooth proper morphism this can be relaxed to add a simple no process divisor if you want that's work in progress Um I look at the space and flat connections that carries some foliation. So there's some differential equation associated to it. Uh which controls uh the um the behavior of of the how the monitor representation changes as I vary move from fiber to fiber. Um and the statement is that if this foliation is closed under P powers in fact even for infinitely many P then all leaves of the is monitor foliation are algebraic. Um so let me uh I understand that this part of the uh the hypothesis is kind of um feels kind of complicated. Let me uh remark on the conclusion which actually has a very simple interpretation. Um so there's a a purely complex analytic way of understanding what it means for all the leaves of the isation to be algebraic. namely well a point of um the point of this mamm is a flat batch bundle which we said is the same as a representation of pi 1 of a five um and uh so I have this map x toss if you this is a fiber bundle uh and what that implies is that the fundamental group s has an outer action on the fundamental group of the fibers and that descends to an action of pi one of the base on conjugacy classes of represent presentations of the fibers. Um and this condition that all leaves of the isomon mutilation are algebraic is the same as saying that this action factors through finite. Um so this is not an obvious uh theorem. It follows from the theory of regular singularities. Um due to the others uh but it's not too hard. Um I should think of this as saying that it's the some kind of nonlinear analog of the fact that uh if you have a differential equation with regular singularities uh its solutions are algebraic if and only if they're finally branching. Somehow this action controls the branches to my differential equation. Okay. So this is some kind of interesting topological statement. You should think of saying this statement is saying some arithmetic condition controls something about the topology of this map. How pi one of the base acts on representations pi one of the fibers. It's some nonlinear version of a monro representation controlled by my nonlinear differential. Okay. Um so I in the last 15 minutes I want to say a little bit about the ideas of the proof. uh you should think of this mostly as a I I want to give you a sense of I told you earlier that there's some kind of structures on nonabilian homology one has to develop some structures on this moduli of um flat bundles or moduli of representations one has to develop to execute this argument and I want to explain some of those structures uh so before I do that uh are there any questions okay Um yeah so let me uh first just give an overview of Cass's from his so uh here's a sort of more fancy way of saying what Cass proved so again I have a smooth proper morphism over a finally generated C algebra sorry are here should be embedded in C um and then the statement is that all flat sections to the gmon connection so the uh uh vector bundle the flat bundle corresponding to the homology of the fibers are algebraic. Uh if and only if uh the action of pi one of the base on the homology of a fiber factors through a finite group. If and only if this p curvature vanishes this differential operator the p power is zero p for infinitely many or infinitely many. Um okay so so so this part is just the reh correspondence plus the regular singularities so it's saying that um I mean it's a sort of deep theorem that this spectra bundle always has regular singularities of infinity we should have mild singularities so to show the splice sections are algebraic it's enough to show they're finitely branching which boils down to this okay so really the hard part is is this implication okay uh so I'm going to explain in a few words the proof of this theorem before I do that let just say the proof of our theorem is to copy cats. Um so cats is going to use a lot of structure on the moology algebraic varieties and we'll need to develop analogs of that structure for these modes. Okay. So what's the the proof of cat theorem? So it has three steps. It has a arithmetic step um an analytic step uh and some kind of topological step. Um so what's the arithmetic step? Um it's the following. So here's the here's the arithmetic input. The p curvature of this differential operator this connection is zero for infinitely many peaks. This just means if you plug in a vector field let's say dd let's say s is an open subset of the fn line. So ddz if you plug in ddz to this connection you get a differential operator on e and you want it to be power to be zero on p for almost infinitely. Um so this vector bundle comes with a filtration the hodge filtration. So this is part of Hajj theory. Um and in general the gaus connection or the monetary representation on the phmology does not preserve that filtration. So it's it's some filtration that's genuinely varying and that's measured by the theory of period maps. But what cats proves is that if the p curvature vanishes then this gbond connection preserves the hodge filtration. This differential equation preserves the hodge filtration. So in other words, there's a basis of flat sections locally which is adapted to the hot filtration. Um okay, this is a consequence of a much deeper theorem the cats prove stal cats's formula which relates the conductor to something called the dyra spenser map the derivative of the period map if you're familiar with that. Okay. Um great. So the rest of the argument is purely analytic and topological. Um so now suppose we're in the setting of the conclusion of step one where the gaspond connection preserves the hot filtration then there's an associated monomy representation which we discussed coming from the fun correspondence or if you'd like analytic continuation of solutions to our differential equation and the statement is that that representation is unitary. So this is some version of the hajindex theorem. Uh so in general uh this group has uh a direct decomposition into subspaces uh called HPQ and the homology of uh differential forms on the fiber and those subspaces uh carry a definite inner that's that's one uh one statement of the hydex theorem. Uh so what this means is that if the monermy preserves that direct sun decomposition that's a consequence of this it also preserves a definite product. So it's Now we're almost done. Uh so the final ingredient is topological input. Um so we now have this representation. So pi one of the base acts on the coology of the fibers of our map. We've shown it's unitary. So it factors through the a unitary group. So for some natural metric on the space, but it also factors through the group of uh automorphisms preserving the integral. Right? Monomy also acts on the integral singularology. Um so here this is a unit this is a compact it's a unitary group this is a discrete group. So a compact discrete set is finite. Um and so uh so we've just shown that this representation factors through a finite group which is one of the things but that was that was uh this equivalent statement. um and then that implies the desired algebraicity statement using some theory of regular singularities. So we'd like to imitate this proof uh in this kind of nonlinear setting. Uh before I I explained that in the last eight minutes, are there any questions about K's proof? Okay. Um so okay so first of all I just want to say like cats used a lot of stuff. So used this hodge filtration and the hodge decomposition used version of the hodge index theorem this cat's formula which related uh the behavior of the gmon connection relative to the hot filtration to the p curvature uh we he used the z structure on singular homology um and we needed analoges of all this filtration so um up until recently Josh and I only understood analoges of this part um and so that that's what appears in print right now and then we recently figured out how to how to handle this part. So that's that's work in progress here soon and the proofs are done. The writing is in progress. Um so okay so I want to just say a few words about what these structures are and how we apply them. So like for example what is the analog the Hudendex theorem in this setting. Uh so first of all uh the analog of the Hodge filtration has been known for a long time. it's due to Simpson and others. Uh so the the analog of the HUD filtration in this setting is called the modul space of lambda connections. So um a connect a connection on a vector bundle is a well it's a vector bundle and a map from E to E tensor omega satisfying some properties. Uh so it's not an linear map. Its failure to be linear is is this this term in the the luminance rule which like the product rule for differentiation in calculus. And a lambda connection is what you get by imposing instead of imposing this relation where f is a function and s is to e you deform it by lambda. Um so uh uh and then you also impose that when you square this operator you get zero. So you get something complex. So a lambda connection is a vector bundle with a map like this satisfying this um this identity and squaring to zero. Here lambda is a number. So, so the space of lambda connections maps to the out. Okay. What are the fibers of this map? Well, if lambda is not zero, you can just divide NAB by lambda and you that that you have to do a little exercise here that makes them one here. Um, so if lambda is one, this is just the usual notion of a flat connection. So the fibers of this map away from zero are just the mod space of flat connections we were studying. um what's the fiber over zero? Well, then this term dies and this just becomes an olinear map. And uh if you if you look at this data where navo is linear and squares to zero, that's what's called a hig bubble. Um so the fiber over zero is the modulus of hig bubbles. You're familiar with that. So this is some analog of the hodge filtration. Uh let me not say why. uh maybe it will become clearer as I continue but you really can put the notion of the hot filration on some kind of equal footing this a little okay so so let me kind of tell you the first theorem we prove so this is some analog of cats formula uh so it's the following so if the isom monotromeation on this space m dam is closed under powers for infinitely many p then it extends across the zero fiber of this modaliz space so it extends to a fully on the hot on on the modul bundles. What does that mean? Well, we had this family every fiber away from zero is mamm. So it carries a foliation. So we get a foliation away from zero. Typically it will blow up at zero. But the the theorem is that if it's if this fation is closed under powers then it in fact extends to collation on the higs space bits. Um so this is the analog of the statement that if the P temperature vanishes then the gas quantum connection preserves the hot filration and I will say I will just call this statement that the isomodonation extends across the zero fiber of this family isomy preserves the concentration this I'm defining this this phrase of quotes and u I won't say much about the proof except to say that it's a consequence of some version of cats's formula which holds even if uh the P temperature is non zero. So it's uh some comparison of the obstruction to the isomeation being closed under Powers and the obstruction to the isome monitor inflation extending across the zero fiber space. Uh the second ingredient is some version of the Hodgeik theorem. Um and what does it say? It says that if the isomon foliation if isomodermy preserves the hot filtration. the consequence of this theorem then uh we had this action of pi 1 of the base on representations of pi one of the fiber so the modized space of representations and the orbits of this action have compact closure in the uklidian okay so why is this some version of the hodgeex theorem so I told you before the hajindex theorem told you some representation is unitary and the unitary group is compact which means that if you take uh an element of a vector space and you take its orbit under a unitary representation it has compact closure. So that's some version of the statement. Okay. And then so the both of these results appear in print. Um and so this is the the really the the result that's new and uh being written up now it's some piatic version of the statement. So if isommony preserves the hodge filtration um so in other words if this p curvature vanishes uh then the orbits of this action on the QP bar points here so the QP bar representations have come by closure in the pic. So this is some uh piic version of the hyenexa uh I don't know what its uh analog is for usual homology. might just be like the fact that uh you know tomology has been search um so I I I feel this is a new result I feel like I don't fully understand the picture it fits into and I think there's a lot of interesting things to say about this um I won't say much about the proofs except to say that they rely on non hops theory and the the theory of maps to build this here this is due to corlet Simpson and others and this is due to grane and and corin So these are are really kind of the theorems of analysis. Okay. So with that uh I in the last minute or so I can finish the main theorem. So let me just unwind it. So suppose we have our hypothesis this is monogrammyation is closed under powers. I know these explicit differential equations I wrote down earlier have some arithmetic property by theorem A that implies the statement that is monodic preserves the hot filtration. And then if you combine theorem B and C that tells you that if you take the orbit of a representation of pi 1 which is defined over a number field or pi one of the base that orbit is compact in the adelic topology on the space of all representations. So all idelic representations um so or sorry it has compact closure um okay uh but it's also discrete right so the the a number field has a its image in the adelis is discreet so what this tells you is that these orbits are both discrete and compact some topology so they're finite um that's a little bit weaker than showing that the this action factors through a finite group but one can deduce it deduce it for that as an exercise. Okay. Uh so maybe I'll stop there. Thank you guys so much for listening.