This is a list of academic visitors I have hosted at the University of Minnesota, Twin Cities (in reversed chronological order, including forthcoming ones), together with a list of titles and abstracts of the talks given by them. (I have omitted many visitors who were mainly invited by my colleagues or were academically too far away from me, even when I was also substantially involved in the hosting of them.)
An invertible matrix is called totally positive if all its minors are positive. In 1994, Lusztig developed the theory of total positivity for arbitrary split real reductive groups and their flag manifolds. He further generalized the theory to arbitrary Kac-Moody groups in 2019. The theory of total positivity has found important applications in different areas: cluster algebras, higher Teichmuller theory, the theory of amplituhedron in physics, etc.
In this talk, we will discuss some remarkable combinatorial, geometric and representation-theoretic aspects of total positivity. This talk is based on some recent works with Huanchen Bao.
It is known that the number of conjugacy classes of a finite group equals the number of irreducible representations (over complex numbers). The conjugacy classes of a finite group give a natural basis of the cocenter of its group algebra. Thus the above equality can be reformulated as a duality between the cocenter of the group algebra and the Grothendieck group of its finite dimensional representations.
The situation becomes more complicated, yet more interesting, if the group algebras are replaced by the Hecke algebras. In the three talks, I will explore some aspects of the theory of cocenters, with applications to arithmetic geometry and representation theory.
In the first talk, I will focus on the structure of the cocenter of affine Hecke algebras, with an explicit example on the affine Hecke algebra for $SL_2$, and discuss some applications to the affine Deligne-Lusztig varieties;
In the second talk, I will discuss the cocenter of the "big" Hecke algebras of reductive $p$-adic groups, and some applications to the representation theory, including the Howe's conjecture and trace Paley-Wiener theorem;
In the third talk, I will discuss a categorical version of the cocenters we discussed in the first two talks, and will explain the semi-orthogonal decomposition of the sheaves on the quotient stack of loop groups and the categorical cocenter of the affine Hecke categories.
Recently, Imai-Kato-Youcis constructed a prismatic F-crystal with G-structure on the integral model of the Shimura variety associated to Abelian type Shimura datum (G, X). This object can be regarded as the prismatic realization of the 'universal G-motive' on a Shimura variety. In this talk, we extend this object to a log prismatic F-crystal on the toroidal compactification of a Shimura variety of Hodge type by establishing log prismatic Dieudonne theory. If time permits, we will also talk about our future work.
A complex variety with a positive first Chern class is called a Fano variety. The question of whether a Fano variety has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. In the last decade, algebraic geometry, or more specifically higher dimensional geometry has played a surprising role in advancing our understanding of this problem. In fact, the algebraic part of this question is one step of a larger project, namely constructing projective moduli spaces that parametrize Fano varieties satisfying the K-stability condition. The latter is exactly the algebraic characterization of the existence of a Kähler-Einstein metric. In the lecture, I will explain the main ideas behind the recent progress of the field.
Braverman and Kazhdan formulated a new approach to establishing conjectures on automorphic L-functions that directly generalizes the Tate thesis and bypasses Langlands' functoriality conjecture. There have been slow but steady advances in this program, with geometric flavors and harmonic analytic flavors. I will report on those advances, including a work in progress in collaboration with Zhilin Luo.
Let G and H be real reductive groups. To any L-homomorphism e : H^L ¡÷ G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of groups which exhibit similar behavior. I will explain how this generalized version of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.
One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the "building blocks" of all unitary representations.
The Fourier transform and Poisson summation formula on a vector space have a venerable place in mathematics. It has recently become clear that they are but the first case of general phenomena. Namely, conjectures of Braverman, Kazhdan, L. Lafforgue, Ngo and Sakellaridis suggest that one can define Fourier transforms and prove Poisson summation formulae whenever the vector space is replaced by a so-called spherical variety satisfying certain desiderata. In this talk I will focus on what has been proven in this direction for a particular family of spherical varieties related to flag varieties. A simple (but nontrivial) example is the zero locus of a nondegenerate quadratic form. Kazhdan believes that these generalized Fourier transforms and Poisson summation formulae will eventually have many applications throughout mathematics. I agree, and to expedite these applications I will present them in a format that is as accessible as possible.
We know what it means to diagonalize an operator in linear algebra. What might it mean to diagonalize a functor?
Given a linear operator f whose characteristic polynomial is multiplicity-free, we can construct projection to each eigenspace as a polynomial in f, using a technique known as Lagrange interpolation. We think of the process of finding a complete family of orthogonal idempotents as the diagonalization of f. After reviewing this we provide a categorical analogue: given a functor F with some additional data (akin to the set of eigenvalues), we construct idempotent functors projecting to "eigencategories." Along the way we'll explain some of the basic concepts in categorification.
Diagonalization is incredibly important in every field of mathematics. I am a representation theorist, so I will briefly indicate some of the important applications of categorical diagonalization to representation theory. I'll also indicate applications to algebraic geometry.
In this talk we will follow a running example involving modules over the ring Z[x]/(x^2 - 1), in other words, the group algebra of the group of size 2. If you know what a complex of modules is (and what chain maps and homotopies are) then you have all the prerequisites needed for this talk.
This is all joint work with Matt Hogancamp.
In this talk, I will discuss the multiplicity formula for spherical varieties, including the definition of the geometric multiplicities for general spherical varieties, the proof of the multiplicity formula in some cases using trace formula, and some application of the multiplicity formula.
Let G be a reductive group and H be a closed subgroup of G. We say H is a spherical subgroup of G if there exists a Borel subgroup B of G such that BH is Zariski open in G. (I will explain what the above terminologies mean in my talk.) One of the fundamental problems in the relative Langlands program is to study the multiplicity problem for the pair (G,H), i.e. to study the restriction of a representation of G to H. In this talk, I will first recall the multiplicity problem in the finite group case and in the Lie group case. Then I will go over the general conjecture and all the known results for the multiplicity problem of spherical varieties. Lastly, I will explain how to use the trace formula method to study this problem.
To a connected reductive group G over a local field F we define a compact topological group \tilde\pi_1(G) and an extension G(F)_\infty of G(F) by \tilde\pi_1(G). From any character x of \tilde\pi_1(G) of order n we obtain an n-fold cover G(F)_x of the topological group G(F). We also define an L-group for G(F)_x, which is a usually non-split extension of the Galois group by the dual group of G, and deduce from the linear case a refined local Langlands correspondence between genuine representations of G(F)_x and L-parameters valued in this L-group.
This construction is motivated by Langlands functoriality. We show that a subgroup of the L-group of G of a certain kind naturally leads to a smaller quasi-split group H and a double cover of H(F). Genuine representations of this double cover are expected to be in a functorial relationship with representations of G(F). We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the theory of endoscopy.
The pioneering work of Langlands has established the theory of reductive algebraic groups and their representations as a key part of modern number theory. I will survey classical and modern results in the representation theory of reductive groups over local fields (the fields of real, complex, or p-adic numbers, or of Laurent series over finite fields) and discuss how they relate to Langlands' ideas, as well as to the various reflections of the basic mathematical idea of symmetry in arithmetic and geometry.
A famous result in number theory is Dirichlet's theorem that there exist infinitely many prime numbers in any given arithmetic progression a, a + N, a + 2 N, ... where a, N are coprime. In fact, a stronger statement holds: the primes are equidistributed in the different residue classes modulo N. In order to prove his theorem, Dirichlet introduced Dirichlet L-functions, which are analogues of the Riemann zeta function which depend on a choice of character of the group of units modulo N.
More general L-functions appear throughout number theory and are closely connected with equidistribution questions, such as the Sato--Tate conjecture (concerning the number of solutions to y^2 = x^3 + a x + b in the finite field with p elements, as the prime p varies). L-functions also play a central role in both the motivation for and the formulation of the Langlands conjectures in the theory of the automorphic forms.
I will give a gentle introduction to some of these ideas and discuss some recent theorems in the area.
The theory of complex multiplication developed in the 19th century for imaginary quadratic fields via the moduli of elliptic curves with complex multiplication, and extended by Shimura, Taniyama and Weil to CM fields, leads to (a partial) explicit class field theory for these base fields. The case where the base field is not a CM field appears more mysterious. I will describe a largely conjectural approach, for the simplest case of real quadratic fields, which rests on replacing modular functions by "rigid meromorphic cocycles". This is an account of ongoing joint work with Alice Pozzi and Jan Vonk.
The third talk will be on the Siegel-Weil formula, a quantitative version of "local-to-global principle" for quadratic forms over integers. We will discuss an arithmetic version, relating the Kudla-Rapoport special cycles on Shimura varieties to the Siegel-Eisenstein series.
In the second talk, we will move to certain high dimensional varieties (such as the product of several elliptic curves) over number fields, where, instead of rational points, we want to search for (or to show finiteness of) algebraic cycles (i.e., parameter solutions) modulo suitable equivalence relations (rational equivalence, Abel-Jacobi or its p-adic variants). In particular, we'll report some recent results on a conjecture of Beilinson/Bloch-Kato and the role of the Gan-Gross-Prasad (or the arithmetic diagonal) cycle in the proof.
A very common theme in number theory (and many other areas) is the study of various "local-to-global principles", such as the Hasse-Minkowski theorem for rational points on a quadric. The three talks will be devoted to several such instances in the context of rational points/algebraic cycles on varieties over global fields and their relation to special values of L-functions. In the first talk (colloquium), we will discuss the Hasse principle for rational points on intersection of two quadrics in P^N over function fields (of algebraic curves over a finite field). Among other things, the new tools here include an on-going work with Zhiwei Yun to establish a function field analog of (a strengthened) Kolyvagin's theorem for elliptic curves and our previous work on a Higher Gross-Zagier formula relating intersection numbers of certain cycles on moduli space of Drinfeld Shtukas to L-functions of elliptic curves.
The Andre-Oort conjecture concerns special points of a Shimura variety S --- points that are in a certain sense "maximally symmetric". It states that if a variety V in S contains a zariski-dense set of such points, then V must itself be a Shimura variety. It is an example of the field now known as "unlikely intersections theory" which seeks to explain "arithmetic coincidences" using geometry. In fact, there is a very natural sense in which the Andre-Oort conjecture can be seen as an analogue of Faltings theorem concerning rational points on curves. The proof of this conjecture involves a wide range of disparate mathematical ideas ---functional transcendence, mondromy, point counting in transcendental sets, upper bounds for arithmetic complexity (heights of special points), and p-adic hodge theory. We will survey these concepts and how they relate to each other in the proof, aiming to give an overview of the relevant ideas. We will also discuss the current status of the field, now spearheaded by the (still extremely open!) Zilber-Pink conjecture, and what is required to make further progress.
The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer in 1968, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. Our proof of this conjecture is based on a new arithmetic algebraization theorem, which has its root in the classical Borel--Dwork rationality criterion. In this talk, we will discuss some ingredients in the proof and a variant of our arithmetic algebraization theorem, which we will use to prove the irrationality of certain 2-adic zeta value. This is joint work with Frank Calegari and Vesselin Dimitrov.
After around 25 years as an algebraic number theorist, I switched fields in 2017 and started to teach number theory to a computer proof system called Lean. I now believe that these systems will be playing an inevitable role in the future of mathematical research. I'll tell the story of what I've been doing and why I think it's important. I will assume no prior knowledge of computer proof systems.
In 1957, Grothendieck defined the K-groups of vector bundles on an algebraic variety to formulate his Riemann-Roch theorem, bringing an algebraic perspective to a theorem in complex analysis. The higher K-groups of varieties were discovered in 1973 by Quillen and have remained a source of both mystery and inspiration in mathematics, and interacts with many vastly different subjects.
In this talk, I will provide a panoramic tour on K-theory and its cousins - cycles and cobordisms. In particular I will explain how the birational geometry of Hilbert schemes (joint with Bachmann) and p-adic Hodge theory (joint with Morrow) can be used to shed new light on these creatures.
Derived categories are a construction used across many mathematical disciplines, including algebraic geometry, topology, and non-commutative algebra. I will introduce derived categories and illustrate how to use them to recover a well-known topological invariant. By discussing some open conjectures revolving around derived categories in algebraic geometry, I will then explain how to decompose derived categories into smaller, more manageable pieces and compare them to one another through my research in geometric invariant theory. To conclude, I will outline my program to solve these conjectures and discuss some works in progress.
I will first review the relationship between the classical Bessel differential equation
z^2f''(z)+zf'(z)+zf(z)=0
and the classical Kloosterman sum
\sum_{x=1}^{p-1} e((x+x*)/p), where e(-)=exp(2\pi i -) and x* is the inverse of x mod p
following the work of Deligne, Dwork and Katz. Then I will discuss a generalization of this story from the point of view of Langlands duality, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, myself, and the recent joint work with Daxin Xu. In particular, the joint work with Xu gives (probably) the first example of a p-adic version of the geometric Langlands correspondence. It allows us to prove a conjecture of Heinloth-Ngo-Yun on the functoriality of some specific automorphism forms.
As an algebraic analogue of micro local analysis, the singular support and characteristic of an etale sheaf on a smooth algebraic variety over a perfect field is defined on the cotangent bundle. We discuss this geometric theory and some recent progress in the arithmetic context.
A pasture is, roughly speaking, a field in which addition is allowed to be both multivalued and partially undefined. I will describe a theorem about univariate polynomials over pastures which simultaneously generalizes Descartes' Rule of Signs and the theory of Newton Polygons. I will also describe a novel approach to the theory of matroid representations which revolves around a universal pasture, called the "foundation", which one can attach to any matroid. This is joint work with Oliver Lorscheid.
p-adic Hodge theory is one of the most powerful tools in modern arithmetic geometry. In this talk, I will review p-adic Hodge theory of algebraic varieties, present current developments in p-adic Hodge theory of analytic varieties, and discuss some of its applications to problems in number theory.
There are many natural sequences of moduli spaces in algebraic geometry whose homology approaches a "limit", despite the fact that the spaces themselves have growing dimension. If these moduli spaces are defined over a field K, this limiting homology carries an extra structure -- an action of the Galois group of K -- which is arithmetically interesting.
In joint work with Feng and Galatius, we compute this action (or rather a slight variant) in the case of the moduli space of abelian varieties. I will explain the answer and why I find it interesting. No familiarity with abelian varieties will be assumed -- I will emphasize topology over algebraic geometry.
In 1916, Ramanujan made a conjecture that can be stated in completely elementary terms: he predicted an upper bound on the coefficients of a power series obtained by expanding a certain infinite product. In this talk, I will discuss a more sophisticated interpretation of this conjecture, via the Fourier coefficients of a highly symmetric function known as a modular form. I will give a hint of the idea in Deligne's proof of the conjecture in the 1970's, who related it to the arithmetic geometry of smooth projective varieties over finite fields. Finally, I will discuss generalisations of this conjecture and some recent progress on these using the machinery of the Langlands program. The last part is based on joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne.
The Weyl group and the nilpotent orbits are two basic objects attached to a semisimple Lie group. The interplay between the two is a key ingredient in the classification of irreducible representations in various contexts. In this talk, I will describe two different constructions to relate these two objects, due to Kazhdan-Lusztig, Lusztig, and myself. I will concentrate on the construction using the loop geometry of the group. The main result is that the two seemingly different constructions give the same maps between conjugacy classes in the Weyl group and the set of nilpotent orbits.
The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Robert Langlands asked whether it is possible to construct a function-theoretic version. Together with Pavel Etingof and David Kazhdan, we have formulated a function-theoretic version as a spectral problem for (a self-adjoint extension of) an algebra of commuting differential operators on the moduli space of G-bundles of a complex algebraic curve.
I will start the talk with a brief introduction to the Langlands correspondence. I will discuss both the geometric and the function-theoretic versions for complex curves, and the relations between them. I will then present some of the results and conjectures from my joint work with Etingof and Kazhdan.
Wiles's proof of Fermat's Last Theorem was published 25 years ago. Wiles's paper introduced many new ideas and methods which have since shaped the field of algebraic number theory. This colloquium talk intends to give a (biased) tour of these developments, especially with regard to questions that might be of interest to non-specialists.
In this talk we will describe a number of new Euler systems and especially try to explain how the ideas from our 2nd lecture fit in. These new examples include Euler systems for Rankin-Selberg products for modular forms (due to Lei, Loeffler, and Zerbes), for Siegel modular forms of genus 2, and for products of unitary groups, among others.
Many of the known examples of Euler systems come from special cycles on Shimura varieties. So algebraic groups are not far in the background. In this talk we will explain how some of the important properties of these Euler systems can be interpreted in terms of the representation theory of the corresponding groups.
The breakthrough work of Kolyvagin on the Birch and Swinnerton-Dyer conjecture introduced the `method of Euler systems.' Euler systems -- when known to exist -- remain one of the most effective tools for studying class groups and Selmer groups and their relations to special values of L-functions. This is an area that has had a recent resurgence of activity, which we hope to describe in these talks. This talk will be an introduction to Euler systems and their main features.
The celebrated Birch and Swinnerton-Dyer (BSD) Conjecture connects the structure of the rational points on an elliptic curve defined over the rationals to the analytic properties of its associated Hasse-Weil L-function. This talk will recall the BSD conjecture (and its various parts) and related conjectures and survey some of the known results toward them, especially recent work.
This talk is intended for a general mathematical audience: no prior acquaintance with elliptic curves or even non-elementary number theory will be assumed.
In this series of two talks, we will introduce the recent progress on Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives of arbitrary rank. We will discuss an important technique used in the proof, namely, the arithmetic level raising for unitary groups of even rank. We will also mention other interesting results we obtained during the course of proof. This is based on a joint work with Y. Tian, L. Xiao, W. Zhang, and X. Zhu.
Over one-dimensional bases, Gabber and Beilinson proved theorems on the commutation of the nearby cycle functor and the vanishing cycle functor with duality. In this talk, I will explain a way to unify the two theorems, confirming a prediction of Deligne. I will also discuss the case of higher-dimensional bases and applications to local acyclicity, following suggestions of Illusie and Gabber. This is joint work with Qing Lu.
A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field. This has applications to the equicharacteristic case of some conjectures on l-independence. I will also discuss the relationship with compatible wild ramification. This is joint work with Qing Lu.
Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.
We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q when p is at least 5. In the first talk, I will present a generalization of Emerton's local-global compatibility result. In the second talk, I will use this compatibility result to make a patching argument for completed homology in this setting.
Every smooth projective variety over a number field yields a Galois representation via etale cohomology, and the Weil conjecture tells that its Frobenius traces are integers. Fontaine and Mazur conjectured that Galois representations satisfying a local condition (de Rham) arise from geometry and hence have a similar finiteness property. In this talk, I will focus on de Rham local systems on algebraic varieties and explain a finiteness of Frobenius traces follows from the Fontaine-Mazur conjecture for Galois representations and the generalized Riemann Hypothesis.
An etale p-adic local system on a rigid analytic variety can be regarded as a family of p-adic Galois representations parametrized by the variety, and p-adic Hodge theory has brought many results and applications on such objects, including a p-adic Riemann-Hilbert correspondence by Diao, Lan, Liu and Zhu. I will discuss constancy of a key invariant (generalized Hodge-Tate weights) of general p-adic local systems.
For a finite type scheme over a field, its motivic cohomology groups were defined by Voevodsky and are an important algebraic invariant. However, the properties of these groups are not well understood, and it is a difficult problem to exhibit explicit classes in motivic cohomology. We will construct such classes in the special fiber of Hilbert modular varieties by using the geometry of the supersingular locus. The construction is related to a geometric realization of the Jacquet-Langlands correspondence, as well as to level raising for Hilbert modular forms. A key ingredient is a form of Ihara's Lemma for compact quaternionic Shimura surfaces.
Affine Deligne–Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport–Zink spaces. Their irreducible components provide an interesting class of cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to count the number of F_q points as q goes to infinity; this boils down to computing a certain twisted orbital integral. After applying techniques from local harmonic analysis, we reduce to computing a particular coefficient of the matrix for the inverse Satake transform. Using an interpretation of this coefficient in terms of a q-analogue of Kostant's partition function, we are able to reduce the problem to the previously known special cases of the conjecture proved by Hamacher–Viehmann and Nie. This is joint work with Yihang Zhu.
Bosch, Lütkebohmert and Raynaud laid down the foundation relating formal and rigid geometry. The type of questions they treat are mostly concerned with going from the rigid side to formal side. In the past, I considered the opposite type of question, namely to what extent properties on the formal side inform us about rigid geometry. More precisely, we will see what geometric consequences one can deduce under the assumption that the rigid space has a projective reduction. In this talk, I shall first say some background of rigid geometry and Raynaud's theory of formal models along with some examples. Then I will state the main theorem and a corollary. If time permitted, I will say something about the proof.
Does a smooth proper variety in positive characteristic know the Hodge numbers of its liftings? The answer is "of course not". However, it's not that easy to come up with a counter-example. In this talk, I will first introduce the background of this problem. Then I shall discuss some obvious constraints of constructing a counter-example. Lastly I will present such a counter-example and state a few questions of similar flavor for which I do not know an answer.
The local (and global) Langlands conjectures attempt to bridge the major areas of harmonic analysis and number theory by forming a correspondence between representations which naturally appear in both areas. A key insight due to Langlands and Kottwitz is that one could attempt to understand such a conjectural correspondence by comparing the traces of natural operators on both sides of the bridge. Moreover, it was realized that Shimura varieties present a natural means of doing this. For global applications, questions of reduction type (at a particular prime p) for these Shimura varieties can often be avoided, and for this reason the methods of Langlands and Kottwitz focused largely on the setting of good reduction. But, for local applications dealing with the case of bad reduction is key. The setting of bad reduction was first dealt with, for some simple Shimura varieties, by Harris and Taylor which they used, together with the work of many other mathematicians, to prove the local Langlands conjecture for GL_n. A decade later Scholze gave an alternative, more geometric, way to understand the case of bad reduction for certain Shimura varieties and was able to reprove the local Langlands conjecture for GL_n. In this talk we will discuss an extension of the ideas of Scholze to a wider class of Shimura varieties, as well as the intended application of these ideas to the local Langlands conjectures for more general groups.
In my colloquium talk on 4/26, I discussed the following result, giving a weak analogue of Hodge theory with torsion coefficients: if a smooth and proper complex variety X specializes to a smooth and proper variety X_p in characteristic p, then the mod p singular cohomology of X is controlled by the de Rham cohomology of X_p. The key innovation of this work is the construction of a p-adic cohomology theory that interpolates between known cohomology theories in p-adic geometry (such as p-adic etale cohomology and de Rham cohomology). In this talk, I shall discuss this construction and explain why it is closely related to certain periodicity theorems (such as Bott's and Bokstedt's) in homotopy theory. Joint work with Matthew Morrow and Peter Scholze.
In the late 60's, Hochster formulated the direct summand conjecture (DSC) in commutative algebra, which is the following innocuous looking assertion: a finite extension A --> B of commutative rings admits an A-module splitting if A is regular and noetherian. A few years later, Hochster himself proved the DSC when the ring contains a field; this and related ideas eventually had a significant impact on the development of the theory of F-singularities.
In the mixed characteristic setting, the case of dimension <= 3 was settled by Heitmann in the 90's. The general case was resolved beautifully by Yves André in late 2016 using perfectoid geometry.
In this talk, I'll present a simplification of André's proof of DSC. I will also explain why similar ideas help establish a derived variant of the DSC put forth by de Jong; the latter roughly states that regular rings have rational singularities. One of my main goals in this talk to explain why passing from a mixed characteristic ring to a perfectoid extension is a useable analogue of the passage to the perfection (direct limit over Frobenius) in characteristic p.
The integral cohomology groups of a complex algebraic variety are one of the most fundamental invariants associated to the variety. The ranks of these groups are well understood in terms of the equations defining the variety, thanks to Hodge theory. However, the torsion tends to be more "transcendental" in nature and is not easily accessible via algebraic techniques. Torsion cohomology classes have played a pivotal role in many recent advances in number theory, algebraic geometry, and representation theory, so it is important to better understand torsion from an algebraic perspective. In this talk, I'll discuss my recent work with Morrow and Scholze that explains how to bound the torsion explicitly in terms of the equations defining the variety.
Perfectoid geometry is a relatively newly uncovered corner of arithmetic geometry. It provides a context where we can fruitfully treat a prime number like a variable, thus opening the door to systematically applying algebro-geometric techniques to problems in arithmetic. Consequently, these spaces have already solved important problems not only in number theory, but also in algebraic geometry, representation theory, commutative algebra, and even homotopy theory. In this talk, I will introduce the basic notions of perfectoid geometry. The goal is to introduce the background necessary to follow the applications discussed in the following talks.
Since the beginning of the century, several approaches to Langlands functoriality conjecture have been proposed by Langlands himself, by Braverman-Kazhdan and Lafforgue, .... In this lecture I will explain how these ideas may be combined and connected to recents works on singularities of certain arc spaces.
We study certain normalized special values of L-functions associated to elliptic curves and real quadratic fields. Under certain hypothesis, we are able to show that these are squares of rational numbers. This result can be regarded as instances of the rank zero case of the Birch and Swinnerton-dyer conjecture modulo squares, and is related to a theorem of Bertolini-Darmon on rationality of Stark-Heegner points over genus fields of real quadratic fields.
Given a path-connected topological space X and two points x and y, there is typically no distinguished homotopy classes of paths between x and y. If X is a normal algebraic variety over the complex numbers, however, there is a distinguished linear combination of paths between x and y; there is an analogous statement for a variety over any local field. I'll make this precise and describe many applications to arithmetic and geometry: for example, to explicit descriptions of Galois actions on fundamental groups, and to the study of the geometry of Selmer varieties. Some of the work described is joint with Alexander Betts.
Let X be an algebraic variety -- that is, the solution set to a system of polynomial equations. Then the *fundamental group* of X has several incarnations, reflecting the geometry, topology, and arithmetic of X. This talk will discuss some of these incarnations and the subtle relationships between them, and will describe an ongoing program which aims to apply the study of the fundamental group to classical problems in algebraic geometry and number theory.
The celebrated subrepresentation theorem of Casselman states that every irreducible Harish-Chandra module can be imbedded into a principal series representation. It is one of the fundamental theorems of representation theory of real groups. In the talk, I will explain a new approach to Casselman's subrepresentation theorem based on the theory of D-modules and the geometry of the wonderful compactification of symmetric spaces. I will also discuss how this new approach leads to the second adjointness conjecture for real groups. The talk is based on joint works with A. Yom Din and D. Gaitsgory
In his famous paper "stable bundles and integrable systems", Hitchin constructs a completely integrable systems on the moduli space of Higgs bundles over a Riemann surface. This system can be presented as a map from the moduli space of Higgs bundles to an affine space, known as the Hitchin fibration. The Hitchin fibration has a rich structure and plays a role in many different areas such as gauge theory, Kähler geometry, non-abelian Hodge theory, and most recently mirror symmetry and Langlands duality. In the talk, I will first give an introduction to the Hitchin fibration and then discuss some recent developments, including applications to Langlands duality and higher dimensional generalizations of the Hitchin fibration. The talk is based on joint works with B. C. Ngô and X. Zhu.
Initiated by Langlands, the problem of comparing the Hasse-Weil zeta functions of Shimura varieties with automorphic L-functions has received continual study. The strategy proposed by Langlands, later made more precise by Kottwitz, is to compare the Grothendieck-Lefschetz trace formula for Shimura varieties with the trace formula for automorphic forms. Recently the program has been extended to some Shimura varieties not treated before. In the particular case of orthogonal Shimura varieties, we discuss the proof of Kottwitz's conjectural comparison (between the intersection cohomology of their minimal compactifications and the stable trace formulas). Key ingredients include point counting on these Shimura varieties, Morel's theorem on intersection cohomology, and explicit computation in representation theory mostly for real Lie groups.
Gamma sheaves on reductive groups, introduced by Braverman and Kazhdan, are generalizations of Deligne's Kloosterman sheaves on torus. In the talk I will give an introduction to Braverman and Kazhdan's construction of gamma sheaves and then explain the motivation and a proof of their acyclicity conjecture of gamma sheaves.
I will discuss some recent results on Serre weight conjectures in dimension > 2, based on the study of certain tame type deformation rings. This is joint work with (various subset of) D. Le, B. Levin and S. Morra.
Representation stability is a relatively new field that studies somewhat exotic algebraic structures and exploits their properties to prove results (often asymptotic in nature) about objects of interest. I will describe some of the algebraic structures that appear (and state some important results about them), give a sampling of some notable applications (in group theory, topology, and algebraic geometry), and mention some open problems in the area.
Given a smooth projective variety X over a number field K, we construct two canonical O_K-lattices in the algebraic de Rham cohomology of X. The first is constructed using the p-adic comparison theorems (for all p). The second is constructed geometrically, but the proof that it is a lattice uses the p-adic comparison theorems. Both constructions have more elementary analogs in complex geometry that I will discuss first. This is joint work with Bhargav Bhatt.
I will discuss, by way of examples, how arithmetic invariant theory seems to play a non-trivial role in the theory of integral representations of L-functions. The examples include results of many people, such as Andrianov, Avner Segal, and myself. With these examples as motivation, I will then discuss my recent work on twisted versions of some of the orbit parametrization theorems of Bhargava. The main technical ingredient in the proof of these parametrizations is a "lifting law", which is a way of relating elements in the open orbit of one prehomogeneous vector space with elements in the minimal orbit of another prehomogeneous vector space.
Arithmetic invariant theory is, roughly, the study of the orbits of groups like GL_n(Z) on lattices inside the finite dimensional representations of GL_n(R). While simply stated, these orbit problems turn out to be delicate and interesting. I will give an introduction to and partial survey of this field. In particular, I will highlight "Gauss composition" on binary quadratic forms, and some of the seminal contributions of Bhargava on "Higher composition laws".
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
In this course, I will review the construction of the Hodge-Tate spectral sequence following Faltings' approach and I will show that it bears a certain analogy with the conjugate spectral sequence in characteristic p. I will focus on one of the main ingredients, namely, Faltings' fundamental comparison theorem which is, in his approach, the basis of all comparison theorems between the p-adic étale cohomology and other p-adic cohomologies. The course is based on a joint work with Michel Gros (http://arxiv.org/abs/1509.03617).
This colloquium is an introduction to my lecture series on the same topic. The general theme is p-adic Hodge theory, a branch of arithmetic geometry initiated by Tate in his 1967 article on p-divisible groups and developed by Faltings, Fontaine, Hyodo, Kato, Messing, Tsuji and many others. It has many important applications to Fermat's last theorem, Taniyama-Shimura-Weil conjecture, Sato-Tate conjecture, aspects of the Langlands program, are among the best known.
There are many strong relations between p-adic and complex Hodge theories. The p-adic theory was inspired by the complex theory. It also provided a new and purely algebraic proof of one of the deepest results in complex Hodge theory. The goal of this lecture is to review this result and to introduce p-adic Hodge theory through this application.
Grothendieck's standard conjecture of Künneth type asserts that the Künneth idempotents (which give rise to the grading in the total cohomology) on the _rational_ cohomology of a smooth projective variety are algebraic.
A. Venkatesh raised the question, in the context of torsion automorphic forms: Can the analogous statement hold with _torsion_ coefficients? We first explain the motivations for this question, and then give answers (in the negative in general, in the affirmative in special cases).
I'll first give Kobayashi's formulation of the plus/minus main conjecture for supersingular elliptic curves, and discuss the applications. Then I'll present the proof.
We will give a survey on some recent progress on Iwasawa theory and BSD conjecture, and briefly discuss some techniques involved in the proof.
I will discuss some foundational results on the geometry of the Hodge-Tate period morphism, including its construction for Shimura varieties of Hodge type, the Newton stratification on the corresponding flag variety, and how to identify its fibers above a given Newton stratum with perfectoid Igusa varieties in the PEL case. I will also explain how the geometry of the Hodge-Tate period morphism can be used to understand the generic part of the cohomology of compact unitary Shimura varieties with torsion coefficients. This is joint work with Peter Scholze.
The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.
I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.
The case of an orbit of a (nonlinear) group of affine polynomial maps arise in the theory of cubic Markoff like surfaces. While the points are scarce they still obey strong approximation. Critical to the proof is a study of the finite orbits of these groups on points with algebraic coordinates - in specific cases this is closely related to the determination of all Painlave VI's which are algebraic.
The Diophantine analysis of orbits of a group generated by integral matrices arises in many contexts. "Thin matrix groups" arise in geometric constructions (eg Apollonian Packings) and monodromy groups of families of varieties. Quantitative strong approximation is now known for these and has Diophantine applications such as to the affine sieve.
Strong approximation is concerned with the extent to which integral points on a variety approximate real or p-adic points. In the case of quadratic forms in four variables such quantitative strong approximationis closely connected with the Ramanujan Conjectures and and we explain how this may used to construct optimal universal quantum gates.
After reviewing various vanishing theorems, we present new Kawamata-Viehweg type vanishing theorems, that allow general coefficients (polarisable variations of Hodge structures and mixed Hodge modules). Some applications will be discussed, and also perhaps some ideas in the proof.
Grothendieck's theory of homological motives is based on the conjectural existence of algebraic cycles, among others the Künneth projectors; this is the standard conjecture of Künneth type. The sign conjecture is a weaker version of this, still strong enough for the construction of the Tannakian category of homological motives.
We deduce the conjecture for certain Shimura varieties from recent results in the theory of automorphic forms, principal among them Arthur's conjectures. This is joint work with Sophie Morel.
The Grothendieck-Katz p-curvature conjecture predicts the topological monodromy of an arithmetic differential equation in terms of its reductions modulo finite primes. The conjecture is still open, but attempts to solve it have repeatedly produced beautiful mathematics, and some striking partial results. In this talk, I will try to give a sense of what the conjecture is about and survey some of the progress towards it.
This is a report on joint work in progress with Julee Kim and Nicolas Templier. Irreducible smooth representations of a p-adic reductive group are said to be supercuspidal if they do not appear in any induced representation from a proper parabolic subgroup. While it is still an open problem to obtain a precise character formula for them (apart from some special cases), I will explain how to obtain a reasonable upper bound and a limit formula as the formal degree tends to infinity, for a large class of supercuspidal representations. As an application we establish an equidistribution result and a low-lying zero statistics for L-functions in a new kind of families of automorphic representations.
In this talk I will prove an Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions for a general modular form and a CM form such that the CM form has higher weight, using Eisenstein series on U(3,1), under the assumption that the CM form is ordinary (no ordinary conditions on the general modular form). This has many arithmetic applications including proving an anticylotomic main conjecture in the sign=-1 case (formulated by Perrin-Riou) and the rank one BSD formula. In view of the Beilinson-Flach elements this gives one way of proving the Iwasawa main conjecture for supersingular elliptic curves formulated in different ways by Kato, Kobayashi and Perrin-Riou.
The theory of reductive groups is very useful and beautiful, and over perfect fields one can reduce general questions about linear algebraic groups to the reductive case (where the rich structure theory provides a lot of insight and technique). However, over imperfect fields (such as local and global function fields over finite fields) there is a substantial gap between general linear algebraic groups and reductive ones, and for problems sensitive to the ground field it is the class of pseudo-reductive groups that relate better to the general case.
In the first half of the talk, I will discuss why it is useful to think about general linear algebraic groups and survey some earlier results with Gabber and G. Prasad on pseudo-reductive groups, both examples/applications and especially a general structure theorem. The structure theorem comes with a catch: in characteristic 2 the ground field k must satisfy [k:k^2] = 2. This degree restriction is genuine insofar as there is a blizzard of phenomena that arise if and only if [k:k^2] 2.
In the second half of the talk, I will present examples to illustrate what goes wrong when [k:k^2] 2, indicate some new constructions that only exist in those cases, and formulate a general structure theorem over all such fields, thereby completing the search for an adequate structure theorem over pseudo-reductive groups over all imperfect fields.
Symmetry is evident in many forms from ancient architecture to classical art; however, not as obvious is the mathematical theory of symmetry behind modern applications, such as Rubik's cube, the art of M.C. Escher, and the security of financial transactions on the Internet. These three topics are not as unrelated as they may initially seem to be. During the lecture, the mathematical ideas behind symmetry will be developed from scratch and illustrated with pictures and numerical examples.
The ABC Conjecture, formulated in the mid-1980's by Oesterle and Masser, is one of the most important conjectures in number theory. It has many deep consequences, but its basic formulation can be given in entirely elementary terms. In September 2012, a 500-page solution was announced by Shinichi Mochizuki (building on several thousand pages of work he has carried out over the last 20 years). In this talk, I will explain what the conjecture asserts, some evidence that supports it, and a few of its consequences.
An old question of Serre (which also appeared in Deligne's Weil II) asks whether the odd-degree ℓ-adic Betti numbers of any proper smooth variety (of any characteristic) are necessarily even. I'll give an answer in the affirmative, and then discuss subsequent, more refined questions by Serre and by Katz.
We will report on a construction of the local Langlands correspondence for general tamely-ramified p-adic groups and a class of wildly ramified supercuspidal Langlands parameters that have emerged in recent works of Gross-Reeder and Reeder-Yu. The ramification of these parameters introduces two new arithmetic phenomena which were not present in the case of real groups or in the case of tamely-ramified supercuspidal parameters for p-adic groups. We will discuss how these phenomena can be handled and, time permitting, we will give an indication of how the various compatibilities expected of a Langlands correspondence are proved. These include in particular Shahidi's tempered L-packet conjecture, stability, endoscopic transfer, and compatibility with GL_n.
The work of Galois in the early 19th century showed that the arithmetic of the rational numbers is reflected in a mysterious set of symmetries, known as the Galois group of Q. In the late 19th century, Felix Klein in his Erlangen program introduced the notion of symmetry into geometry. This geometric symmetry arises in the form of Lie-groups, seemingly quite different in nature from the Galois groups. In the 1960's, Robert Langlands developed a tantalizing web of conjectures relating the two notions of symmetry. He postulated that the Galois group of Q bears a close and specific elation to certain Lie groups well known in geometry, including the linear, symplectic, and orthogonal groups. An example of the fruitful and surprising results arising from such a relationship would be that the different algebraic extensions of Q of degree n are governed by infinite-dimensional representations of the general linear group GL_n.
I will introduce some of the ideas surrounding Langlands' conjectures and then focus on more recent developments involving the so called local side of these conjectures, in which the rational numbers are replaced by the p-adic numbers.
These activities have been partially supported by an Alfred P. Sloan Research Fellowship (academic years 2014–2018), the NSF Grants DMS-1258962 (academic years 2012–2014) and DMS-1352216 (CAREER) (academic years 2014–2019), and the University of Minnesota, Twin Cities. Any opinions, findings, and conclusions or recommendations expressed above do not necessarily reflect the views of the funding organizations.