## Spring 2024 Speakers

### Jeremy Hahn (MIT)

**Title: **Exotic spheres from p-adic cohomology theories**Abstract: ** Part 1: An n-dimensional exotic sphere is a smooth manifold homeomorphic, but not diffeomorphic, to S^n. I will give a leisurely introduction to the telescope conjecture in stable homotopy theory (recently resolved in joint work with Burklund, Levy, and Schlank), by explaining its implications for the diffeomorphism classification of exotic spheres. In the end, understanding the telescope conjecture reduces to a computation in algebraic K-theory.

Part 2: I will explain recent computational developments in algebraic K-theory, which among many other applications lead to new constructions of exotic spheres. This will be a survey highlighting many scholars’ recent works, but with a focus on connections to cohomology theories of p-adic formal schemes.

### David Helm (Imperial)

**Title: **Finiteness for Hecke algebras of p-adic reductive groups.**Abstract: **Part 1: Let F be a p-adic field, and G the F-points of a reductive group over F. For any compact open subgroup U of F, one can form the Hecke algebra C[U\G/U] of complex valued functions on G, invariant under left and right translation by U. This space has the structure of a noncommutative ring under convolution. A celebrated result of Bernstein from the mid-’80s proves that this ring is Noetherian, and in fact proves a stronger statement: that the center of such a Hecke algebra is a finitely generated C-algebra, and that the full Hecke algebra is finitely generated as a module over its center. We will discuss Bernstein’s proof of this result, and some of its consequences.

Part 2: Bernstein’s result over the complex numbers makes is natural to ask if one still has strong finiteness results for Hecke algebra R[U\G/U], in which the complex coefficients have been replaced by a ring R such as Z_{\ell} or even Z[1/p]. Such questions have natural arithmetic implications, but turn out to be much more difficult, and in full generality this question has remained open from the ’80s until quite recently. We show how one can use the Fargues-Scholze geometrization of the local Langlands correspondence to finally settle the question. This is joint work with Jean-Francois Dat, Rob Kurinczuk, and Gil Moss.

### Karl Schwede (Utah)

**Title: **Singularities in Mixed Characteristic via Alterations.**Abstract: **Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively. In characteristic zero, multiplier ideals are computed by a sufficiently large blowup by comparing the canonical module of the base and the resolution. In characteristic p > 0, test ideals were originally defined via Frobenius, but under moderate hypotheses, can be computed via a sufficiently large alteration again via canonical modules. In mixed characteristic (for example over the p-adic integers) we show that various mixed characteristic analogs of multiplier/test ideals can be computed via a single sufficiently large alteration, at least when one builds in a small perturbation term. This perturbation is particularly natural from the perspective of almost mathematics, from the theory of tight closure in characteristic p, or the theory of pairs from birational algebraic geometry. Besides unifying the three pictures, this has various applications. This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron and Jakub Witaszek.

## Fall 2023 Speakers

### Sean Howe (Utah/IAS)

**Title: **Differential topology for diamonds.**Abstract: **This is a gentle introduction to a new theory assigning Tangent Bundles to many non-classical objects in p-adic geometry, including the period domains and covering spaces that arise naturally when studying the p-adic cohomology of p-adic varieties. We work in Scholze’s category of diamonds, which provides a robust framework for modern p-adic geometry, but these Tangent Bundles are an additional differentiable structure on top of the underlying “topological” structure of a diamond, similar to how a differentiable atlas can be viewed as an additional structure on a topological manifold. In the spirit of differential topology over the real numbers, we expect this differentiable structure to encode interesting topological features, and this is what is meant by Differential Topology for Diamonds. The prototypical example of such a result is the Fargues-Scholze Jacobian Criterion, but our constructions of Tangent Bundles apply much more generally and it is not necessary to know anything about the Fargues-Scholze Jacobian Criterion to follow these talks! Instead, we will focus on the special case of the infinite level modular curve as an illustrative and important first example that goes beyond the Fargues-Scholze construction. In the first talk, we give an example-based overview of the theory of Banach-Colmez spaces. This is a category of topological Q_p vector spaces that arises naturally in p-adic Hodge theory and allows for interesting mixes of finite dimensional vector spaces over Q_p and finite dimensional vector spaces over C_p, the completion of an algebraic closure of Q_p. This theory provides the natural home for our Tangent Spaces. In the second talk, we explain how to construct a Banach-Colmez Tangent Bundle for the infinite level modular curve, why this construction is natural, and why it is an interesting thing to do.

### Eric Larson and Isabel Vogt (Brown)

**Title: **Interpolation for Brill–Noether Curves.**Abstract: **The interpolation problem is one of the oldest problems in mathematics. In its most broad form it asks: When can a curve of a given type be passed through a given number of general points? In this two-part talk, we will discuss our recent joint work that completely solves this problem for curves of general moduli (“Brill–Noether curves’”).

### Xinwen Zhu (Stanford/IAS)

**Title: **The unipotent categorical local Langlands correspondence.**Abstract: **I will discuss a conjectural categorical form of the local Langlands correspondence for p-adic groups and establish the unipotent part of such correspondence (for characteristic zero coefficient field).

## Spring 2023 Speakers

### Ben Elias (Oregon)

**Title: **Categorical diagonalization.**Abstract: **We give an introduction to the circle of ideas known as categorification, guided by the following question: what might it mean to diagonalize a functor?

Given a category, one can forget most of the information and just remember skeletal information about the objects up to isomorphism, a process known as decategorification. For example, the decategorification of the category of vector spaces is the natural numbers, where a vector space is only remembered by its dimension. What one forgets in this process is the rich structure of morphisms between objects. Meanwhile, categorification is the art of taking something you know and love, and realizing that it was secretly, all along, the decategorification of an interesting category! In this talk, we discuss the case of an invertible functor from a category to itself, which corresponds to a diagonalizable operator acting on the decategorification. What rich structure has been forgotten, and how do we put it back in? We demonstrate examples in the category of modules over the ring **Z**[x]/(x^2-1). We also wave our hands at some important examples in projective algebraic geometry.

### Lillian Pierce (Duke)

**Title: **Number-theoretic methods to produce counterexamples for questions motivated by PDE’s.**Abstract: **In 1980 Carleson posed a question in PDE’s: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation (as time goes to zero)? After being studied by many authors over nearly 40 years, this celebrated question was recently resolved by a combination of two results: one by Bourgain, whose counterexample construction proved a necessary condition, and later a complementary result of Du and Zhang, who proved a sufficient condition.

Bourgain’s counterexample was particularly interesting for two reasons: first, it generated a necessary condition that contradicted what everyone had expected, and second, it was a number-theoretic argument. In this talk we will describe how number theory plays a role, first in the context of Bourgain’s counterexample. Then we will describe a new, far more flexible number-theoretic method for constructing counterexamples, which opens the door to studying convergence questions for other dispersive PDE’s, where many questions remain open. Along the way we’ll see why no mathematics we learn is ever wasted, and how the boundary from one mathematical area to another is not always clear. While the motivation for the problems we describe will come from PDE’s, the talk will not assume specialist knowledge in analysis, and will be aimed for students and researchers in algebra and number theory.

### Sug Woo Shin (Berkeley)

**Title: **Automorphic Galois representations for classical groups**Abstract: **It is a theorem due to many people, most recently by Harris-Lan-Taylor-Thorne and Scholze, that there exist Galois representations associated with regular cuspidal automorphic representations of GL(n) over totally real or CM fields. This may be thought of as one direction of the global Langlands correspondence for GL(n). I will explain what we need, especially from the theory of automorphic forms, to go from there to obtain similar results for classical groups. Some open questions will also be discussed.

## Fall 2022 Speakers

### Tony Feng (Berkeley)

**Title for part 1: **A brief history of theta series**Abstract: ** Theta series were introduced by Jacobi as generating functions for counting lattice vectors. They turn out to enjoy a symmetry properties called modularity, which has many interesting applications. Kudla introduced an incarnation of theta series in arithmetic geometry, called arithmetic theta series, which are also expected to enjoy modularity, although this is conjectural in general. I will survey this story, as well as recent function field version which is joint work with Zhiwei Yun and Wei Zhang, wherein we construct generalizations that we call “higher theta series”, and conjecture their modularity properties.

**Title for part 2:** Derived Fourier analysis and modularity**Abstract: ** The modularity of classical theta series can be proved using the Poisson summation formula, a tool in Fourier analysis. I will explain an approach to the modularity conjecture for higher theta series, which is joint with Zhiwei Yun and Wei Zhang. It is based on a sheaf-cycle correspondence generalizing the classical sheaf-function correspondence, plus a theory of Fourier analysis on derived vector spaces.

### Michael Larsen (Indiana)

**Title for Part 1: **Character bounds for finite simple groups

**Abstract:** The absolute value of any character value of a finite group is bounded above by the degree of the character. There has been a concerted effort to find stronger bounds, with an eye to applications in probability theory and algebraic group theory. I will talk about recent work on character bounds for finite simple groups and say something about typical applications.

**Title for Part 2:** Strongly dense subgroups of algebraic groups

**Abstract:** Breuillard, Green, Guralnick, and Tao introduced the notion of strongly dense free subgroups of algebraic groups and proved that for every simple algebraic group G over an algebraically closed field of sufficiently high transcendence degree, G has subgroups of this kind. I will describe this work and some recent improvements and generalizations.

### Burt Totaro (UCLA)

**Title: **Algebraic varieties with extreme behavior**Abstract: **The “volume” is the basic discrete invariant for an algebraic variety of general type, analogous to the genus of a curve. We construct varieties of general type with the smallest known volume. These can be viewed as varieties that are “barely” of general type, generalizing curves of genus 2. We also construct algebraic varieties of several other types (such as Fano and Calabi-Yau varieties) with extreme behavior. (Joint with Louis Esser and Chengxi Wang.)

## Spring 2022 Speakers

### Vesselin Dimitrov (Toronto)

Title: Arithmetic algebraization: holonomy bounds and applications.

Abstract: The arithmetic algebraization method uses ideas from Diophantine approximations and Arakelov geometry in order to recognize when a formal-analytic scheme or morphism is in fact algebraic. The first part of the talk will be an introductory talk to set up the scene and then prove a basic ‘arithmetic holonomy bound’ from a joint recent work with Frank Calegari and Yunqing Tang.

In the second part, we will look into some applications of arithmetic holonomy bounds to number theory. Our two case study examples will be the solutions of the unbounded denominators conjecture on noncongruence modular forms (joint with Calegari and Tang), and the Hall-Ruzsa conjecture on pseudo polynomials.

### Dennis Gaitsgory (Harvard)

Title: Restricted geometric Langlands, and connection to the classical theory via the categorical trace of Frobenius.

Abstract: The two existing forms of the geometric Langlands conjecture say that (1) In the de Rham setting, the category of (ind)-coherent sheaves on the stack of Ǧ-local systems is equivalent to the category of D-modules on BunG and (2) In the de Betti setting, the category of (ind)-coherent sheaves on the stack of Ǧ-local systems is equivalent to the category of Betti sheaves on BunG with nilpotent singular support.

We propose yet another version, which makes sense in an arbitrary constructible sheaf-theoretic context: the category of (ind)-coherent sheaves on the stack of Ǧ-local systems with restricted variation is equivalent to the category of ind-constructible sheaves on BunG with nilpotent singular support.

It turns out that (when working over a finite field), the latter category can be directly related to the classical space of unramified automorphic functions via the operation of categorical trace of Frobenius.

Combined with the above version of the geometric Langlands conjecture,this gives an explicit description of the space of automorphic functions in terms of Langlands parameters.

### Giulia Saccà (Columbia)

Title: Hyper-Kähler manifolds and Lagrangian fibrations.

Abstract: Compact hyper-Kähler manifolds are one of the building blocks of compact Kähler manifolds with trivial first Chern class and are the natural higher dimensional analogue of K3 surfaces. Lagrangian fibrations are the natural generalization of elliptic K3 surfaces. In the first talk I will give an introduction to hyper-Kähler manifolds and Lagrangian fibrations, while in the second talk I will discuss some recent results on the topic.

## Fall 2021 Speakers

### David Ben-Zvi (University of Texas at Austin)

**Title: **Shearing and Geometric Arthur Parameters.**Abstract: **Shearing is a name for the symmetry of the derived category of graded vector spaces which simultaneously shifts weights and cohomological degrees. I will explain some games one can play with shearing that play a role in my joint work with Yiannis Sakellaridis and Akshay Venkatesh. In particular I’ll explain the construction of the “spectral exponential sheaf”, a coherent analog of the exponential D-module or Artin-Schreier sheaf. Using this object we define a functor of “spectral Whittaker induction” on moduli of local systems on curves, and in particular a geometric form of Arthur parameters – local systems (Langlands parameters) which are sheared and twisted using a commuting action of SL_2.

### Jayce Getz (Duke University)

**Title: **The Poisson summation conjecture for generalized Schubert varieties.**Abstract: **(Joint with Y. Choie.) Conjectures of Braverman and Kazhdan, L. Lafforgue, Ngo and Sakellaridis state that spherical varieties admit Schwartz spaces, a Fourier transform or transforms, and corresponding Poisson summation formulae. I refer to this collection of conjectures as the Poisson summation conjecture. If the Poisson summation conjecture were known in general then the functional equation and meromorphic continuation of Langlands L-functions would follow. By converse theory, this in turn would imply much of Langlands functoriality.

I will define a family of generalized Schubert varieties and then outline the proof of the Poisson summation conjecture for these varieties. I will also explain how this allows one to prove a conjecture of Bump and Choie on the meromorphic continuation of Schubert Eisenstein series.

### Ivan Loseu (Yale University)

**Title: **Unipotent Harish-Chandra bimodules.**Abstract: **Unipotent representations of semisimple Lie groups is a very important and somewhat conjectural class of unitary representations. Some of these representations for complex groups (equivalently, Harish-Chandra bimodules) were defined in the seminal paper of Barbasch and Vogan from 1985 based on ideas of Arthur. From the beginning it was clear that the Barbasch-Vogan construction doesn’t cover all unipotent representations. The main construction of this talk is a geometric construction of Harish-Chandra bimodules that should exhaust all unipotent bimodules. A nontrivial result is that all unipotent bimodules in the sense of Barbasch and Vogan are also unipotent in our sense. The proof of this claim is based on the so called symplectic duality that in our case upgrades a classical duality for nilpotent orbits in the version of Barbasch and Vogan. Time permitting I will explain how this works. The talk is based on a joint work with Lucas Mason-Brown and Dmytro Matvieievskyi.

### Mona Merling (University of Pennsylvania)

**Title: **Scissors congruence for manifolds via K-theory**Abstract: **The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic.

### Carl Wang-Erickson (University of Pittsburgh)

**Title: **A fully faithful alternative to the Montreal functor.**Abstract: ** Let p be a prime number. The Montreal functor of P. Colmez sends p-adic representations of GL 2 (Q_p) to p-adic modules for Γ, where Γ denotes the absolute Galois group of Q_p . One result of V. Paškūnas’s study of the Montreal functor, which was a crucial step toward the p-adic Langlands correspondence for GL_2 (Q_p), is that its failure to be fully faithful boils down to the fact that it sends the trivial representation to zero. In this talk, for p greater than 3, we introduce a fully faithful alternative to the Montreal functor. It has a different target category: a derived category of modules over the stack of 2-dimensional p-adic representations of Γ. This is joint work with Christian Johansson and James Newton.

## Spring 2021 Speakers

### Jarod Alper (Washington)

**Title:** Iwahori decompositions and S-completeness. (Slides.)**Abstract: **We present a short and self-contained proof of Iwahori decompositions for reductive groups. The existence of Iwahori decompositions is the key algebraic input in the proof of the Hilbert–Mumford criterion in geometric invariant theory. We then relate the existence of Iwahori decompositions for a group G to a property of the classifying stack BG, which we refer to as S-completeness. We will discuss some of the spectacular properties of S-completeness and highlight its applications to moduli theory. This is joint work with Daniel Halpern-Leistner and Jochen Heinloth.

### Chandrashekhar Khare (UCLA)

**Title:** An analog of Serre’s conjecture for reducible mod p Galois representations.**Abstract: **Serre’s conjecture asserts that 2-dimensional irreducible odd representations G_Q —> GL_2(k), with G_Q the absolute Galois group the rationals Q and k a finite field, arise from modular forms. As has been remarked by Wiles one can ask for an analog for reducible odd representations. In joint work with Fakhruddin and Patrikis we prove (almost all of) this analog, building on earlier work of S. Hamblen and R. Ramakrishna. I would like to explain the relevance of lifting mod p representations to geometric characteristic zero representations to the proofs of Serre’s conjecture (in earlier joint work with J-P. Wintenberger), and its analog for reducible representations.

### Bao Le Hung (Northwestern)

**Title:** Moduli of Fontaine-Laffaille modules and mod p local-global compatibility.**Abstract: **The localized mod *p* cohomology of locally symmetric spaces for definite unitary groups at infinite level is expected to realize the mod *p* local Langlands correspondence for GL_*n*. In particular, one should be able to extract the component at *p* of the associated Galois representation from the action of the *p*-adic group. I will explain how one achieves this when the local Galois representation is Fontaine-Laffaille under mild assumptions. The key input is the study a collection of `Hecke functions” on the moduli of Fontaine-Laffaille modules.

This is joint work in progress with D. Le, S. Morra, C. Park and Z. Qian.

### Emily Norton (Clermont Auvergne)

**Title:** The problem of decomposition numbers of finite classical groups. (Slides.)**Abstract: **A basic problem in modular representation theory of finite groups is to understand decomposition numbers, that is, how an irreducible representation of a group in characteristic 0 decomposes into irreducible representations over a field of positive characteristic. This problem is open even for symmetric groups. I will discuss the case of finite groups of Lie type in non-defining characteristic, in particular types B and C, for which this problem is still wide open. Based on joint work with Olivier Dudas.

### Lue Pan (Chicago)

**Title:** The Hecke action on overconvergent modular forms.**Abstract: **Serre introduced the notion of *p*-adic modular forms in the study of congruences between modular forms. It is well-known that to get a better spectral theory of the Hecke operator at $p$ (U_p-operator), one should consider the subspace of overconvergent modular forms. In this talk, we will show that Hecke operators away from $p$ also have better convergence acting on overconvergent modular forms. As a direct consequence, we will present a new way to determine the Hodge-Tate-Sen weights of the Galois representation attached to an overconvergent eigenform, which previously was obtained by myself and Sean Howe independently by totally different methods.

## Fall 2020 Speakers

### Irina Bobkova (Texas A&M)

**Title:** Elliptic curves and chromatic homotopy theory**Abstract: **Computation of the stable homotopy groups of spheres is a long-standing open problem in algebraic topology, which has deep connections to number theory and derived algebraic geometry. I will introduce chromatic homotopy theory and explain how it splits this problem into simpler chromatic pieces, which can be understood using the theory of formal group laws and their deformations. Then I will talk about recent results and work in progress at the second chromatic level. In particular, I will talk about a self-dual decomposition of the sphere at the second chromatic level, where the decomposition pieces are constructed using the supersingular elliptic curves. This talk is based on joint work with M. Behrens, D. Culver and P. VanKoughnett.

### Will Sawin (Columbia)

**Title:** The Shafarevich conjecture for hypersurfaces in abelian varieties. (Slides.)**Abstract: **Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension n, defined over the rational numbers, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will broadly discuss the strategy of the proof, which combines ideas from multiple fields of algebraic geometry and number theory.

### Chao Li (Columbia)

**Title:** On the Beilinson-Bloch conjecture for unitary Shimura varieties. (Slides.)**Abstract: **For certain automorphic representations π on unitary groups, we show that if L(s, π) vanishes to order one at the center s=1/2, then the associated π-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary Shimura varieties, which generalizes the BSD conjecture. Assuming the modularity of Kudla’s generating series of special cycles, we further prove a precise height formula for L'(1/2, π). This proves the conjectural arithmetic inner product formula, which generalizes the Gross-Zagier formula to Shimura varieties of higher dimension. We will motivate these conjectures and discuss some aspects of the proof. This is joint work with Yifeng Liu.

### Gordan Savin (Utah)

**Title:** Bernstein projectors for positive depth Moy Prasad types.**Abstract: **Let G be a p-adic reductive group. For a fixed Moy-Prasad type, every smooth representation of G can be written as a direct sum of two summands where one summand is generated by vectors in the type. We use the refined building of G to write down an explicit idempotent in

the Bernstein’s center that gives this decomposition for every smooth

representation. (Joint with A. Moy)

### Sam Raskin (UT Austin)

**Title:** Geometric Langlands for *l*-adic sheaves. (Slides.)**Abstract: **In celebrated work, Beilinson-Drinfeld formulated a categorical analogue of the Langlands program for unramified automorphic forms. Their conjecture has appeared specialized to the setting of algebraic D-modules: non-holonomic D-modules play a prominent role in known constructions.

In this talk, we will discuss a categorical conjecture suitable in other geometric settings, including l-adic sheaves. One of the main constructions is a suitable moduli space of local systems. We will also discuss applications to unramified automorphic forms for function fields. This is joint work with Arinkin, Gaitsgory, Kazhdan, Rozenblyum, and Varshavsky.

## Fall 2019 Speakers

### Henri Darmon (McGill University)

**Title:** Incoherent *p*-adic families of Eisenstein series and the RM values of rigid meromorphic cocycles**Abstract:** I will express the Fourier coefficients of the ordinary projections of certain incoherent *p*-adic families of Hilbert Eisenstein series in terms of the RM values of appropriate rigid meromorphic cocycles. This transposes a seminal calculation of Gross and Zagier to a setting where the place infinity is replaced by a finite prime *p*, and (more importantly) where genus characters of real quadratic fields are replaced by arbitrary (odd) ring class characters.

This is joint work with Alice Pozzi amd Jan Vonk.

### Mark Andrea de Cataldo (Stony Brook University)

**Title:** The Hodge numbers of OG10 via Ngô strings**Abstract: **I report on joint work with A. Rapagnetta (U.Rome) and G. Saccà (Columbia U.), where we compute the Hodge numbers of the hyperkähler manifolds in the deformation class of O’Grady’s 10-dimensional example by using the Ngô support theorem.

### Akhil Mathew (University of Chicago/IAS)

**Title:** *p*-adic deformations of algebraic cycle classes and topological cyclic homology**Abstract:** Let *X* be a smooth projective scheme over the ring of integers in a *p*-adic field. The *p*-adic deformation problem (a weaker version of a conjecture of Fontaine-Messing) asks when a class in *K _{0}* of the special fiber can be lifted “infinitesimally” to

*K*. This question was considered by Bloch-Esnault-Kerz: in the unramified case (and with hypotheses on the dimension), the condition is exactly that the crystalline Chern classes should live in an appropriate step of the Hodge filtration. We use recent advances in the theory of topological cyclic homology to extend the Bloch-Esnault-Kerz theorem to arbitrary

_{0}(X)*p*-adic fields, and refine results of Beilinson on relative continuous

*K*-theory and cyclic homology. Joint with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus.

### Evangelia Gazaki (University of Virginia)

**Title:** Zero-cycles over arithmetic fields**Abstract:** The Chow group of zero-cycles of a smooth projective variety is a generalization to higher dimensions of the Picard group of a curve. For a smooth projective variety *X* over a field *k*, this group provides a fundamental geometric invariant, but unlike the case of curves very little is known about its structure, especially when *k* is a field of arithmetic interest. In the mid 90’s Colliot-Thélène formulated a conjecture about zero-cycles over *p*-adic fields. A weaker form of this conjecture has been established, but the general conjecture is only known for very limited classes of varieties. In this talk I will present some joint work with Isabel Leal, where we prove this conjecture for a large family of products of elliptic curves, and discuss some work in progress about certain cases when serious obstructions seem to appear.

### Hang Xue (University of Arizona)

**Title:** Towards a factorization of linear periods**Abstract:** I will explain how to relate local root numbers to the existence of linear models of representation of *GL(2n)*. I will also explain how to make use of this to prove conjectures of Sakellaridis and Venkatesh on the Plancherel formula for *GL(n, E) \ GL(2n, F)* and on the canonical factorization of linear periods.

## Spring 2019 Speakers

### David Hansen (University of Notre Dame)

**Title:** Families of perverse sheaves**Abstract:** Given a morphism of varieties *X* -> *Y*, is there a good definition of a “family of perverse sheaves *F _{y}* on the fibers

*X*” as

_{y}*y*runs through the closed points of

*Y*? It turns out that such a definition has been lurking in the geometric Langlands literature, albeit in an obscured form, for almost twenty years. In this talk, I’ll explain this definition and highlight the many remarkable properties enjoyed by families of perverse sheaves. I’ll also discuss some applications and some open questions.

### June Huh (IAS/Princeton University)

**Title:** Lorentzian polynomials**Abstract:** I will give a gentle overview of my work with Petter Brändén on Lorentzian polynomials, which link continuous convex analysis and discrete convex analysis via tropical geometry. The class contains homogeneous stable polynomials, volume polynomials of convex bodies and projective varieties, as well as some partition functions considered in statistical physics. No specific background will be needed to enjoy the talk.

### Kiran Kedlaya (UCSD/IAS)

**Title:** Étale and crystalline companions**Abstract:** Deligne’s “Weil II” paper includes a far-reaching conjecture to the effect that for a smooth variety on a finite field of characteristic *p*, for any prime *l* distinct from *p*, *l*-adic representations of the étale fundamental group do not occur in isolation: they always exist in compatible families that vary across *l*, including a somewhat more mysterious counterpart for *l *= *p* (the “petit camarade cristallin”). We explain in more detail what this all means, describe some key ingredients in the proof (particularly the role of the Langlands correspondence for function fields), and mention some concrete applications.

### David Nadler (UC Berkeley)

**Title:** Traces, characters, and… loops?**Abstract:** Traces of matrices, and more generally characters of representations, are indispensable tools in algebra. This will be an introductory talk about their universal nature from the perspective of topology. We will see how this perspective explains and predicts various current research streams.

### Bianca Viray (University of Washington)

**Title:** Isolated points on modular curves**Abstract:** Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that all but finitely many algebraic points on a curve arise in families parametrized by **P**^{1} or positive rank abelian varieties; we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on *X*_{1}(*n*) push down to isolated points on a modular curve whose level is bounded by a constant that depends only on the *j*-invariant of the isolated point. This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.

## Fall 2018 Speakers

### Charlotte Chan (Princeton University)

**Title:** Affine Deligne-Lusztig varieties at infinite level for GL_{n}**Abstract: **Affine Deligne-Lusztig varieties have been of interest for some time because of their relation to Shimura varieties and the Langlands program. In this talk, we will construct a tower of affine Deligne Lusztig varieties for GL_{n} and its inner forms. We prove that its limit at infinite level is isomorphic to the semi-infinite Deligne-Lusztig variety of Lusztig and that its cohomology realizes certain cases of automorphic induction and Jacquet-Langlands. This is joint work with A. Ivanov.

### Tsao-Hsien Chen (University of Chicago)

**Title:** Kostant-Sekiguchi homeomorphisms**Abstract: **The Kostant-Sekiguchi correspondence is a remarkable bijection between the real and symmetric nilpotent orbits. It is one of the fundamental theorems in Lie theory and plays an important role in representations of real groups. In my talk, I will explain a new approach to the Kostant-Sekiguchi correspondence which provides a lift of the correspondence to a stratified homeomorphism between the real and symmetric nilpotent cones. This new approach is inspired by the geometric Langlands theory. If time permits, I will discuss applications to Springer theory. Joint works with David Nadler.

### Sándor Kovács (University of Washington)

**Title:** Liftable local cohomology and deformations**Abstract:** This is a report on joint work with János Kollár. We introduce a lifting property for local cohomology, which implies that the cohomology sheaves of the relative dualizing complex of a flat morphism are flat and commute with base change. We also establish that several well-known classes of singularities have this property and derive various consequences for moduli spaces and moduli functors. In particular, this implies, among other consequences, that a stable non-Cohen-Macaulay singularity, e.g., a cone over an abelian surface, is never smoothable. All of these results work in arbitrary characteristic.

### Baiying Liu (Purdue University)

**Title:** On Langlands functoriality and converse theorems**Abstract: **Converse theorems have been playing important roles in the theory of Langlands functoriality. In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GL_{n}(*F*), where *F* is a non-archimedean local field. This is a joint work with Hervé Jacquet. Then I will briefly introduce our recent progress on establishing Langlands functorial descent (joint with Joseph Hundley) and the study of converse theorems (joint with Qing Zhang) for the split exceptional group of type G_{2}.

### Romyar Sharifi (UCLA)

**Title:** Iwasawa theory in higher codimension**Abstract:** Classically speaking, Iwasawa theory concerns the growth of *p*-parts of class groups in towers of number fields of *p*-power degree. One considers the inverse limit of such groups as a finitely generated, torsion module over the Iwasawa algebra, which is a completed group ring for the Galois group of the tower. Often, this growth can be slow enough that the support of this unramified module lies in codimension two and higher, while invariants like characteristic ideals which one hopes might be described through *L*-values don’t measure anything beyond codimension one. The typical way around this is to allow ramification at enough primes over *p* to make the module larger. Yet, the original modules are certainly of arithmetic interest. I’ll discuss how, over CM fields, these unramified Iwasawa modules and others satisfy nice “reflection principles” and how *p*-adic *L*-functions tell us something about them. This is joint work with Bleher, Chinburg, Greenberg, Kakde, and M. Taylor, and in part with Pappas as well.

## Spring 2018 Speakers

### Eyal Goren (McGill University)

**Title:** Theta operators for unitary modular forms**Abstract:** This is joint work with Ehud De Shalit (Hebrew University). We shall consider *p*-adic modular forms on a unitary Shimura variety associated to a quadratic imaginary field, where *p* is inert in the field, and the mod *p* reduction of this variety. In this case, theta operators were constructed by Eischen and Mantovan, and by De Shalit-Goren, independently and using different approaches. I will describe our approach that makes heavy use of Igusa varieties. The main two theorems are (i) a formula for the effect of a theta operator on *q*-expansions and (ii) its analytic continuation from the ordinary locus to the whole Shimura variety in characteristic *p*. Along the way interesting questions about filtrations of automorphic vector bundles arise and, to the extent time allows, I will discuss these questions in light of our work on foliations on unitary Shimura varieties.

### Christopher Hacon (University of Utah)

**Title:** Birational boundedness of algebraic varieties**Abstract:** The minimal model program (MMP) is an ambitious program that aims to classify algebraic varieties. According to the MMP, there are 3 building blocks: Fano varieties, Calabi-Yau varieties and varieties of general type. In this talk I will recall the general features of the MMP and discuss recent advances in our understanding of Fano varieties and varieties of general type.

### Gonçalo Tabuada (MIT)

**Title:** A noncommutative approach to the Grothendieck, Voevodsky, and Tate conjectures**Abstract:** The Grothendieck standard conjectures, the Voevodsky nilpotence conjecture, and the Tate conjecture, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of these celebrated conjectures remains elusive. The aim of this talk is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.

### Yunqing Tang (Princeton University)

**Title:** Exceptional splitting of reductions of abelian surfaces with real multiplication**Abstract:** Zywina showed that after passing to a suitable field extension, every abelian surface *A* with real multiplication over some number field has geometrically simple reduction modulo 𝔭 for a density one set of primes 𝔭. One may ask whether its complement, the density zero set of primes 𝔭 such that the reduction of *A* modulo 𝔭 is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod 𝔭 isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.

## Fall 2017 Speakers

### Pierre Colmez (CNRS, Université Pierre et Marie Curie)

**Title: ***p*-adic étale cohomology of the Drinfeld tower and *p*-adic local Langlands correspondence**Abstract: **It is now classical that the *l*-adic étale cohomology of the Drinfeld tower, for *l* not *p*, encodes both the local Langlands and Jacquet-Langlands correspondences. I will explain that, in dimension 1, the *p*-adic étale cohomology of this tower encodes part of the *p*-adic local Langlands correspondence (this is joint work with Gabriel Dospinescu and Wieslawa Niziol).

### Davesh Maulik (MIT)

**Title:** Gopakumar-Vafa invariants via vanishing cycles**Abstract: **Given a Calabi-Yau threefold *X*, one can count curves on *X* using various approaches, for example using stable maps or ideal sheaves; for any curve class on *X*, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map). Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.

In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi. Conjecturally, these should agree with the invariants as defined by stable maps. I will also explain how to prove the conjectural correspondence in various cases. This is joint work with Yukinobu Toda.

### Wieslawa Niziol (CNRS, ENS-Lyon/IAS)

**Title:** Cohomology of *p*-adic Stein spaces**Abstract: **I will discuss a comparison theorem that allows us to recover *p*-adic (pro-)étale cohomology of *p*-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. Present applications include a computation of the *p*-adic étale cohomology of the Drinfeld half-space in any dimension and of its coverings in dimension 1. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

### Nicolas Templier (Cornell University)

**Title:** Mirror symmetry for minuscule flag varieties**Abstract: **We prove cases of Rietsch mirror conjecture that the quantum connection for projective homogeneous varieties is isomorphic to the pushforward *D*-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. In particular we link the purity of Berenstein-Kazhdan crystals to the Ramanujan property of certain Hecke eigensheaves. The talk will keep the prerequisite knowledge to a minimum by introducing the above concepts of “mirror” and “crystal” with the examples of **CP**^{1}, projective spaces and Grassmannians. Work with Thomas Lam.

### Yihang Zhu (Columbia University)

**Title:** Arithmetic fundamental lemma in the minuscule case**Abstract: **The arithmetic Gan-Gross-Prasad conjecture generalizes the Gross-Zagier formula to Shimura varieties associated to unitary or orthogonal groups. The arithmetic fundamental lemma (AFL), formulated by Wei Zhang in the unitary case, is a key local ingredient in the relative trace formula approach towards arithmetic GGP. The AFL compares arithmetic intersection numbers on Rapoport-Zink spaces with derivatives of orbital integrals. We prove an explicit formula for the arithmetic intersection numbers in both unitary and orthogonal cases, under a minuscule assumption. In particular, our work gives a new proof of the theorem of Rapoport-Terstiege-Zhang on the AFL in the unitary case. This is joint work with Chao Li.

## Spring 2017 Speakers

### Benjamin Antieau (UIC)

**Title: **On the Brauer group of the moduli stack of elliptic curves**Abstract: **Mumford proved that the Picard group of the moduli stack of elliptic curves is a finite group of order 12, generated by the Hodge bundle of the universal family of elliptic curves. After giving background on Brauer groups and on the moduli of elliptic curves, I will talk about recent work with Lennart Meier, which computes the Brauer group of the moduli stack over various arithmetic base schemes and shows in particular that the Brauer group of the integral moduli stack vanishes. This talk will focus on the concrete computational and arithmetic aspects of the proof.

### Joel Kamnitzer (University of Toronto)

**Title:** Monodromy of Bethe vectors and crystals**Abstract: **Let *g* be a semisimple Lie algebra. A long-standing problem is to decompose of tensor products of representations of *g*. One approach to this problem is the Gaudin system, which provides a family of commuting operators acting on tensor product multiplicity spaces — the eigenvectors for this Gaudin system are called Bethe vectors. The Gaudin system depends on a parameter which lives in the moduli space of genus 0 curves and we can study how the Bethe vectors change as we vary this parameter. This provides the action of the cactus group, a certain finitely-generated group analogous to the braid group. We prove that this action of this cactus group can be obtained combinatorially, through the theory of crystals.

### Kai-Wen Lan (University of Minnesota)

**Title:** Nearby cycles of automorphic étale sheaves**Abstract: **I will explain that, in many cases where integral models are available in the literature, the automorphic étale cohomology of a (possibly noncompact) Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles in positive characteristics. If time permits, I will also talk about some applications or related results. (This is joint work with Stroh.)

### Christian Schnell (Stony Brook University)

**Title:** Pluricanonical bundles and maps to abelian varieties**Abstract: **Suppose we have a morphism from a smooth projective variety *X* to an abelian variety *A* (over the complex numbers). If we push forward the canonical bundle of *X*, we get a coherent sheaf on *A* with many special properties, all coming from Hodge theory. In the talk, I am going to explain what happens for pluricanonical bundles (= powers of the canonical bundle), based on joint work with Luigi Lombardi and Mihnea Popa.

## Fall 2016 Speakers

### Jennifer Balakrishnan (Boston University)

**Title: **Iterated *p*-adic integrals and rational points on curves**Abstract:** I will discuss some new relationships between iterated p-adic line integrals (Coleman integrals), motivated by the problem of explicitly finding rational points on curves. In particular, I will describe the link between *p*-adic heights and double integrals and give a few classes of hyperelliptic curves where “quadratic Chabauty” gives us a finite set of *p*-adic points containing all rational points. I will also briefly discuss new identities between triple Coleman integrals. This is joint work with Netan Dogra.

### Frank Calegari (University of Chicago)

**Title:** Ramanujan, *K*-theory, and modularity**Abstract:** The Rogers-Ramanujan identity:

1 + *q*/(1-q) + *q*^{4}/(1-*q*)(1-*q*^{2}) + *q*^{9}/(1-*q*)(1-*q*^{2})(1-*q*^{3}) + … = 1/(1-*q*)(1-*q*^{4})(1-*q*^{6})(1-*q*^{9})…

says that a certain *q*-hypergeometric function (the left hand side) is equal to a modular form (the right hand side). To what extent can one classify all *q*-hypergeometric functions which are modular? We discuss this question and its relation to conjectures in knot theory and *K*-theory. This is joint work with Stavros Garoufalidis and Don Zagier.

### George Pappas (Michigan State University)

**Title:** On certain moduli spaces of *p*-divisible groups**Abstract:** We will discuss two constructions of moduli spaces of *p*-divisible groups with additional structures that give integral models for certain Rapoport-Zink *p*-adic analytic spaces. These constructions use integral models of Shimura varieties and a group theoretic version of Zink’s Witt vector displays.

### Ben Webster (University of Virginia)

**Title:** Representation theory of symplectic singularities**Abstract:** Since they were introduced about 2 decades ago, symplectic singularities have shown themselves to be a remarkable branch of algebraic geometry. They are much nicer in many ways than arbitrary singularities, but still have a lot of interesting nooks and crannies.

I’ll talk about these varieties from a representation theorist’s perspective. This might sound like a strange direction, but remember, any interesting symplectic structure is likely to be the classical limit of an equally interesting non-commutative structure, whose representation theory we can study. While this field is still in its infancy, it includes a lot of well-known examples like universal enveloping algebras and Cherednik algebras, and has led a lot of interesting places, including to categorified knot invariants and a conjectured duality between pairs of symplectic singularities. I’ll give a taste of these results and try to indicate some interesting future directions.

## Spring 2016 Speakers

### Alexis Bouthier (UC Berkeley)

**Title: **Hitchin-Frenkel-Ngô’s fibration and fundamental lemma**Abstract:** The fundamental lemma for the spherical Hecke algebra is obtained by a combination of analytic results of Waldspurger and Ngô’s geometric proof of the fundamental lemma for Lie algebras. The latter makes a crucial use of Hitchin’s fibration to link orbital integrals with some counting problems on moduli spaces. In this context, we will explain that also in the group case, such a fibration exists, introduced by Frenkel and Ngô and that we can perform a geometric proof of the corresponding fundamental lemma as well as giving some new insights on the computation of transfer factors. Moreover, these new objects are expected to have other applications towards the geomtrization of the trace formula.

### Michael Hill (UCLA)

**Title:** Modular forms, duality, and equivariant homotopy**Abstract:** The Goerss-Hopkins-Miller theory of topological modular forms has introduced elliptic curves and modular forms into stable homotopy theory. Under this, natural properties of moduli spaces of elliptic curves can be translated into properties of spectra. In this talk, I’ll focus on a particular example: the *C*_{2}-Galois cover “elliptic curves with a point of order 3” over “elliptic curves with a subgroup of order 3”. This connects in an interesting way to *C*_{2}-equivariant homotopy, allowing a clear dictionary between the algebraic geometry and homotopy theory.

### Aaron Pixton (MIT)

**Title:** The tautological ring of the moduli space of curves**Abstract:** The tautological ring of the moduli space of smooth curves of genus *g* is the subring of its Chow ring generated by the kappa classes. This subring was introduced by Mumford in the 1980s in analogy with the cohomology of Grassmannians. Work of Faber and Faber-Zagier in the 1990s led to two competing conjectural descriptions of the structure of the tautological ring. After reviewing these conjectures and discussing their current status, I will state two new conjectures about the ranks of this ring.

### Jacob Tsimerman (University of Toronto)

**Title:** Counting Abelian Varieties over Finite Fields**Abstract:** (Joint with M. Lipnowski) A long standing open question is to count smooth, proper curves of genus *g* over a fixed finite field, at least in an asymptotic sense. At the moment, there is not even a consensus on whether the growth should be exponential or factorial. We consider the analogous question for principally polarized abelian varieties. The answer turns out to be very surprising: The number of isomorphism classes of abelian varities of dimension *g* grows exponenitally in *g ^{2}*, but the number of polarizations grows as fast as

*g*. One consequence is that most abelian varieties of dimension

^{g^2}*g*over a fixed finite field are essentially powers of elliptic curves, and do not obey any sort of Cohen-Lenstra heuristics.

## Fall 2015 Speakers

### Pramod Achar (LSU)

**Title: **Modular perverse sheaves and applications in representation theory**Abstract:** Perverse sheaves (with coefficients in **C** or **Q**_{ℓ}) have been a powerful tool in representation theory for over 35 years, but modular perverse sheaves (i.e., with coefficients in a field of positive characteristic) were poorly understood and little used. In the past five or six years, that has begun to change: a slew of new tools has emerged that makes the study of modular perverse sheaves accessible, and is leading to concrete new advances in the modular representation theory of algebraic groups and related objects. I will discuss as many examples as time permits, perhaps including flag manifolds, affine Grassmannians, and nilpotent cones. Various results in this talk are joint with A. Henderson, D. Juteau, C. Mautner, S. Riche, and L. Rider.

### Florian Herzig (University of Toronto)

**Title:** On mod *p* local-global compatibility for GL_{3}(**Q**_{p})**Abstract:** I will discuss the hypothetical mod *p* Langlands correspondence for GL_{3}(**Q**_{p}), which should associate to a 3-dimensional mod *p* representation of the Galois group of **Q**_{p} one (or many) smooth mod *p* representation(s) of GL_{3}(**Q**_{p}). We focus on the expected global realisation of this correspondence in the cohomology of unitary groups of rank 3. In the special case when ρ is upper-triangular and maximally non-split, its “extension class” is classified by an invariant in **F**_{p}-bar. We will show (under suitable assumptions) that this Galois-theoretic invariant is determined by the GL_{3}(**Q**_{p})-action on the above cohomology spaces. On the way we prove results about Serre weights and a mod *p* multiplicity one result. This is joint work with D. Le and S. Morra.

### Wei Ho (University of Michigan)

**Title:** Distributions of ranks and Selmer groups of elliptic curves**Abstract:** In the last several years, there has been significant theoretical progress on understanding the average rank of all elliptic curves over **Q**, ordered by height, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as discuss generalizations in many directions (e.g., to other families of elliptic curves, higher genus curves, and higher-dimensional varieties) and some corollaries of these types of theorems. We will also describe recently collected data on ranks and Selmer groups of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).

### Brandon Levin (University of Chicago)

**Title:** The weight part of Serre’s conjecture**Abstract:** Serre’s modularity conjecture (now a Theorem due to Khare-Wintenberger and Kisin) states that every odd irreducible two dimensional mod *p* representation of the absolute Galois group of **Q** comes from a modular form. I will begin with an overview of the Serre’s original conjecture on modular forms focusing on the weight part of the conjecture. Herzig gave a generalization of the conjecture for *n*-dimensional Galois representations which predicts the modularity of so-called shadow weights. After briefly describing Herzig’s conjecture, I will discuss joint work with D. Le, B. Le Hung, and S. Morra where we prove instances of this conjecture in dimension three.

## Spring 2015 Speakers

### Andrew Blumberg (University of Texas at Austin)

**Title:** *K*-theoretic Tate-Poitou duality and the fiber of the cyclotomic trace**Abstract:** Our understanding of the algebraic *K*-theory of the sphere spectrum boils down to studying the fiber of the cyclotomic trace (an analogue of the Chern character) from *K*(*S*) to a topological analogue of cyclic homology. This fiber can in turn be studied in terms of the *p*-completion map in étale cohomology. I will explain this story and describe joint work with Mike Mandell that characterizes the fiber in terms of a kind of Poincare duality, proving a conjecture of Calegari.

### Kartik Prasanna (University of Michigan)

**Title:** Extensions of the Gross-Zagier formula**Abstract:** I will first give an introduction to the general conjectural picture relating algebraic cycles to *L*-functions and discuss some extensions of the Gross-Zagier formula involving *p*-adic *L*-functions. This leads naturally to the question of constructing algebraic cycles corresponding to the vanishing of Rankin-Selberg *L*-functions at the center of symmetry. I will also outline some new constructions of such cycles, based on work in progress with A. Ichino.

### Claire Voisin (CNRS/IAS)

**Title:** Decomposition of the diagonal and stable birational invariants**Abstract:** The Lüroth problem asks whether a unirational variety is rational. It has a negative answer starting from dimension 3 and can be attacked by various geometric approaches. For the stable Lüroth problem, where “rational” is replaced by “stably rational,” only the Artin-Mumford approach had been used up to now to solve the problem in dimension 3. Using the notion of decomposition of the diagonal, we exhibit many unirational threefolds which are not stably rational while their Artin-Mumford invariant is trivial.

### Liang Xiao (University of Connecticut)

**Title:** Basic loci of Shimura varieties and the Tate conjecture**Abstract:** We explain a global description of the basic locus of a Shimura variety (of PEL type for now), in terms of a union of families of affine Deligne-Lusztig varieties (in mixed characteristic d’après X. Zhu) parameterized by zero-dimensional Shimura varieties. Under a certain genericity condition, we show that the irreducible components of the basic locus generate all Tate classes of the special fiber of the Shimura variety, and therefore verify the Tate conjecture in this setting. This is a joint on-going project with Xinwen Zhu.

## Fall 2014 Speakers

### Benjamin Howard (Boston College)

Professor, Boston College**Research Interests: **Number theory and arithmetic geometry**Website:** https://www2.bc.edu/~howardbe/**Title: **Supersingular points on some orthogonal and unitary Shimura varieties**Abstract:** I’ll describe the locus of supersingular points on some orthogonal and unitary Shimura varieties. This is joint work with G. Pappas.

### Allen Knutson (Cornell University)

Professor, Cornell University**Rearch Interests: **Algebraic geometry and algebraic combinatorics**Website:** http://www.math.cornell.edu/~allenk/**Title:** *SO*(3)-multiplicities of *SL*_{3}(**R**)-representations**Abstract:** Given a topological representation of a noncompact real Lie group like *SL*_{3}(**R**), one classically constructs an algebraic replacement called a (\mathfrak g,*K*)-module, and from there a geometric replacement called a \mathcal{D}_{G/B}-module, which is supported on a *K*-orbit closure on *G*/*B* (of which there are finitely many). I’ll recall this story in some detail.

When the *K*-orbit closure is smooth (and the “infinitesimal character” is integral), I’ll use equivariant localization to compute the *K*-multiplicities in the representation. This generalizes Blattner’s conjecture (Schmid’s theorem).

Then I’ll refine this alternating sum to a combinatorial formula, in the case of the *SO*(3)-multiplicities in the four types of *SL*(3,**R**)-irreps, and explain how and why the four formulae fit together.

### Andrew Snowden (University of Michigan)

Assistant Professor, University of Michigan**Research Interests:** Number theory, algebra/algebraic geometry**Website:** http://www-personal.umich.edu/~asnowden/**Title: **Constructing elliptic curves from Galois representations**Abstract: **Given an elliptic curve *E* over a finitely generated field *K*, the Tate module *V _{p}*(

*E*) is a representation of the Galois group

*G*that determines

_{K}*E*up to isogeny. It is an interesting problem to determine which representations of

*G*arise from this construction. I will speak on joint work with J. Tsimerman where we give a solution to this problem, excluding the case where the Galois representation is isotrivial. (The isotrivial case falls under the still-unsolved Fontaine-Mazur conjecture).

_{K}### Kirsten Wickelgren (Georgia Tech)

Assistant Professor, Georgia Institute of Technology**Research Interests:** Algebra, geometry, and topology**Website:** http://people.math.gatech.edu/~kwickelgren3/**Title:** A computational approach to the section conjecture**Abstract:** Grothendieck’s section conjecture predicts that rational points on hyperbolic curves *X* over number fields *k* are in bijection with conjugacy classes of sections of pi_1(*X*) → pi_1(*k*). Part of this conjecture reduces to *X* = **P**^{1} – {0,1,infty}. Conjugacy classes of sections are pi_0 of a mapping space of étale homotopy types. We resolve the étale homotopy type of **P**^{1} – {0,1,infty} to study these sections.

## Spring 2014 Speakers

### Ana Caraiani (Princeton/IAS)

Veblen Research Instructor, Princeton University

NSF Postdoctoral Fellow**Research Interests: ** Classical and *p*-adic Langlands programs, Shimura varieties, and arithmetic geometry**Website:** https://web.math.princeton.edu/~caraiani/**Title: **Patching and *p*-adic local Langlands**Abstract:** The *p*-adic local Langlands correspondence is well understood for GL_{2}(**Q*** _{p}*), but appears much more complicated when considering GL

*(*

_{n}*F*), where either

*n*> 2 or

*F*is a finite extension of

**Q**

*. I will discuss joint work with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas and Sug Woo Shin, in which we approach the*

_{p}*p*-adic local Langlands correspondence for GL

*(*

_{n}*F*) using global methods. The key ingredient is Taylor-Wiles-Kisin patching of completed cohomology. This allows us to prove many new cases of the Breuil-Schneider conjecture. If time permits, I will also discuss joint work in progress with Matthew Emerton, Toby Gee and David Savitt concerning certain instances of local-global compatibility and relating the geometry of local Galois deformation rings to local models.

### Tyler Lawson (University of Minnesota)

Associate Professor, University of Minnesota**Research Interests: ** Algebraic topology, particularly in interactions between multiplicative structures, number theory, and arithmetic geometry**Website:** http://www.math.umn.edu/~tlawson/**Title:** Topological modular forms and level structures**Abstract:** Modular forms made a surprising appearance in algebraic topology through work of Witten, and cohomological data about the moduli of elliptic curves has turned out to have a close connection to the stable homotopy groups of spheres. In this talk I’ll discuss how this connection came about through the theory of “topological modular forms,” and discuss joint work with Hill on generalizing it to include versions with level structure.

### Yiannis Sakellaridis (Rutgers-Newark)

Assistant Professor, Rutgers-Newark**Research Interests: ** Automorphic forms, representation theory and number theory**Website:** http://math.newark.rutgers.edu/~sakellar/**Title:** On the *L*-functions of affine spherical varieties**Abstract:** The study of period integrals of automorphic forms suggests that one should be able to attach *L*-functions (or rather, *L*-values) to (many) affine spherical varieties, but the meaning of these *L*-values remains mysterious. I will discuss several different ways in which these *L*-values come up in local harmonic analysis. Parts of the talk will be based on ongoing joint work with Delorme and Harinck, and with Ngô.

### Zhiwei Yun (Stanford University)

Assistant Professor, Stanford University**Research Interests: ** Geometric Representation Theory**Website:** http://www.stanford.edu/~zwyun/**Title: **Rigid local systems coming from automorphic forms**Abstract:** We will give a survey of recent progress on constructing local systems over punctured projective lines using techniques from automorphic forms and geometric Langlands. Applications include solutions of particular cases of the inverse Galois problem and existence of motives with exceptional Galois groups.

### Wei Zhang (Columbia University)

Associate Professor, Columbia University**Research Interests: ** Number theory, automorphic forms and related area in algebraic geometry**Website:** http://www.math.columbia.edu/~wzhang/**Title: **Selmer groups and the divisibility of Heegner points**Abstract:** This talk is about the proof of Kolyvagin’s conjecture in 1991 on *p*-indivisibility of (derived) Heegner points over ring class fields for ordinary primes *p* > 3 with some ramification conditions, with some application to the arithmetic of elliptic curves.

## Fall 2013 Speakers

### Yifeng Liu (MIT)

C.L.E. Moore Instructor, MIT**Research Interests: ** Number Theory, Algebraic Geometry**Website**: http://math.mit.edu/~liuyf**Title: **Arithmetic of Heegner points**Abstract: **In this talk, I will recall the construction of Heegner points on elliptic curves, or more generally, on Abelian varieties of GL(2)-type. These points are closely related to the arithmetic properties of the Abelian varieties from various aspects. After a brief summary of major developments in this direction, I will focus on a new work, joint with Shouwu Zhang and Wei Zhang, on the study of *p*-adic logarithm of Heegner points and the construction of some new *p*-adic *L*-functions of Rankin-Selberg type via the so-called universal *p*-adic Waldspurger periods.

### Melanie Matchett Wood (University of Wisconsin-Madison)

Assistant Professor, University of Wisconsin-Madison

American Institute of Mathematics Five Year Fellow**Research Interests: ** Number Theory, Algebraic Geometry**Website**: http://www.math.wisc.edu/~mmwood/**Title:** Semiample Bertini theorems over finite fields**Abstract:** When a hypersurface over a finite field is chosen randomly in a large multiple of an ample linear system, Poonen estimated the probability that such a hypersurface would be smooth by showing that smoothness at various points was independent. We answer this question for linear systems that are only large in a semiample direction, e.g. curves in **P**^{1}x**P**^{1} that have bidegree which is small in one component but large in the other. In these cases, smoothness at various points is no longer independent but we exactly characterize the dependence. Applications include finding the probability of smoothness of and distribution of points on curves in Hirzebruch surfaces over finite fields, a counterexample to Bertini over finite fields for embeddings into arbitrarily large projective spaces, and finding curves over finite fields with no points.

### Jason Starr (Stony Brook)

Associate Professor, Stony Brook**Research Interests:** Algebraic geometry**Website**: http://www.math.sunysb.edu/~jstarr/**Title:** Rational curves and rational points over global function fields**Abstract: **For a variety *X* over a global function field, e.g., *K *= *F _{q}*(

*t*), one obstruction to existence of a

*K*-point is the “elementary obstruction”

*e*(

*X*). Assuming

*e*(

*X*) vanishes, what “geometric” conditions guarantee existence of a

*K*-point? Building on earlier work with A. J. de Jong and Xuhua He, and using work of H. Esnault in an essential way, Chenyang Xu and I prove that “rationally simply connected” varieties, and specializations thereof, have

*K*-points if

*e*(

*X*) vanishes. Using this, we give uniform proofs and some extensions of early results of Tsen-Lang (

*K*is

*C*), Brauer-Hasse-Noether (period equals index for division algebras over

_{2}*K*) and Harder (the split case of Harder’s general proof of Serre’s “Conjecture II” for

*K*).

### Michael Thaddeus (Columbia University)

Associate Professor, Columbia University**Research Interests: ** Algebraic Geometry**Website**: http://www.math.columbia.edu/~thaddeus/**Title: **Group compactifications and principal bundles on nodal curves**Abstract:** Any group acts on itself on the left and right. For an algebraic group *G*, we may seek a compactification of *G* such that all *G*x*G*-orbit closures are smooth. For reductive *G*, such compactifications were classified by De Concini and Procesi. I will explain how such compactifications — and their generalizations in the orbifold setting — appear as moduli spaces of principal *G*-bundles on rational nodal curves. Then I will indicate how such compactifications may be used to study principal *G*-bundles on curves of higher genus.

## Spring 2013 Speakers

### Mark Behrens (MIT)

Associate Professor, MIT**Research Interests:** Algebraic topology**Website**: http://math.mit.edu/~mbehrens/**Title: **A Lie algebra model for unstable *v _{n}*-periodic homotopy

**Abstract:**Quillen-Sullivan rational homotopy theory encodes an unstable rational homotopy type in a commutative DGA, or equivalently, a DG Lie algebra. The former encodes the rational cohomology of the space, and the latter encodes the rational homotopy groups. I will describe an analogous theory for unstable

*v*-periodic homotopy, in the special case of spheres. This theory will be applied to relate unstable

_{n}*v*-periodic homotopy groups of spheres to the study of level structures on the Lubin-Tate formal group. This is joint work with Charles Rezk.

_{n}### Tasho Kaletha (Princeton University)

Veblen Research Instructor, Princeton University and IAS member**Research Interests:** the Langlands program, endoscopy, *p*-adic representation theory, harmonic analysis**Website**: https://web.math.princeton.edu/~tkaletha/**Title:** Epipelagic *L*-packets and rectifying characters**Abstract: **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, the Hiraga-Ichino-Ikeda formal degree conjecture, stability, endoscopic transfer, and compatibility with *GL _{n}*.

### Max Lieblich (University of Washington)

Associate Professor, University of Washington**Research Interests:** Algebraic geometry**Website**: http://www.math.washington.edu/~lieblich/**Title:** Recent progress on K3 surfaces**Abstract: **There has been an explosion of work on K3 surfaces in positive characteristic over the last several years, leading to significant progress on their derived categories, the structure of their moduli, and the Tate conjecture. I will discuss some of these results and a few of the main ideas that have helped break the logjam.

### Sophie Morel (Princeton University)

Professor, Princeton University**Research Interests:** the Langlands program**Website**: https://web.math.princeton.edu/~smorel/**Title: **Yet another application of the Arthur conjectures: the sign conjecture for Shimura varieties**Abstract: **The sign conjecture is a weakening of the Künneth standard conjecture; it predicts that there exists motives representing the even and odd parts of the cohomology of a smooth proper variety. I will explain this conjecture and some of its consequences, and then show how, for Shimura varieties, it can be deduced from Arthur’s conjectures and the work of Adams-Johnson and Vogan-Zuckerman on cohomological representations. This is joint work with Junecue Suh.

## Fall 2012 Speakers

### Bhargav Bhatt (IAS)

Member, IAS**Research Interests:** Algebra/algebraic geometry**Website**: http://www.math.ias.edu/~bhattb/**Title:** *p*-adic derived de Rham cohomology**Abstract:** A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The *p*-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine’s conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry — specifically, derived de Rham cohomology — and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.

### John Francis (Northwestern University)

Assistant Professor, Northwestern University**Research Interests:** Algebraic topology**Website**: http://www.math.northwestern.edu/~jnkf/**Title:** Factorization homology of topological manifolds**Abstract:** Factorization homology, or the topological chiral homology of Lurie, is a homology theory for manifolds conceived as a topological analogue of the homology of Beilinson & Drinfeld’s factorization algebras. I’ll describe an axiomatic characterization of factorization homology, generalizing the Eilenberg-Steenrod axioms for usual homology. These homology theories are determined by an interesting algebraic structure, that of an *n*-disk algebra, examples of which arise from commutative algebra, loop spaces, configuration spaces, Hochschild cohomology, metric Lie algebras, and Poisson geometry. An appropriate action of a 3-disk algebra on an associative algebra gives rise to a knot homology theory, studied in work joint with David Ayala & Hiro Lee Tanaka. I’ll conclude with some sample classes of calculations.

### Keerthi Madapusi Pera (Harvard University)

Benjamin Pierce Fellow, Harvard University**Research Interests:** Integral models of Shimura varieties and their compactifications, Hodge cycles on abelian varieties, Integral p-adic Hodge theory, Logarithmic Dieudonne theory**Website**: http://www.math.harvard.edu/~keerthi/**Title:** The Tate conjecture for K3 surfaces over fields of odd characteristic**Abstract:** The classical Kuga-Satake construction, over the complex numbers, uses Hodge theory to attach to each polarized K3 surface an abelian variety in a natural way. Deligne and Andre extended this to fields of characteristic zero, and their results can be combined with Faltings’s isogeny theorem to prove the Tate conjecture for K3 surfaces in characteristic zero. Using the theory of integral canonical models of Shimura varieties of orthogonal type, we extend the Kuga-Satake construction to odd characteristic. We can then deduce the Tate conjecture for K3s in this situation as well (with some exceptions in characteristic 3).

### Jared Weinstein (Boston University)

Assistant Professor, Boston University

Research Interests: Number Theory, Arithmetic Geometry, Automorphic Forms, Representation Theory

Website: http://math.bu.edu/people/jsweinst/

Title: Formal vector spaces

Abstract: Let K be a nonarchimedean local field with ring of integers OK. A formal OK-module is a collection of multivariate power series which behaves as if it were an OK-module. Formal OK-modules are indispensable to the study of the Galois representations of K, with Lubin-Tate theory being the most basic example. Following Faltings and Fargues-Fontaine, we introduce the notion of a formal K-vector space. The theme of the talk is that formal K-vector spaces are much simpler than formal OK-modules (as one might expect), even though one must sacrifice the Noetherian property. Our main theorem is a simple description of the Lubin-Tate tower at infinite level.