## 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 (https://arxiv.org/abs/1902.03719), 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

**Reasearch 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

Reasearch 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.