Past Speakers

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) + q4/(1-q)(1-q2) + q9/(1-q)(1-q2)(1-q3) + … = 1/(1-q)(1-q4)(1-q6)(1-q9)…

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 C2-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 C2-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 g2, but the number of polarizations grows as fast as gg^2. One consequence is that most abelian varieties of dimension 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) 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 GL3(Qp)
Abstract: I will discuss the hypothetical mod p Langlands correspondence for GL3(Qp), which should associate to a 3-dimensional mod p representation of the Galois group of Qp one (or many) smooth mod p representation(s) of GL3(Qp). 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 Fp-bar. We will show (under suitable assumptions) that this Galois-theoretic invariant is determined by the GL3(Qp)-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
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
Title: SO(3)-multiplicities of SL3(R)-representations
Abstract: Given a topological representation of a noncompact real Lie group like SL3(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
Title: Constructing elliptic curves from Galois representations
Abstract: Given an elliptic curve E over a finitely generated field K, the Tate module Vp(E) is a representation of the Galois group GK that determines E up to isogeny. It is an interesting problem to determine which representations of GK 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).

Kirsten Wickelgren (Georgia Tech)

Assistant Professor, Georgia Institute of Technology
Research Interests: Algebra, geometry, and topology
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 = P1 – {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 P1 – {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
Title: Patching and p-adic local Langlands
Abstract: The p-adic local Langlands correspondence is well understood for GL2(Qp), but appears much more complicated when considering GLn(F), where either n > 2 or F is a finite extension of Qp. I will discuss joint work with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas and Sug Woo Shin, in which we approach the p-adic local Langlands correspondence for GLn(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
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
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
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
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
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
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 P1xP1 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
Title: Rational curves and rational points over global function fields
Abstract: For a variety X over a global function field, e.g., K = Fq(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 C2), Brauer-Hasse-Noether (period equals index for division algebras over 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
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 GxG-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
Title: A Lie algebra model for unstable vn-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 vn-periodic homotopy, in the special case of spheres. This theory will be applied to relate unstable vn-periodic homotopy groups of spheres to the study of level structures on the Lubin-Tate formal group. This is joint work with Charles Rezk.

Tasho Kaletha (Princeton University)

Veblen Research Instructor, Princeton University and IAS member
Reasearch Interests: the Langlands program, endoscopy, p-adic representation theory, harmonic analysis
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 GLn.

Max Lieblich (University of Washington)

Associate Professor, University of Washington
Reasearch Interests: Algebraic geometry
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
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
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
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 (Havard 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
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
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.