Seminars Sorted by Series

Studying the \’etale cohomology of Shimura varieties with Hecke and Galois actions provides an avenue toward understanding the Langlands correspondence. 
While the structure of the rational cohomology groups is predicted conjectures of Kottwitz and...

The analytic de Rham stack is a new construction in Analytic Geometry whose theory of quasi-coherent sheaves encodes a notion of p-adic D-modules. It has the virtue that can be defined even under lack of differentials (eg. for perfectoid spaces or...

Kazhdan-Lusztig theory provides a pattern of applying tools of algebraic geometry,
such as the theory of Frobenius or Hodge weights, to numerical problems of representation theory.
These techniques have been used in representation theory over a field...

Ohta described the ordinary part of the 'etale cohomology of towers of modular curves in terms of Hida families. Ohta's approach crucially depended on the one-dimensional nature of modular curves. In this talk, I will present joint work with Chris...

We show how the language of motivic non-archimedean homotopy theory can be used to define p-adic cohomology theories and prove new results about them. For example, we show how to define solid relative rigid cohomology and deduce a version of

I will discuss a theorem obtained in joint work with Wyss and Ziegler, which is devoted to describing the Frobenius traces for the IC sheaf on moduli space of objects in symmetric abelian categories linear over a finite field. The formula is...

Pretannakian categories, that is k-linear abelian categories with finite dimensional Hom groups, given with a commutative and associative tensor product, with a unit object (such that End(1)=k) and duals, can be viewed as generalizations of linear...

Let X be a smooth rigid-analytic space over C_p. In contrast to algebraic geometry, it turns out that there are many pro-etale Q_p local systems on X that do not admit any Z_p-lattice. Furthermore, cohomology of these local systems often fail to be...

Arthur proposed a description of automorphic forms in terms of tempered automorphic forms for centralizers of SL2 homomorphisms. I will explain a point of view on the Arthur parameterization in the setting of function fields coming from relative...

Gromov--Witten invariants and Welschinger invariants count curves over the complex and real numbers. In joint work with J. Kass, M. Levine, and J. Solomon, we gave arithmetically meaningful counts of rational curves on smooth del Pezzo surfaces over...

Let X be a complex algebraic variety, and X à D a proper morphism to a small disk which is smooth away from the origin. In this setting, the higher direct images of the constant sheaf form a local system on the punctured disk, and the Local...

I will talk about a proof of local-global compatibility at p for higher coherent cohomology mod p in weight one for Hilbert modular varieties at an unramified prime, assuming we are not in middle degree. I will discuss some key ingredients to the...

If X is a smooth proper rigid variety over C_p and L is a Z_p-local system on X, the cohomology groups H*(X,L) are finitely generated Z_p-modules by a basic result of Scholze. If L is merely a Q_p-local system, its cohomology groups are still finite...

The inverse Galois problem asks for finite group G, whether G is a finite Galois extension of the rational numbers. Malle’s conjecture is a quantitative version of this problem, giving an asymptotic prediction of how many such extensions exist with...

A top degree differential form \omega on a smooth algebraic variety X over a local field K gives rise to a (real valued) measure on X(K). The Serre duality yields a natural isomorphism between 1-dimensional spaces of global top degree forms on an...

The Arithmetic Quantum Unique Ergodicity (AQUE) conjecture predicts that the $L^2$ mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact...

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of...

Given a compact hyperbolic surface together with a suitable choice of orthonormal basis of Laplace eigenforms, one can consider two natural spectral invariants: 1) the Laplace spectrum $\Lambda$, and 2) the 3-tensor $C_{ijk}$ representing pointwise...

Teichmuller dynamics give us a nonhomogeneous example of an action of SL_2(R) on a space H_g preserving a finite measure. This space is related to the moduli space of genus g curves. The SL_2(R) action on H_g has a complicated behavior: McMullen...

Let w be a word in a free group. A few years ago, Magee and I, relying on a work of Calegari, discovered that the stable commutator length of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier...

In this talk we will discuss some aspects of the question: given a group, what is the range of Furstenberg entropy of ergodic stationary actions of it? For the special linear group and its lattices, constraints on this spectrum come from Nevo-Zimmer...

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple planar graphs on n vertices, answering a question from 1969. The proof uses a connection with continued fractions and advances towards Zaremba’s...

We will discuss recent results towards the quantum unique ergodicity conjecture of Rudnick and Sarnak, concerning the distribution of Hecke--Maass forms on hyperbolic arithmetic manifolds. The conjecture was resolved for congruence surfaces by...

 We prove ''reasonable'' quantitative bounds for sets in $\mathbb{Z}^2$ avoiding the ${polynomial}$ ${corner}$ ${configuration}$ $(x,y), (x+P(z),y), (x,y+P(z))$, where $P$ is any fixed integer-coefficient polynomial with an integer root of...

Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the difference between the largest and second-largest eigenvalues—measures the expansion properties of a...

A common heuristic in Diophantine approximation asserts that different arithmetic expansions (such as digit and continued-fraction expansions) should be independent of one another. In some cases, these elementary heuristics have led to surprisingly...

We discuss what a "typical" short curve on a random large genus hyperbolic surface looks like. In particular, for each $L$, there are finitely many curves of length at most $L$. We find length scales at which such a curve chosen at random is highly...

The Bergelson conjecture from 1996 asserts that the multilinear polynomial ergodic averages with commuting transformations converge pointwise almost everywhere in any measure-preserving system. This problem was recently solved affirmatively for...

Every braid can be thought of as a homeomorphism of a punctured disc. Morally, the more complicated a braid is, the more dynamics is contained in the corresponding homeomorphism, which one can quantify using topological entropy. In particular, one...

The plan of the talk is to describe joint work with A. Brown and Z. Wang on the smooth classification of actions of lattices in $\mathrm{SL}(n,\mathbb{R})$ on $n-1$ dimensional manifolds. The method is also amenable to show rigidity for some...

The fundamental group of an n-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space $H^n$. If n is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions...

Motivated by the quantum unique ergodicity conjecture, we examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic $n$-manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic...

The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in $n \geq$ 3 variables, the image set $Q(Z^n)$ of integral vectors is a dense subset of the real line. Determining the...