Seminars Sorted by Series

It is a classical fact that countable groups of homeomorphisms of the interval and the circle are characterized by purely algebraic properties, namely linear and circular orderability. It remains a difficult and deep problem to understand countable...

Fibered 3-manifolds are those constructed via surface homeomorphisms. Given such a manifold with pseudo-Anosov monodromy, much is already known about how topological data of the mapping class determine geometric information about the hyperbolic 3...

Consider a closed surface M of genus greater than or equal to 2. For negatively curved metrics on M and their corresponding geodesic flow, we can study the topological entropy, the Liouville entropy, and the mean root curvature. In 2004, Manning...

Thurston gave an explicit construction of pseudo-Anosov elements 
in the mapping class group of a compact surface, using Dehn twists in 
pairs of filling multicurves.  We show that the probability that a random 
walk of length $n$ on the mapping class...

The theory of hierarchically hyperbolic groups, due to Behrstock, Hagen, and Sisto, was developed by abstracting work of Masur and Minsky on mapping class groups. Study of the large scale geometry of the outer automorphism group $\operatorname{Out}...

Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on...

Given a bounded sequence of zero mean, we study the problem of its orthogonality to all continuous  observables in topological dynamical systems whose all invariant measures yield measure-theoretic systems belonging to a fixed characteristic class...

Given a one-parameter degenerating family of rational maps on the projective line, it is possible to construct a non-archimedean limit which captures how this family degenerates.  Recently, Luo used ultrafilters to construct limits for an arbitrary...

Let $\Gamma$ be a countable group with a Cayley graph G. Suppose we are running an independent identical probabilistic algorithm on each vertex (or each edge) and vertices are only allowed to communicate with their neighbors. The goal of this...

Let $X$ be a K3 surface. Consider a finitely supported probability measure $\mu$ on $\operatorname{Aut}(X)$ such that $\Gamma_{\mu} = \langle \operatorname{Supp}(\mu)\rangle Aut(X)$ is non-elementary. We do not assume that $\Gamma_{\mu}$ contains...

I will discuss random walks on manifolds and pose questions of stiffness and rigidity of stationary measures.  In the case of conservative random walks, I will discuss criteria that guarantee stiffness of stationary measures and certain rigidity...

Let $q,Q$ be two integral quadratic forms in $m n$
variables. One can ask when $q$ can be represented by $Q$ - that is,
whether there exists an $n \times m$-integer matrix $T$ such that $ Q \circ T = q $.  Naturally, a necessary condition is that...

In this talk I will describe new results establishing the existence of a strong spectral gap for  geometrically finite, n dimensional, hyperbolic manifold with critical exponent greater than $(n-1)/2$. This settles a conjecture of Mohammadi and Oh...

The geodesic flow (for the hyperbolic metric) of an infinite Riemann surface is ergodic if and only if Brownian motion is recurrent, which is also equivalent to the divergence of the Poincaré series. Surfaces with ergodic geodesic flows are most...

Let K be the function field of a smooth projective curve B over the complex numbers and let g be a positive integer. The uniform boundedness conjecture predicts that there exists a constant N, depending only on g and K, such that for any g...

In characteristic zero, Castelnuovo proved that every unirational surface is rational. In positive characteristic, this fails dramatically: there exist many non-rational, often even general-type, surfaces that are nevertheless unirational. In 1977...

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. In a series of papers, Venkatesh and his collaborators proposed an arithmetic reason for...

The theory of Euler systems, first developed by Thaine and Kolyvagin, has become a central tool for proving cases of the Birch–Swinnerton-Dyer and Bloch–Kato conjectures. Many of the known examples are inspired from automorphic period integrals that...

The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In...

In this talk, I will focus on simultaneous non-vanishing results for Dirichlet L-functions at the central point 1/2. Specifically, I will describe non-vanishing results (conditional on GRH) for two L-functions associated to Dirichlet characters in...

TBD

A Diophantine upper bound on the dimensions of certain spaces of holonomic functions was the main ingredient in our proof with Calegari and Tang of the 'unbounded denominators conjecture' (presented by Tang in last year's number theory seminar) from...

Modular forms are degree zero cohomology of certain invertible sheaves on modular curves. One is often led to consider also higher cohomology of automorphic vector bundles on Shimura varieties. We try to define and understand the integral coherent...

The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field...

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture...

Tate reformulated the theory of the Riemann zeta function and its functional equation as the Mellin shadow of the Fourier transform on a certain space of function on the adeles. Conjecturally, Langlands' general automorphic L-functions and their...

The linearization method of Dani-Margulis controls the amount of time a unipotent trajectory spends near invariant subvarieties of a homogeneous space. I will describe a problem in number theory where a similar control is desired for diagonalizable...

Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation $\pi_0$ at infinity and level dividing some finite upper bound. We compute statistics of this family as the...

I will attempt to give a gentle introduction to the categorical p-adic Langlands program and its connections to questions about congruences between modular forms.

Given a set of integers, we wish to know how many primes there are in the set.  Modern tools allow us to obtain an asymptotic for the number of primes, or at least a lower bound of the expected order, assuming certain strength Type-I information...

I will report on a series of joint works with Gehrmann, Guitart and Masdeu about a plectic generalization of Darmon's Stark-Heegner (SH) points. 

We constructed plectic SH points as p-adic points on elliptic curves and we expect them to control the...

The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves.  Here we describe a proof of an asymptotic for the second moment of...

The circle method has been a versatile tool in the study of rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to information about moduli spaces of rational...

The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed...

Orbits of many coregular representations of algebraic groups are closely linked to moduli spaces of genus one curves with extra data. We may use these orbit parametrizations to compute the average size of Selmer groups of elliptic curves in certain...

We prove that at least 2/21 of all integers can be written as a sum of two rational cubes, and at least 1/6 of all integers cannot. More generally, in any cubic twist family of elliptic curves, at least one 1/6 of curves have rank 0 and at least 1/6...

Let p be a prime. I want to explain how to use the geometry of modular curves at infinite level and the Hodge–Tate period map to study regular de Rham p-adic Galois representations appearing in the p-adically completed cohomology of modular curves...

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an...

I discuss the spectral and arithmetic side of the relative trace formula of Kuznetsov type for congruence subgroups of SL(n, Z) with applications to automorphic density theorems. A particular focus is on properties of general Kloosterman sums as...

In 2017, Radchenko-Viazovska proved a remarkable interpolation result for even Schwartz functions on the real line: such a function is entirely determined by its values and those of its Fourier transform at square roots of integers. We give a new...

I will talk about a program aimed at exploiting randomness of certain "graphs of symbols" in determining the precise structure of the pro-nilpotent Galois groups of number fields, and how this has been successfully implemented in nilpotency class 2...

Given an automorphic representation \pi of SO(n,n+1) with certain nice properties at infinity, one can nowadays attach to \pi a p-adic Galois representation R of dimension 2n. The Bloch--Kato conjectures then predict in particular that if the L...

We discuss the integrality of theta liftings of anti-cyclotomic characters to a definite unitary group of two variables. This will allow us to construct a Hida family of the theta liftings and relate the congruence module of which to an anti...

We prove this bound by first using the unitary Ichino-Ikeda formula of N. Harris to relate the central L-value to an automorphic period integral.  There is a `trivial' bound for this integral, which turns out to correspond to the convexity bound for...

In joint work with Denis Benois, we give an étale construction of Bellaïche's p-adic L-functions about θ-critical points on the Coleman–Mazur eigencurve. I will discuss applications of this construction towards leading term formulae in terms of p...

In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland...

Oort conjectured in 1995 that isogeny classes in the moduli space A_g of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. There is a straightforward generalisation of this conjecture...

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this...

The class number formula describes the behavior of the Dedekind zeta function at s=0 and s=1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s=0 and s=1 in...