Seminars Sorted by Series

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...

The Boone-Higman conjecture (1973) predicts that a finitely generated group has solvable word problem if and only if it embeds in a finitely presented simple group. The "if" direction is true and easy, but the "only if" direction has been open for...

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...

Zimmer proposed the study of higher-rank semisimple group actions in the 1980's after showing certain rigidity results, especially about associated cocycles in the measurable category. In this talk, I will describe recent progress towards...

A translation surface is a closed surface that is obtained by gluing edges of a polygon in parallel. The group GL_2(R) acts on the collection translation surfaces of a fixed genus g. For a fixed translation surface S and t>0, we obtain a probability...

The classical Rokhlin Lemma asserts that for an aperiodic measure-preserving transformation $T$ of a probability space, one can find a "tower" of sets on which $T$ acts by translation and which covers almost all of the space. This result is a basic...

There has been considerable activity in establishing effective versions of equidistribution theorems in homogeneous dynamics in recent years. In this talk, we report on new results concerning effective equidistribution for one-parameter unipotent...

In recent years it has become possible to import some ideas and methods from homogeneous dynamics to the general smooth dynamics setting. I will discuss some results and open problems in this area.  

We discuss recent results on distribution of orbits of horocycle flows at subquadratic polynomial times (joint work with M. Radziwill) and at the prime counting function $\Omega$ (joint work with K. Loyd)

We present solutions to two problems on indefinite integral ternary quadratic forms. The first, highlighted by Margulis in 1990, concerns the distribution of the ternary Markoff spectrum associated with minima of forms. The second, initiated by...

In this talk, we show that the pair correlation of the Laplacian eigenvalues on 3-dimensional rectangular flat tori follows Poissonian statistics. Similar to the earlier work of Eskin, Margulis, and Mozes on 2D flat tori, these eigenvalues are...

The normal rank (or weight) of a group G is the smallest number of elements that normally generate G. This plays an important role in 3-manifold topology, but it is poorly understood. It is extremely difficult to give lower bounds apart from looking...

In this talk, we will discuss random dynamical systems and group actions on surfaces given by diffeomorphisms. We will mostly focus on the absolutely continuity of stationary measures, the classification of orbit closures, and the exact...

Rank 1 transformations are a long studied, flexible class of measure preserving transformations. King proved a variety of deep results about rank 1 rank transformations including: A) A rank 1 transformation’s centralizer is the weak closure of its...

The generalized Markoff equation gives rise to a dynamical system via the Markoff group acting on the solution set. This talk considers the resulting dynamics over finite fields, which has strong connections with combinatorial group theory and...

I will explain why a random walk on a simple homogeneous space equidistributes toward the Haar measure with an explicit rate, provided that the walk is not trapped in a finite invariant set and that the distribution driving the walk is Zariski-dense...

A similarity is a map from $R^d$ to itself that uniformly scales distances. When one repeatedly applies randomly chosen contracting similarities, the resulting Markov chain converges to a limiting stationary distribution known as a self-similar...

Let $n>2$. Weigold conjectured that if $G$ is a finite simple group then the product replacement graph on $n$-generating tuples is connected (which also implies that it is an expander for $n>3$). This conjecture is still (wide) open. I will discuss...

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...

We construct the tame part of a split anticyclotomic Euler system in a setting where local multiplicity one does not hold. Instead of the traditional zeta integral approach, we prove the existence of the necessary test vectors using a new method...

The past few years have seen rapid development in arithmetic statistics over function fields over finite fields, questions like:  what is the distribution of the ell-primary part of the class group of a random quadratic extension of F_q(t)?  (Cohen...

Kudla and Millson proved in the 80's that the generating series of special cycles in orthogonal and unitary Shimura varieties are modular forms and a well-established conjecture of Kudla asks about extensions to toroidal compactifications. In this...

Consider a collection of forms of odd degree with rational coefficients. Birch proved in 1957 that if the number of variables is sufficiently large, then the forms must have a nontrivial rational zero. The bounds resulting from Birch's proof...

Jacobi's theta function $\Theta(q) := 1 + 2q + 2q^4 + 2q^9 + \cdots $, and more generally the theta functions associated to positive-definite quadratic forms, have the property that they are modular forms of half-integral weight. The usual proof of...

The local intertwining relation (LIR) is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. It plays a central role in Arthur’s endoscopic classification for...

TBA

The goal of this talk is to discuss a new local-global phenomenon in the Langlands program on automorphic forms and Galois representations. Spaces of modular forms are parametrized by weights, and those same weights parametrize local deformation...