Joint IAS/PU Groups and Dynamics Seminar

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...

The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in n≥

3 variables, the image set Q(Zn) of integral vectors is a dense subset of the real line. Determining the distribution...

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

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 SL(n,ℝ)

on n−1 dimensional manifolds. The method is also amenable to show rigidity for some boundary actions.

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 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...

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...

We prove ''reasonable'' quantitative bounds for sets in ℤ2

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 multiplicity 1. This simultaneously...

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...