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...
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...
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...
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...
The Arithmetic Quantum Unique Ergodicity (AQUE) conjecture
predicts that the L2 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...
