BGU Probability and Ergodic Theory (PET) seminar Ical Atom

Let $\Gamma$ be a discrete countable infinite Group and let $X$ be compact minimal $\Gamma$-space. A $\Gamma$-compactification by $X$ is a compact topology on $\Gamma\cup X$ on which $\Gamma$ acts continuously by left multiplication and the original action on $X$ respectively, and such that $\Gamma$ is dense in $\Gamma\cup X$.

Is there more than one way to “glue” $X$ to $\Gamma$ in such a way? Are there canonical families of $\Gamma$ compactifications? and what all of this has to do with the old and famous Brouwer’s non-retract theorem from classical topology?

Based on a joint work with Yair Hartman, Aranka Hrušková & Mehrdad Kalantar

Let w(x,y) be a word in a free group. For any group G, w induces a word map w:G^2–>G. For example, the commutator word w=xyx^(-1)y^(-1) induces the commutator map. In the setting of finite simple groups, Larsen, Shalev and Tiep showed there exists epsilon(w)>0 (depending only on the word w), such that for all sufficiently large G, the probability that a random pair (g_1,g_2) in G^2 satisfies w(g_1,g_2)=g is smaller than |G|^(-epsilon(w)). They further obtained uniform upper bounds on the L^1- and L^infty-mixing times for the random walks induced by the corresponding word measures.
I will discuss analogous results for the family of unitary groups in all ranks.

(joint work with Tattwamasi Amrutam and Eli Glasner) Let (X,\Gamma) be a minimal equicontinuous (or more generally rigid) topological dynamical system, with a discrete countable acting group. Intermediate C^-algebras of the form C^_r(\Gamma) < \mathcal{A} < C(X) \rtimes \Gamma, can be thought of as non-commutative generalizations of \Gamma-factors X \rightarrow Y as each such factor gives rise to an intermediate algebra of the form \mathcal{A} = C(Y) \rtimes \Gamma. When the group \Gamma is Gromov hyperbolic we show that this is the only possible source of intermediate algebras. The proof relies on a delicate interplay betwee two actions: The given dynamical system (X,\Gamma) and the boundary action (Z,\Gamma).

The theory of characters of infinite groups, initiated by Thoma, is a generalization of the representation theory of finite groups. More explicitly, a character of a group is an (extremal) conjugation invariant positive definite function. A group said to be character rigid if every character of the group is either supported on the center or comes from a finite dimensional representation. Connes conjecture that any irreducible lattice in a higher rank Lie group is character rigid. Surprisingly, this conjecture is a generalization of the celebrated Margulis normal subgroup theorem and of the Stuck-Zimmer conjecture on IRS rigidity. I will discuss a recent joint work with Alon Dogon, Yuval Gorfine, Liam Hanany, and Arie Levit showing that any nonuniform higher rank lattice is character rigid, proving the Stuck-Zimmer conjecture for such lattices.

There are problems of optimization for which one can optimize locally or one can have a finitely additive measurable outcome, but never both together. This is a connection to the Banach-Tarski Paradox and non-amenable group and semi group action.

The Frenkel-Kontorova model is a standard model in solid state physics describing particles having nearest neighbor interactions. Mathematical analysis of this model leads to studying standard-like twist maps. In the 80’s Suris found a remarkable family of potentials for this model with integrable dynamics. In some sense this is similar to the role that ellipses play in planar billiards. In the talk we will highlight this connection, via the action-angle coordinates of the two systems. Then we will also show that an integrable pertrubation of a Suris potential has to be a Suris potential itself. This is in spirit of local results proven for the Birkhoff conjecture in billiards. The proof relies heavily on Fourier anlaysis, as well as construction of a suitable basis for L2 wihch captures the dynamics of the system. Joint work with Corentin Fierobe

We will discuss questions of invertibility of large random matrices, and estimating the smallest singular value of random matrices in the non-asymptotic setting, via methods of asymptotic geometric analysis. We will discuss a construction of an efficient net on the high-dimensional sphere, and its applications to the questions of invertibility of inhomogeneous and log-concave random matrices.

The study of Fourier decay for stationary measures is a classical problem, with roots in Erdős’ 1939 work on Bernoulli convolutions. Over the years, tools and ideas from number theory, smooth dynamics, and arithmetic combinatorics have led to substantial progress, yet several fundamental questions remain open. After introducing the problem and its history, I will present recent joint work with Federico Rodriguez Hertz and Zhiren Wang, establishing essentially optimal results for a large class of stationary measures with respect to smooth nonlinear maps.

We count the number of minimally ramified quartic dihedral exten- sions of the rational numbers by discriminant unconditionally, and by the product of ramified primes assuming the Generalized Riemann Hypothe- sis.