פעילויות השבוע

A self-interacting random walk is a random process evolving in an environment which depends on its history. In this talk, we will discuss a few examples of these walks including the Lorentz gas, the mirror walk, the once-reinforced walk and the cyclic walk in the interchange process. I will present methods to analyze these walks in high dimensions and prove that they behave diffusively.

The talk is based on joint works with Allan Sly, Felipe Hernandez, Antoine Gloria, Gady Kozma and Lenya Ryzhik.

We shall rewrite in the simplicial language the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the standard definitions based on syntax, making them formally unnecessary. However, in the lectures that I shall explain these definitions both in standard language, and in the simplicial, diagram-chasing language. We shall assume only basic familiarity with category theory and model theory.

In this approach we view a complete first-order theory as a symmetric simplicial object in the category of profinite sets and open continuous maps, defined by the functor sending a finite set of variables into the Stone space of complete types in those variables. A model of a complete first-order theory is then a morphism from a representable simplicial set satisfying certain lifting properties reminiscent of, but weaker then, those in the definition of a fibration. The class of simplicial profinite sets corresponding to complete first order theories is characterised by the same lifting properties required of the map from the simplicial covering space (decalage) forgetting the extra degeneracy.

In a concise manner our simplicial reformulations are presented in the notes

A $C^\ast$-probability space is a pair $(A,\tau)$ consisting of a $C^\ast$-algebra and a tracial state $\tau$ on $A$. For any two $C^\ast$-probability spaces, there‘s a definition of a reduced free product $C^\ast$-algebra $(A,\tau) \ast_r (B,\sigma)$. This is a generalization of the case of reduced group $C^\ast$-algebras: if $G$ and $H$ are discrete groups, then the reduced free product of $C^\ast_r(G)$ and $C^\ast_r(H)$ is the reduced group $C^\ast$-algebra of the free product $G \ast H$. We show that if $A$ decomposes as a nontrivial reduced free power of infinitely many copies of separable $C^\ast$-probability spaces, then $C([0,1]) \ast_r A$ is isomorphic to $A$. Several other related isomorphism theorems are obtained as well. I will review some background and outline the proof. This is joint work with N. Christopher Phillips.

Harish Chandra developed representation theory on real and p-adic groups using analytic objects such as matrix coefficients and his distributional characters. He found that density properties of these distributions in the class of all invariant distributions plays an important role in Establishing basic results of Harmonic analysis on the group. Moving to G-Homogeneous spaces, spherical characters are distributions that play an important role in the relative trace formula. These objects were studied extensively in special cases and important results were obtained by Rallis and Rader who formulated natural density problems regarding these distributions. In a joint work with A. Aizenbud (Weizmann) and J. Bernstein (Tel-Aviv), we introduce some algebraic methods based on the concept of Cohen-Macaulay and Bernstein‘s theory of representations of p-adic groups to tackle some of these density problems in the -adic case. In my presentation, I will not assume knowledge of Bernsein‘s theory or knowledge of Harmonic analysis on p-adic groups.

(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).