פעילויות השבוע

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).

The central limit theorem (CLT) and related results for stationary weakly dependent sequences of random variables have been extensively studied in the past century, starting from a pioneering work of Berenstien (1927). However, in many physical phenomena there are external forces, measurement errors and unknown variables (e.g. storms, the observer effect, the uncertainty principle etc.). This means that the local laws of physics depend on time, and it leads us to studying non-stationary sequences.

The asymptotic behaviour of non-stationary sequences have been studied extensively in the past decades, but it is still developing compared with the theory of stationary processes. In this talk we will focus on inhomogeneous Markov chains. For sufficiently well contracting Markov chains the CLT was first proven by Dobrushin (1956). Since then many results were proven for stationary chains. In 2021 Dolgopyat and Sarig proved local central limit theorems (LCLT) for inhomogeneous Markov chains. In 2022 Dolgopyat and H proved optimal CLT rates in Dobrusin‘s CLT. These results closed a big gap in literature concerning the non-stationary case.

An open problem raised by Dolgopyat and Sarig in their 2021 book concerns limit theorems for Markov shifts, that is when the underlying sequence of functions that forms the partial sums depend on the entire path of the chain. Two circumstances where such dependence arises are products of random matrices and random iterated functions, and there are many other instances when the functionals depend on the entire path.

In this talk we will present our solution to the above problem. More precisely, we prove CLT, optimal CLT rates and LCLT for a wide class of sufficiently well mixing Markov chains and functionals with infinite memory. Even though the inhomogeneous case is more complicated, our results seem to be new already for stationary chains.

ב-1911, Brouwer הוכיח (כמסקנה ממשפט נקודת השבת שלו) שמרחבים אוקלידיים מממדים שונים אינם הומאומורפיים. תוצאה זו הובילה למושג של ממד טופולוגי שהוא מספר טבעי שמוגדר לכל מרחב מטרי. נציג גישות שונות להגדרת ממד טופולוגי ותוצאות קלאסיות בתורת הממד.