פעילויות השבוע

Let $x \to x+ \alpha$ be a rotation on the circle and let $\varphi$ be a function with bounded variation. Denote by $S_n(\varphi, x) := \sum_{j=0}^{n-1} \varphi(x+j \alpha)$ the ergodic sums.

For a large class of $\alpha$‘s including irrationals with bounded partial quotients, we show decorrelation inequalities between the ergodic sums at time $q_k$, where the $q_k$‘s are the denominators of $\alpha$.

This allows to study the asymptotic distribution of the ergodic sums $S_n(\varphi, x)$ after normalization, in particular for some step functions, along subsequences.

We will give also an application to a geometric model, the billiard flow in the plane with periodic rectangular obstacles when the flow is restricted to special directions.

The family of high rank arithmetic groups is class of groups which is playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more. A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity. We will talk about a new type of rigidity : ”first order rigidity“. Namely if D is such a non-uniform characteristic zero arithmetic group and E a finitely generated group which is elementary equivalent to it ( i.e., the same first order theory in the sense of model theory) then E is isomorphic to D. This stands in contrast with Zlil Sela‘s remarkable work which implies that the free groups, surface groups and hyperbolic groups ( many of which are low-rank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them. Joint work with Nir Avni and Chen Meiri.

I will discuss the KdV integrable hierarchy, and its tau functions and wave functions.

Witten conjectured that the tau functions are the generating functions of intersection numbers over the moduli of curves (now Kontsevich‘s theorem). Recently the following was conjectured: The KdV wave function is a generating function of intersection numbers on moduli of ”Riemann surfaces with boundary“ (Pandharipande-Solomon-T,Solomon-T,Buryak).

I will describe the conjecture, its generalization to all genera (Solomon-Tessler), and sketch its proof (Pandharipande-Solomon-T in genus 0, T,Buryak-T for the general case). If there will be time, I‘ll describe a conjectural generalization by Alexandrov-Buryak-T.