Activities This Week
BGU Probability and Ergodic Theory (PET) seminar
Asymptotic distributions for normalized ergodic sums over rotations
Nov 21, 11:00—12:00, 2017, 201
Speaker
Jean-Pierre Conze (Rennes)
Abstract
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.
Colloquium
First order rigidity of high-rank arithmetic groups
Nov 21, 14:30—15:30, 2017, Math -101
Speaker
Alex Lubotzky (Hebrew University)
Abstract
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.
Algebraic Geometry and Number Theory
Integrable hierarchies, wave functions and open intersection theories
Nov 22, 15:10—16:30, 2017, Math -101
Speaker
Ran Tessler (ETH)
Abstract
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.
A night of mathematics and Jazz
Nov 26, 18:00—22:00, 2017, Math -101
All faculty and students are cordially invited.
Program
18:00 Undergraduate math club
Barak Weiss, Tel Aviv University
The illumination problem
The following elementary problem in geometry is still open: given a polygon $P$ in the plane, say that points $x$ and $y$ in $P$ see each other if there is a billiard path from $x$ to $y$. Is there a polygon in which infinitely many pairs of points do not see each other?
Such problems turn out to be easy to state but very difficult to solve. I will explain this and related questions in greater detail, and describe some recent progress which relies on well-known work of Eskin and Mirzakhani (part of the late Maryam Mirzakhani’s Field medal citation).
19:00 A Jazz show by Tsachik and the perverse sheaves
Ehud Ettun on bass
Haruka Yabuno on Piano
Tsachik Gelander on drums
Operator Algebras and Operator Theory
Completely Positive Noncommutative Kernels
Nov 27, 16:00—17:00, 2017, -101
Speaker
Gregory Marx (BGU)
Abstract
It is well known that a function $K: \Omega \times \Omega \to \mathcal{L}(\mathcal{Y})$ (where $\mathcal{L}(\mathcal{Y}$) is the set of all bounded linear operators on a Hilbert space$\mathcal Y$) being (1) a positive kernel in the sense of Aronszajn (i.e. $\sum_{i,j=1}^N \langle K(\omega_i , \omega_j) y_j, y_i \rangle \geq 0$ for all $\omega_1, \dots, \omega_N \in \Omega$, $y_1, \dots, y_N \in \mathcal Y$, and $N=1,2,\dots$) is equivalent to (2) $K$ being the reproducing kernel for a reproducing kernel Hilbert space $\mathcal H (K)$, and (3) $K$ having a Kolmogorov decomposition $K(\omega, \zeta)=H(\omega)H(\zeta)^*$ for an operator-valued function $H: \Omega \to \mathcal{L}(\mathcal X, \mathcal Y)$ where $\mathcal X$ is an auxiliary Hilbert space.
In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative functions with the target set $\mathcal L ( \mathcal Y)$ of $K$ replaced by $\mathcal L (\mathcal A, \mathcal L (\mathcal Y))$ where $\mathcal A$ is a $C^*$-algebra. In my talk, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring’s dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.