Activities This Week

It is well known that a compact metric space embeds in a finite-dimensional Euclidean space if and only if it has finite Lebesgue covering dimension. In 1999, Gromov introduced the notion of mean dimension, a fundamental invariant for dynamical systems that could be considered as a dynamical analogue of Lebesgue covering dimension.

A (discrete time) dynamical system is a pair (X,\Phi), where $\Phi:X\to X$ be a homeomorphism of a compact metric space $X$. The \emph{shift embeddability} or \emph{sampling-rate problem} asks: Under what conditions do there exists a finite number of continuous real-valued functions $f_1,\ldots,f_d:X \to \mathbb{R}$ so that a point $x \in X$ can be uniquely recovered by sampling the values of the $f_1,\ldots,f_d$ along the orbit of $x$?

In this talk I will describe historical developments around the shift embeddability problem and some exciting recent developments.

We will not assume any specialized background (in particular, we will recall the definition of Lebesgue covering dimension).

בהרצאה נעסוק במרחבי הילברט של פונקציות בעלי גרעין משחזר. נבנה את יסודות התיאוריה ונבחן את הדוגמה המרכזית – מרחב הארדי על הדיסק. באמצעות הכלים שנפתח, ובמעט אלגברה לינארית, נראה כיצד ניתן לשחזר באופן אלגנטי שני משפטים מרכזיים באנליזה מרוכבת. אם יתיר הזמן, נדון גם בכמה שאלות יפות שפתרונן נשען על רעיונות מתוך התיאוריה. ההרצאה מתאימה לסטודנטים משנה ב׳ ומעלה.

I will provide an overview of the model theoretic notions of internality and internal covers, and their relation to definable groupoids. I will then indicate a generalization of these notions to higher homotopical dimension

This is the continuation of the talk from the previous week

The classical Fourier transform is an ubiquitous operator acting on L^2(V) for a finite-dimensional quadratic space V. We study it from the point of view of representation theory. Together with other operators it forms a remarkable representation of a metaplectic group on L^2(V), that has minimal functional dimension. Minimal representation of other groups, often have models on L^2(X) for a cone X. We shall see how to define generalized Fourier transforms on L^2(X) and discuss their properties.

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