Activities This Week

The talk focuses on the general behavior of discrete weakly o-minimal structures, and their definable sets and functions. Comparisons are drawn with the dense weakly o-minimal case and with the discrete o-minimal case. We develop tools to describe definable maps in these structures, and ultimately state a prove a monotonicity theorem. We also state, using the results given so far, how we hope to prove that no definable groups exist in the discrete, weakly o-minimal setting.

In this talk I will give an overview of Popa’s deformation/rigidity theory for von Neumann algebra factors. The main motivation for this theory is the question of when an isomorphism of factors arising from group actions comes from an isomorphism of the groups. After providing some background on early results, examples of using deformation/rigidity to prove structural results will be given. Topics discussed in this talk include deformations, Cartan subalgebras, and intertwining of subalgebras.

There are problems of optimization for which one can optimize locally or one can have a finitely additive measurable outcome, but never both together. This is a connection to the Banach-Tarski Paradox and non-amenable group and semi group action.

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.