Activities This Week
BGU Probability and Ergodic Theory (PET) seminar
Intermediate subalgebras of commutative crossed products of discrete group actions. Online
Oct 21, 11:10—12:00, 2021, Building 34, room 14
Speaker
Tattwamasi Amrutam (Ben-Gurion University)
Abstract
In this talk, we shall focus our attention on intermediate subalgebras of $C(X)\rtimes_r\Gamma$ (and $L^{\infty}(X,\nu)\ltimes\Gamma$). We begin by describing the construction of the commutative crossed product $C(X)\rtimes_r\Gamma$ and how the group contributes to its structure. We shall talk about various (generalized) averaging properties in this context. As a first application, we will show that every intermediate $C^*$-subalgebra $\mathcal{A}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{A}\subseteq C(X)\rtimes_r\Gamma$ is simple for an inclusion $C(Y)\subset C(X)$ of minimal $\Gamma$-spaces whenever $C(Y)\rtimes_r\Gamma$ is simple. We shall also show that, for a large class of actions of $C^*$-simple groups $\Gamma\curvearrowright X$, including non-faithful action of any hyperbolic group with trivial amenable radical, every intermediate $C^*$-algebra $\mathcal{A}$, $C_r^*(\Gamma)\subset \mathcal{A}\subset C(X)\rtimes_r\Gamma$ is a crossed product of the form $C(Y)\rtimes_r\Gamma$, $C(Y)\subset C(X)$ is an inclusion of $\Gamma$-$C^*$-algebras.
Non-commutative Analysis Seminar
Generalized Powers’ averaging for commutative crossed products
Oct 26, 11:00—12:00, 2021, seminar room -101
Speaker
Tattwamasi Amrutam (BGU)
Abstract
In 1975, Powers proved that the free group on two generators is a $C^{\star}$-simple group. The key insight in Powers’s proof of the $C^\star$-simplicity is that the left regular representation of $\mathbb{F}_2$ satisfies Dixmier type averaging property. Using the pioneering work of Kalantar-Kennedy, it was shown by Haagerup and Kennedy independently that the $C^\star$-simplicity of the group $\Gamma$ is equivalent to the group having Powers’ averaging property. In this talk, we introduce a generalized version of Powers’ averaging property for commutative crossed products. Using the notion of generalized Furstenberg boundary introduced by Kawabe and Naghavi (independently), we show that the simplicity of the commutative crossed products $C(X)\rtimes_r\Gamma$ (for minimal $\Gamma$-spaces $X$) is equivalent to the crossed product having generalized Powers’ averaging. As an application, we will show that every intermediate $C^\star$-subalgebra $\mathcal{A}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{A}\subseteq C(X)\rtimes_r\Gamma$ is simple for an inclusion $C(Y)\subset C(X)$ of minimal $\Gamma$-spaces whenever $C(Y)\rtimes_r\Gamma$ is simple. This is a joint work with Dan Ursu.
Colloquium
Integral geometry and valuation theory in pseudo-Riemannian spaces
Oct 26, 14:30—15:30, 2021, Math -101
Speaker
Dmitry Faifman (Tel Aviv University)
Abstract
We will discuss the Blaschke branch of integral geometry and its manifestations in pseudo-Riemannian space forms. First we will recall the fundamental notion of intrinsic volumes, known as quermassintegrals in convex geometry. Those notions were extended later to Riemannian manifolds by H. Weyl, who discovered a remarkable fact: given a manifold M embedded in Euclidean space, the volume of the epsilon-tube around it is an invariant of the Riemannian metric on M. We then discuss Alesker’s theory of smooth valuations, which provides a framework and a powerful toolset to study integral geometry, in particular in the presence of various symmetry groups. Finally, we will use those ideas to explain some recent results in the integral geometry of pseudo-Riemannian manifolds, in particular a collection of principal Crofton formulas in all space forms, and a Chern-Gauss-Bonnet formula for metrics of varying signature. Partially based on joint works with S. Alesker, A. Bernig, G. Solanes.
אשנב למתמטיקה
מהלך מקרי על הילוכים מקריים Online
Oct 26, 18:10—19:30, 2021, בניין 32 חדר 309 וכן במרשתת
Speaker
אריאל ידין
Abstract
נציג את המושג של הילוך מקרי, ונספר על חלק מהתוצאות הקשורות למושג זה.
נקודת ההתחלה שלנו היא משפט של Polya שמוכיח שאדם שיכור יחזור מתישהו לביתו, אבל טייס חללית שיכור יעלם לנצח בהסתברות סבירה. בהתאם לזמן, נתאר את הקשרים לרשתות חשמליות ואולי גם לתורת חבורות גיאומטרית.
אשתדל שהכל יהיה נגיש לתלמידי שנה ב ומעלה.