פעילויות השבוע

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.

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.

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