Vorträge des SFB 1442

Cartan subalgebras in operator algebras have been an active research area since their introduction in the 1970s (by Vershik and Feldman-Moore for von Neumann algebras, and Renault and Kumjian for C*-algebras). While on the side of von Neumann algebras some deep rigidity results have been established, much less is known about the variety of Cartan subalgebras that a C*-algebra can admit, with only a few exceptions of certain subhomogeneous C*-algebras. In this talk, we discuss another case of C*-algebras whose Cartan subalgebras can be parametrized in full, namely unital essential extensions of continuous functions on the circle by the compact operators (which includes the Toeplitz algebra). As we shall see, in the non-trivial extension case, these C*-algebras admit no C*-diagonals, all of their Cartan subalgebras have the one-point compactification of the natural numbers as their spectrum, and, up to conjugacy, these are parametrized by certain "near symmetries" of the natural numbers. This is joint work with Philipp Sibbel.

Abstract: In the higher-categorical setting, Galois theory extends beyond finite groups: one can study Galois extensions for an arbitrary E_1-group. We develop such a theory for higher semiadditive categories, where finite groups are naturally replaced by pi-finite groups. Under suitable height assumptions, we construct an n-truncated pro-pi-finite "absolute Galois group" classifying all Galois extensions. We also relate abelian extensions to the Picard spectrum through a higher version of Kummer theory. Time permitting, we will compute the absolute Galois group in some chromatic and categorified settings.

Given an action of a countable group on the Cantor space, clopen subsets can be compared by cutting and moving through the action or by evaluating them on invariant measures. Dynamical comparison asserts that the measure-theoretic notion controls the dynamical one. Since its first appearance, dynamical comparison has been verified for a wide variety of actions. In this talk, I will survey some recent positive results in this direction and present the construction of the first examples of topologically free minimal actions that fail dynamical comparison, obtained in a joint work with Akshara Prasad.

In the Langlands program, theta functions are used as integral kernels to construct automorphic forms through theta lifting, and the Rallis inner product formula relates the inner product of two theta lifts to special values of L-functions. In the Kudla program, arithmetic theta functions with values in Chow groups are used to construct algebraic cycles through arithmetic theta lifting, while the arithmetic inner product formula relates the Beilinson?Bloch height pairing of two arithmetic theta lifts to the central derivative of an L-function and could be used to give evidence for the Beilinson?Bloch conjecture. In this talk, we propose a cohomological framework that could be used to give a uniform construction of arithmetic theta functions and of motivic theta functions with values in higher Chow groups. These motivic theta functions are used to define motivic theta lifting, which provides a new way to construct motivic classes beyond those arising from modular units. The resulting motivic theta lifts satisfy an identity relating their Beilinson regulators to non-critical L-values. This identity can be viewed as a Hodge?Deligne analogue of the Rallis inner product formula and provides evidence for Beilinson?s conjectures. This is joint work in progress with Wenxuan Qi, Liang Xiao, and Yichao Zhang.