Vorträge des SFB 1442

Abstract: Embedding group rings into division rings provides methods for tackling problems which superficially seem unrelated. One way of constructing such embeddings (in characteristic 0) is by proving that they satisfy the Atiyah Conjecture. I will give a survey of the status of the Atiyah Conjecture, as well as applications to other problems, and discuss joint work with Sam Fisher on proving the conjecture for congruence subgroups of mapping class groups and outer automorphism groups of RAAGs.

Real rank was introduced by Brown and Pedersen in 1991 as a variant of stable rank, generalizing a characterization of covering dimension in terms of the instability of tuples of continuous functions. Nuclear dimension, introduced by Winter and Zacharias in 2010, generalizes a different characterization of covering dimension, based on the colorability of refinements of open covers. The two notions agree for commutative C*-algebras, where they coincide with the covering dimension of the spectrum, but differ already for homogeneous C*-algebras. In this talk, I will recall both notions of dimension, discuss examples illustrating their differences, and sketch the proof of the main result: the real rank of a C*-algebra is bounded above by its nuclear dimension.

We generalize the DeBacker--Reeder/Kaletha construction of depth-0 supercuspidal L-packets to all tame covers defined by étale cohomology. This can be seen as the "regular depth-0" part of the local Langlands correspondence for covers. This talk is based on joint work with Nhat Hoang Le, which is in turn based on earlier works by Luozi Shi and myself.

In this talk, I will discuss ongoing work with R. Neagu and G. Szabó concerning weights on crossed product C*-algebras arising from actions of locally compact abelian groups. Several constructions in literature produces KMS weights or tracial weights on crossed product C*-algebras from suitable invariant weights of the underlying C*-algebra. When the acting group is the real numbers, one important such construction was introduced by Kishimoto and Kumjian. I will present new results on constructing tracial weights on crossed product C*-algebras and illustrate that these new results naturally generalize and unifies several constructions in literature, including that of Kishimoto and Kumjian. Subsequently, I will discuss the following question: under what sufficient and necessary conditions on the C*-dynamical system will all extremal tracial weights on the crossed product arises from our construction?