Oberseminare und sonstige Vorträge

The category of synthetic spectra is a strong tool for understanding the Adams-Novikov spectral sequence and acts as a 1-parameter deformation between spectra and quasi-coherent sheaves on the moduli stack of formal groups. One perspective on the Adams-Novikov spectral sequence is that it is the descent spectral sequence for the moduli stack of formal groups. In this talk I will present an approach which, to any geometric (non-connective) spectral stack X, produces a category of "synthetic" quasi-coherent sheaves on this stack. This acts as a 1-parameter deformation of quasi-coherent sheaves and provides a natural place to study the descent spectral sequence of X. If time permits, I will discuss the relationship with synthetic spectra and the even stack of Hahn-Raksit-Wilson.

Abstract: In this talk, we explore geometric and analytic aspects of metric spaces that are homeomorphic to smooth manifolds. Such spaces, often called metric manifolds, arise naturally in geometry, geometric group theory, and analysis on metric spaces. We introduce the notion of a metric fundamental class, which provides an analytic representative of the topological fundamental class, and investigate conditions under which it exists. We then discuss some of the motivations behind studying metric fundamental classes and highlight applications to Lipschitz volume rigidity and isoperimetric inequalities.

Roughly speaking, the classical groups are groups of linear bijections of vector spaces over division rings, and subgroups singled out by requiring that some form (quadratic, bilinear, hermitian) be invariant. So each one of those groups is determined by some recipe, and some ingredients (a division ring, a number, a form, ...). E.g., a famous recipe named "A" takes a division ring K and a number n to produce the group SL(n,K), along with its relatives GL(n,K), PSL(n,K). Another recipe (no name given) takes a field R, a number n, and a non-degenerate quadratic form of Witt index i in n variables from R to produce the corresponding group of isometries.Conversely, the produced group determines the recipe and its ingredients, in general. However, there are some exceptions, including the reason why physicists study SL(2,C) while doing special relativity.The talk will report about such exceptions, and ways to understand deeper reasons for their occurrence.

Given a finitely generated group G, a non-trivial element g\in G, and a minimal action of G on a compact space X, with amenable neighborhood stabilizers, we prove sufficient conditions in terms of various growth/decay functions for topological freeness of the action of g on X. We apply our results to the case of the (Furstenberg) boundary action of G, to conclude C*-simplicity under (sub)rapid decay conditions. In this context, we also give a description of the support of stationary states on the reduced C*-algebra of G. This is joint work with Nazmul Alam, Joseph Gondek, and Randy Pham.