Sonstige Vorträge

Abstract: The Dehn invariant is known to many as the satisfying solution to Hilbert?s 3rd problem: a three-dimensional polyhedron P can be cut into pieces and reassembled into a polyhedron Q if and only if Q and P have not only the same volume, but also the same Dehn invariant. Generalised versions of Hilbert?s 3rd problem concern the so-called scissors congruence groups of euclidean, hyperbolic and spherical geometry in varying dimensions, and in these contexts one can define a generalised Dehn invariant. In the spherical case, Sah showed that the Dehn invariant makes the scissors congruence groups into a graded Hopf algebra. Zakharevich has shown that one can lift the scissors congruence group to a K-theory spectrum. In this talk I will discuss a lift of the Dehn invariant to the spectrum level, and we will see how it gives rise to a spectral version of Sah?s Hopf algebra. This talk is based on joint work in progress with Inbar Klang, Cary Malkiewich, David Mehrle and Thor Wittich.

Abstract: Clausen?Scholze's condensed mathematics provides a framework for doing homological algebra in the presence of topology. A key notion is solidification, a completion process for condensed spectra (roughly, spectra equipped with a topology). In this talk, we consider the solidification of the algebraic K-theory of condensed algebras. I explain results that connect it to topological and semitopological K-theory. I then discuss Rosenberg's conjecture on the algebraic K-theory of operator algebras and suggest how condensed mathematics may offer new perspectives.

In this talk I will explain how derived analytic geometry, both Archimedean and non-Archimedean, can be realised as algebraic geometry relative to the monoidal category of bornological abelian groups of convex type. I will compare with existing models of derived analytic geometry, and suggest some applications.