Abstracts for Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

The goal of this talk is to introduce $E$-theory for $C^*$-algebras, which is a slight variant of $C^*$-algebraic $KK$-theory, and explain why $E$-theory is dualizable.

In this talk, I will initially introduce the definition of smooth and proper category, and then explore how it can be used in algebraic and analytic geometry. This definition fits very well within the algebraic setting, as it allows for the characterization of smooth and proper algebraic varieties by examining their category of quasi-coherent sheaves. In the analytic context, such a characterization is more challenging to achieve; during the talk, I will aim to explain the difficulties and some results that we can obtain in this setting.

Among various features of algebraic $K$-theory, there is known to be covariance with respect to finite flat morphisms of schemes. In this talk we will see, in which sense $K$-theory is universal as a cohomology theory with such covariance. Time permitting, we will discuss an analogous universality property for hermitian $K$-theory. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro.