Algebraic geometry, categories and trace forumulas

Abstract: In this talk I will present an approach to non-commutative algebraic geometry based on the following principle: a non-commutative variety simply is a category. This principle seems naive at first sight, but it has been very fruitful during the last 20 years, thanks to the works of many authors such as Kapranov, Bondal-Orlov, Rosenberg, Van den Bergh, Artin-Zhang, Konstevich-Soibelman, Keller, etc.

The purpose of this lecture is to explain how we can (or can not) “do geometry” with categories with a particular focus on cohomological and numerical aspects: Euler characteristic, Lefschetz’s type trace formula, etc. I will illustrate this by exploring its interactions with singularity theories, both in the classical complex analytic situation and in more arithmetic settings. In the last part of the lecture I will explain how this approach to non-commutative geometry can be used in order to make progress on the, still conjectural, Bloch’s conductor formula.