Yonatan Harpaz - New perspectives in hermitian K-theory I

Abstract: Yonatan Harpaz (Paris 13): New perspectives in hermitian K-theory
In this lecture series we will describe an approach to hermitian K-theory which sheds some new light on classical Grothendieck-Witt groups of rings, especially in the domain where 2 is not assumed to be invertible. Our setup is higher categorical in nature, and is based on the concept of a Poincaré ∞-category, first suggested by Lurie. We will explain how classical examples of interest can be encoded in this setup, and how to define the principal invariants of interest, consisting of the Grothendieck-Witt spectrum and L-theory spectrum, within it. We will then describe our main abstract results, including additivity, localization and universality statements for these invariants and their relation to each other and to algebraic K-theory via the fundamental fibre sequence.

We will then specialize again to the case of rings and extract a variety of applications, including a generalization of Karoubi's fundamental theorem, a solution of the homotopy limit problem for number rings, a comparison range between quadratic and symmetric Grothendieck-Witt groups for rings of bounded dimension, a finite generation statement for Grothendieck-Witt groups of number rings and an essentially complete calculation of the quadratic and symmetric Grothendieck-Witt groups of the integers. Our approach makes systematic use of the fundamental fibre sequence for reducing questions about Grothendieck-Witt groups to questions about L-groups, which in turn can be very efficiently accessed via the machinery of algebraic surgery, as will be explained in the companion lecture series of Markus.