Oberseminar Wintersemester 2018/19-Oliver Bräunling

Oliver Bräunling will speak on

$K$-theory and the question "Is there an equivariant Haar measure?"

Abstract: Algebraic $K$-theory, once used in number theory, has often found invariants which we knew before, e.g. class group and unit group of a number field show up in $K_0$ and $K_1$. However, the genuine idèle class group of class field theory has never naturally shown up so far. In 2017 Clausen had the idea to look at locally compact topological modules instead of untopologized ones. This changes everything. I will present some of my own contributions to this beautiful, amazing idea of his:

- the idèle class group shows up as $K_1$, including the infinite places(!!!!) (Clausen)
- the universal determinant functor (in the sense of Deligne's "La déterminant de la cohomologie") of locally compact modules turns out to be essentially the Haar measure.

It appears that, equivariantizing, perhaps these ideas might be fruitful for the equivariant Tamagawa number conjecture (ETNC) in the non-commutative setting of Burns-Flach. I would like to explain the picture I propose for this, plus what's proven.

Partially joint work with Peter Arndt.