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# CRC-Colloquium

## June 6 at 15:30 – 18:30 CEST

15:20 Coffee (or earlier)

15:30 – 16:30 Georg Tamme (Universität Mainz): Higher algebra and K-theory

16:30 Coffee and Cake

17:15 – 18:15 Otmar Venjakob (Universität Heidelberg): Explicit Reciprocity Laws in Number Theory

18:45 Dinner

Abstract G. Tamme:

Algebraic K-theory, more precisely the group K_0(X) of a scheme X, was first introduced by Grothendieck in order to formulate and prove a very general Riemann-Roch theorem. Subsequently, lower (i.e. K_i for i<0) and higher (i.e. K_i for i>0) K-groups have been introduced by Bass, Milnor, Quillen, and Thomason. They have turned out to be a rich invariant, playing an important role in various fields such as number theory and topology. Although explicit computations of K-groups remain challenging, modern methods from higher algebra and derived algebraic geometry have led to several breakthrough structural results concerning these groups. In the talk, I will give an introduction to these topics and, time permitting, discuss some recent applications.

Abstract O. Venjakob:

The quadratic Reciprocity Law for the Legendre or Jacobi-Symbol forms the starting point of all Reciprocity Laws as well as of class field theory. It is closely related to the product formula of the quadratic Hilbert-Symbol over local fields. Various mathematicians have established higher explicit formulae to compute higher Hilbert-Symbols. Analogs were found for formal (Lubin-Tate) groups. Eventually Perrin-Riou has formulated a Reciprocity Law, which allows the explicit computation of local cup product pairings by means of Iwasawa- and p-adic Hodge Theory. In this talk I shall try to give an overview of these topics, at the end I will explain recent developments in this regard.