Abstracts for MPI-Oberseminar

We discuss a novel nano material structure of a Double Gyroid (DG)
wire network. We start by introducing the geometrical structure of the DG
and its
fabrication as background. We then use methods of commutative and
 non-commutative geometry
to describe this quantum wire network. Its non--commutative geometry
is closely related to non-commutative 3-tori as we discuss in detail.
This is joint work with R. Kaufmann and S. Khlebnikov.

We start with a sketch a fragment of non-abelian homological algebra adopting an intermediate level of generality, which
allows to begin along the lines of Grothendieck's Tohoku lectures replacing abelian categories by right (or left)
exact categories with initial (resp. final) objects. Further analysis leads to the notion of the stable
category of a left exact category and to the notions of quasi-suspended and quasi-triangulated categories.

We define K_0 of a right exact category, introduce a structure of right exact category on the category

Functional determinants provide a way, using zeta
functions, of defining the determinants of certain elliptic
differential operators on compact manifolds. In this lecture
we shall study how such determinants depend on the geometric
data and explain how this is connected to representation theory
of semisimple Lie groups. Functional determinants play a role
in string theory and other physical theories, and the
representation theory involves inequalities satisfied by
Knapp-Stein intertwining operators.
 

R. Thomason has proved a concentration theorem for algebraic
equivariant K-theory on the schemes which are endowed with the action
of a diagonalisable group scheme. As usual, such a concentration
theorem induces a fixed point formula of Lefschetz type which can be
used to calculate the equivariant Euler-Poincare characteristics of
equivariant vector bundles. In this talk, I will try to explain how to
generalize Thomason's results to Arakelov geometry.