Vorträge des SFB 1442

Let A be a suitably nice C*-algebra (for example, quasidiagonal with countable K-homology). I will explain how the K-homology of A can be realized by almost multiplicative ucp maps to matrices. I?ll sketch some applications to K-homology of spaces, and to approximate representation theory of groups.

Abstract: The signature of a closed manifold is an important invariant in geometric topology. Recently, Higson, Xie, and Schick proved an invariance theorem for the K-theoretic signature in codimension 2. However, the L-theoretic counterpart of this result remains an open problem. In this talk, I will describe how to achieve the L-theoretic counterpart of their result and provide some examples. I will also explain the basic ideas of the proof, highlighting the key steps and challenges involved.

Kac-Moody groups form an infinite-dimensional generalization of reductive groups, constructed from infinite root systems. For instance, loop groups of reductive groups are Kac-Moody groups. Whereas their behavior over finite or complex fields is very similar to the reductive setting, many constructions fall apart over p-adic or function fields. In this first talk, I will present a building theoretic approach to this problem and introduce the Kac-Moody analog to affine Weyl groups. Even though these are no longer Coxeter groups, they present combinatorial similarities which I will discuss.

Harmonic morphisms are maps $\phi:(M,g)\to (N,h)$ between Riemannian manifolds pulling back harmonic functions on N to harmonic functions on M. They have been characterised as being harmonic maps which are horizontally conformal. Under certain natural conditions they induce a conformal foliation on M with minimal leaves. In this talk we will give a short introduction to the general theory. Then we will described what is known in the important cases when M is a Lie group foliated in this way by the left translations of a semisimple subgroup of M.

The Lagrangian of the Standard Model of particle physics consists of four sectors: the Yang-Mills, the Higgs, the Dirac, and the Yukawa sector. In this talk we study the Dirac-Yang-Mills (DYM) sector in isolation in the Riemannian setting. Variation of the Lagrangian leads to a system of partial differential equations, called the DYM equations, and whose solutions are referred to as DYM pairs. Our goal is to study spherically symmetric DYM pairs on with structure group SU(2), on Riemannian manifolds of dimension 3. In particular, we will systematically obtain the most general spherically symmetric ansatz, which turns the DYM equations into a system of non-autonomous ODEs. We will then construct several different families of solutions to this system, leading to new examples Dirac-Yang-Mills pairs (most notably also on closed manifolds). This is joint work with Adam Lindström.

Harmonic morphisms are maps $\phi:(M,g)\to (N,h)$ between Riemannian manifolds pulling back harmonic functions on N to harmonic functions on M. They have a non-trivial existence theory and it is therefore interesting to construct manifolds allowing for the existence of solutions to this non-linear problem. Special well-known cases are the holomorphic maps from Kähler manifolds to Riemann surfaces. Investigations have shown that the strong Kähler conditions can be relaxed in interesting cases. In this talk we will construct eight-dimensional Hermitian Lie groups foliated by SU(2)xSU(2)-leaves allowing for the existence of harmonic morphisms.