Course overview -- Wintersemester 2013/2014 (Archive)

For surfaces without conjugate points, Eberhard Hopf's method leads to optimal rigidity results. The question arises of whether (a generalization of) the method is equally powerful in higher dimensions. We will investigate the case of conformally flat cylinders.

We briefly describe the definition of curvature-dimension bounds in the sense of Lott-Villani/Sturm (CD(K,N)) and recent results about Riemannian differential calculus for metric measure spaces. Then we present the following theorem: the cone over a metric measure space satisfies CD(KN,N-1) if and only if the underlying space satisfies CD(N-1,N). As consequence of this result and the Cheeger-Gigli-Gromoll splitting theorem we obtain a maximal diameter theorem in the context of metric measure spaces.

This will be an overview of the Kähler-Einstein problem, including its very recent resolution in the positive case.

In this talk I will give a short introduction to the exceptional Lie-group G2. Then I will motivate G2-structure with G2-manifolds, i. e. Riemannian manifolds with holonomy group G2, that are an exceptional case in the classification of Berger and allegedly have applications in physics. On these G2-structures Crowley and Nordström defined the nu-invariant with values in Z/48Z, which I will explain. The mod 3 reduction of the nu-invariant is a G2-bordism invariant and has been interpreted as a universal eta-invariant in my thesis.

In this talk we will discuss how to use eta-invariant to find some geometric models for some algebraic K-groups.

I will introduce some of the main mathematical ideas behind N=2 superconformal field theory, using the N=2 minimal models as a concrete example, and explore some of their connections to geometry.

To illustrate these ideas I will focus on the so-called "chiral rings", which arise from basic Lie algebra representation theory plus ideas from quantum field theory. I show how the chiral rings relate to Calabi-Yau geometry, then I'll describe how chiral rings appear in Cecotti and Vafa's famous ADE singularity classification of a special subclass of N=2 minimal models (those with "space-time supersymmetry"), and we'll briefly reacquaint ourselves with some other prominent appearances the ADE pattern in maths and physics.

Finally I'll present the chiral rings of the entire class of N=2 minimal models (when the extra assumption of space-time supersymmetry is dropped) and report on ongoing work with K. Wendland concerning a geometric classification of these theories.

This talk will give a short introduction how the differential of the eta form gives the difference between the cohomological and the analytical index. Then we'll look at an example where the dimension of the kernel of the Dirac operator changes and see how this affects the eta-form.

By the van Est isomorphism, continuous cohomology of simple Lie groups vanishes in degree greater than the dimension of the associated symmetric space. Monod conjectured that a similar vanishing theorem should hold for continuous bounded cohomology. In this talk, we will present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove Monod's conjecture for SL(2,R) in degree four and discuss perturbations of the Spence-Abel functional equation for the dilogarithm function. This is joint work with Tobias Hartnick.