Course overview -- Wintersemester 2012/2013 (Archive)

I will give a short overview about higher torsion invariants for families of compact manifolds. Then I will sketch a generalisation of Igusa-Klein torsion that has similar properties as Bismut-Lott torsion. This is the first step in the comparison of the two invariants.

In his proof of the Arnold conjecture about the number of fixed points of Hamiltonian symplectomorphisms, A. Floer developed a new infinite-dimensional version of Morse homology. While the chain groups are generated by these fixed points, the differential counts pseudoholomorphic sections, which in turn provides a link to Gromov-Witten theory. In my talk I will introduce the basic concepts of Floer homology and its application to the Arnold conjecture. At the end I plan to sketch the main results of my recent paper 1206.1564, where I was able to complete the correspondence between rational Gromov-Witten theory and Floer theory of symplectomorphisms.

1984 Vafa and Witten developed a method to obtain upper bounds for the first eigenvalue of a Dirac-Operator on a compact manifold. I will explain the method and the related geometry in detail. We will see an example of an optimal Vafa-Witten bound on the projective space obtained by geometric arguments. Furthermore, I will give a connection between the Vafa-Witten method and L-infinity variational problems.

The formalism of free fermions used for construction of tau-functions of classical integrable hierarchies is reviewed. We give a derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear identity and the generalized Wick's theorem. We also consider various examples of tau-functions and give their fermionic realization.

Many beautiful geometric structures, such as Calabi-Yau (CY) manifolds and their moduli spaces, arise naturally in the context of string theory. In heterotic string theory a generalization of CY geometry is given by holomorphic vector bundles over CY spaces. This talk is concerned with more general heterotic geometries: vector bundles over complex but typically non-Kaehler manifolds. I will review this geometric structure and its origin in string theory and then focus on a characterization of the first order deformation space of these geometries.

Metrics with holonomy contained in G_2 on a 7-manifold are induced by so-called positive 3-forms which are torsionfree, i.e. satisfy a certain nonlinear 1st order PDE. I will discuss a parabolic flow on positive 3-forms whose stationary points are precisely given by the torsionfree ones. This is joint work with F. Witt. If time permits I will describe a generalization to a spinorial setting (joint with B. Ammann and F. Witt).

A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (for which orbifolds and their crepant resolutions are an example), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.

I will give an introduction to special Kähler structures and some of their aspects. A special Kähler structure is a connection on the tangent bundle of a Kähler manifold with a number of "special" properties. They appear naturally in the context of certain integrable systems and of moduli spaces of Calabi-Yau threefolds.