Course overview -- Wintersemester 2011/2012 (Archive)

Monotone variational recurrence relations such as the Frenkel- Kontorova lattice, arise in solid state physics and as Hamiltonian twist maps. Aubry-Mather theory guarantees the existence of minimizers of every rotation number. They constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected - a foliation, or a Cantor set - a lamination. It turns out that when the rotation number of a minimal foliation is Liouville (easy to approximate by rational numbers) the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation.

Symplectic field theory (SFT) assigns to each symplectomorphism (of a closed symplectic manifold) an infinite-dimensional Hamiltonian system with an infinite number of symmetries. While the latter are obtained using descendants, for Hamiltonian symplectomorphisms this leads to the well-known integrable hierarchies of Gromov-Witten theory. In joint work with P. Rossi we study the algebraic structure of descendants, both to find richer geometrical invariants and to answer the question of integrability for general symplectomorphisms.

A Calabi-Yau orbifold in complex dimension 3 is locally modeled on C^3/G with G a finite subgroup of SL(3,C). When G acts with an isolated fixed point on C^3, a crepant resolution has the structure of an asymptotically locally euclidean (ALE) manifold. Using index theory techniques we derive a geometrical interpretation of the McKay correspondence which relates the geometry of the crepant resolution to the representation theory of the finite group G. This extends a result of Kronheimer and Nakajima to this higher dimensional case.

Some of generating functions in enumerative geometry are known to be related to matrix integrals and classical integrable hierarchies of KP/Toda type. In recent years it become clear that the partition functions of the this still not completely described set of models posses other nice properties such as cut-and-join-type representations, random partition descriptions and Virasoro-type constraints. I will explain some of the aforementioned properties and relations between them for three important models, namely Hermitian matrix model, Kontsevich-Witten tau-function and generating function of Hurwitz numbers.

We give an overview on Gromov-Witten theory of Calabi-Yau manifolds. In particular, we study generating functions of Gromov-Witten invariants and explain their reformulations by physicists. Our focus will lie on those cases for which these functions turn out or are expected to be modular forms.

We look at a special type of elliptically fibered Calabi-Yau 3-folds arising via the heterotic/F-theory string-string duality. We describe the imprint of this duality on the geometry and topology of the Calabi-Yaus. This is joint work with Katrin Wendland.

We'll give an overview of the index theorem in different situations and then we'll concentrate on families of closed manifolds. We are interested in the eta-form and its convergence at infinity and zero. Our presentation will be based on "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne.

In meinem Vortrag konstruiere ich nach Brylinski-Mc Laughlin Kozykeldarstellungen für charakteristische Klassen in der glatten Deligne-Kohomologie.