Course overview -- Sommersemester 2016 (Archive)

Γ-structures are weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting Γ-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of Γ-structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups.

Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products given by geodesic symmetries define Γ-structures. This extends work of Albers, Frauenfelder and Solomon on Γ-structures on Lagrangian Grassmannians.

We will introduce eta invariants, which are spectral invariants of Dirac operators, and the notion of adiabatic limits. Then we present some known results by Bismut, Cheeger and Dai before we give a partial answer in a more general setting.

I will give an introduction to my PhD thesis topic. My emphasis will be on explaining the problem by stating known resutls of interest to a broader audience of differential geometers. Most of the talk should be understandable without a detailed knowledge of K3 surfaces.

Starting with the relation of infinite loop spaces and generalized cohomology theories, we use some simple examples to illustrate some special homotopy invariant properties of infinite loop spaces. Then we go on and introduce various delooping machines. In the end of the talk, a description of infinite loop space by Gamma-space will be given.

The study of global analysis of spaces with (isolated) cone-like singularities has started with work of Cheeger in the 80s and has seen a rich development since. One important result is the generalisation of the Chern-Gauss-Bonnet theorem for these spaces, which is due to Cheeger. It establishes a relation between the $L^2$-Euler characteristic of the space, the integral over the Euler form and a local contribution $\gamma$ of the singularities. The ``Cheeger invariant'' $\gamma$ is a spectral invariant of the link manifold.

The aim of this talk is to establish a local index formula for the intersection Euler characteristic of a cone. This is done by studying local index techniques as well as the spectral properties of the model Witten Laplacian on the infinite cone. As a result we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, one of which is Cheeger's invariant $\gamma$.

Since Milnor discovered exotic 7-spheres in 1956, it is known that a topological manifold can have several non-diffeomorphic smooth structures. The aim of smoothing theory is to calculate the set S(X) of smoothings of a given topological manifold X in terms of a homotopy theoretical property: S(X) turns out to be in bijection with the set of lifts of a certain classifying map.

In my talk I will introduce all necessary concepts such as mircobundles and piecewise linear manifolds and try to illustrate their properties. Then the fundamental theorem will be stated and important parts of the proof will be sketched. In the end I hope to give some practical advices on how to calculate structure sets.

Morse theory is concerned with the relationships between the structure of the critical set of a function and the topology of the ambient space where the function is defined. The applications of Morse theory are ubiquitous in mathematics, since objects of interest (geodesics, minimal surfaces etc) are often critical points of a functional (length, area etc). In this talk I will review basic concepts in Morse theory, and will focus on Hamiltonian dynamics where the applications emerge from the fact that periodic solutions of Hamilton's equations are critical points of the action functional. I will explain how to define a local Morse homology of the action functional at an isolated periodic orbit which takes into account the symmetries associated to time-reparametrization, and serves a well-defined alternative to local contact homology. Then I will explain dynamical applications. This is joint work with Doris Hein (Freiburg) and Leonardo Macarini (Rio de Janeiro).

In this talk we give an overview of the Higson-Roe exact sequence for discrete groups, also known as the analytic surgery sequence, and explain its relation with secondary invariants of type rho. Using the machinery of equivariant Roe-algebras we shall also outline a proof of some rigidity results of $\ell^2$-rho-invariants, generalizing earlier work of Higson and Roe. This is joint work with M.-T. Benameur.