Course overview -- Sommersemester 2018 (Archive)

We present a state-sum construction of TFTs on r-spin surfaces which uses a combinatorial model of r-spin structures. We give an example of such a TFT which computes the Arf invariant for r even. We use the combinatorial model and this TFT to calculate diffeomorphism classes of r-spin surfaces with parametrized boundary.

A natural way of constructing Calabi-Yau manifolds is to build them as fibrations whose fibers are Calabi-Yau manifolds of lower dimension. For example, elliptic curves can be thought of as fibrations over the projective line by pairs of points, and K3 surfaces fibered over the projective line by elliptic curves form a large class of interesting K3 surfaces. In higher dimensions, the question of computing the Hodge numbers of such Calabi-Yau manifolds becomes a non-trivial one. I’ll talk about a method of computing Hodge numbers starting from the Picard-Fuchs differential equation of a family of Calabi-Yau manifolds. Applying this to families of lattice-polarized K3 surfaces provides a key ingredient in the classification of fibered Calabi-Yau threefolds.

I will report on the search for stable, closed geodesics on a K3 surface. In particular, I will sketch the Bourguignon-Yau proof of why the Riemann tensor vanishes along such geodesics, and I will explain how to find totally geodesic tori in highly symmetric Kummer K3s.

I will talk about joint work with Bernd Siebert, which aims to give a general construction of mirrors to either log Calabi-Yau manifolds or maximal degenerations of Calabi-Yau manifolds. The construction goes by way of building the coordinate ring of the mirror as an abstract ring whose multiplication law is governed by counting curves on the original (log) Calabi-Yau.

I will report on general results and questions about spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds, and present recent joint work with Michael Wiemeler. In particular, we construct the first classes of manifolds for which these spaces have non-trivial rational homotopy, homology and cohomology groups.

Wave equations are usually studied in globally hyperbolic regions of spacetimes. However, to approach the famous strong cosmic censorship conjecture, this is not sufficient. One needs to understand the behavior of waves close to the boundary of the globally hyperbolic region, the Cauchy horizon. The purpose of this talk is to discuss the characteristic Cauchy problem with initial data on a compact Cauchy horizon. We prove an energy estimate close to compact non-degenerate Cauchy horizons which implies existence and uniqueness results for wave equations. In particular, we overcome the essential remaining difficulty in proving the Moncrief-Isenberg conjecture in the non-degenerate case. This can be seen as a special case of the strong cosmic censorship conjecture.

The standard notion of equivalence of field theories roughly requires the Euler-Lagrange loci of the two associated variational problems to be diffeomorphic, possibly modulo the action of the respective symmetry distributions, in some appropriate framework. This can be made stated more precisely in the Batalin-Fradkin-Vilkovisky setting, where some cohomological presentation of said locus is constructed. I will discuss a series of examples, all related to General Relativity in different space-time dimensions, that suggest that higher codimension data should play a role in defining (and refining) equivalence between classical theories, and raise the question of whether (and how) this picture carries over to quantisation.

This talk will be an introduction to contact geometry and dynamics. In particular, I will explain the relation between Hamiltonian systems and contact dynamics. The main examples are level sets of Hamiltonians from classical mechanics and contact manifolds, ob which the Reeb flow induces an S^1-action with reasonably nice quotients.

The Weyl Denominator Identity arises usually as a special case of the Weyl Character Formula of a complex semisimple Lie algebra. Even though it prominently features the roots of the Lie algebra, the original proof provides limited insight into the structure of the root system. I will therefore present a direct proof of the Denominator Identity. This alternative approach explicitly uses connections between the roots and the Weyl Group and might provide a different perspective on the Denominator Identity as well as the structure of the root system.