Course overview -- Sommersemester 2017 (Archive)

An attempt to axiomatize locality of path integrals leads to the notion of functorial quantum field theory (usually known as Atiyah-Segal field theory). In this talk, we will review this notion and briefly indicate how it predicts the gluing relation for the zeta regularized determinant of Laplacian. We will also discuss how to construct a functorial quantum field theory for the scalar field theory. Time permitting, we will outline a construction of functorial quantum field theory arising from two dimensional perturbative quantum scalar field theories.

In this talk, we explain how to use Kirby calculus and characteristic sublinks to describe spin structures on 3-manifolds and the obstruction to extending a given spin structure on the boundary of a 4-dimensional cobordism. We will illustrate this approach with some concrete examples.

We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tangent sheaf are established. As a consequence, we show that up to finite quasi-étale covers, varieties with strongly stable tangent sheaf are either Calabi-Yau (CY) or irreducible holomorphic symplectic (IHS). Finally, finiteness properties of fundamental groups of CY and IHS varieties are established.

Connective spaces are a generalization of both graphs and topological spaces. They carry a structure that is somewhat weaker than a topology, yet strong enough to support a sort of "algebraic topology". We shall look at their properties, define suitable morphisms and introduce a homotopy theory. In the end we will see a theorem that allows us to directly compare the homotopy groups of manifolds and graphs.

Mirror symmetry of Calabi-Yau manifolds was discovered from physics in 90's. Since then, one way to describe the symmetry is to look at suitable moduli spaces of Calabi-Yau manifolds. In this talk, I will start with a brief summary of mirror symmetry, and then I will show two interesting examples of Calabi-Yau manifolds given as complete intersections. In these examples, I will observe that birational geometry of Calabi-Yau manifolds are nicely encoded in the moduli spaces of mirror Calabi-Yau manifolds in terms of monodromy properties. In particular, I will identify Picard-Lefschetz type monodromy which corresponds to flops. This is based on collaborations with Hiromichi Takagi.

After reviewing the classical notions of moment maps in symplectic and hyperkähler geometry, we discuss several generalizations to multisymplectic geometry, where a closed differential form higher degree takes the place of the symplectic form. We describe how these generalizations are related and give further examples for moment maps on manifolds with G_2 or Spin(7)-structures.

The Dirac equation has been widely used to build up relativistic models of particles. Recently it made its (somehow unexpected) appearance in Condensed Matter Physics. New two-dimensional materials possessing Dirac fermions low-energy excitations have been discovered, the most famous being the graphene (2010 Nobel Prize in Physics awarded to Geim-Novoselov). In this talk I will give an overview about the role of the Dirac operator in some condensed matter systems, with particular emphasis on some models and related analytical problems.

In this talk we introduce certain ways to think about the group K^1(X): For a compact space X, the group K^1(X) is the Grothendieck group of the monoid of fnite rank complex vector bundles over the suspension SX.
There are several classifying spaces for the gorup K^1(X). We study here two of them: the infinite dimensional unitary group and the space of selfadjoint fredholm operator. Both of them are connected via the Cayley transform. Finally we will consider the kernel dimension of self-adjoint fredholm operators. Using the Fredholm operators as classifying space of K^1 we get an obstruction for high kernel dimension.

After a short introduction to hermitian symmetric spaces, I will explain the classical statement that the moduli space of complex elliptic curves is isomorphic to the Siegel modular variety of genus 1. In analogy Clingher and Doran proved that the moduli space of certain lattice polarized K3 surfaces is isomorphic to the Siegel modular variety of genus 2. Finally I will introduce a compactification for this moduli space and show that it is isomorphic to the Baily-Borel compactification of the Siegel modular variety of genus 2.

Analytically computing the spectrum of the Laplacian is impossible for all but a handful of classical examples.  Consequently, it can be tricky business to determine which geometric features are spectrally determined; such features are known as geometric spectral invariants.  Weyl demonstrated in 1912 that the area of a planar domain is a geometric spectral invariant.  In the 1950s, Pleijel proved that the perimeter is also a spectral invariant.  Kac, and McKean & Singer independently proved in the 1960s that the Euler characteristic is a geometric spectral invariant for smoothly bounded domains.  At the same time, Kac popularized the isospectral problem for planar domains in his article, "Can one hear the shape of a drum?''  Colloquially, one says that one can "hear'' spectral invariants.  Hence the title of this talk in which we will show that the presence, or lack, of corners is spectrally determined.  This talk is based on joint work with Zhiqin Lu. 

In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt on the asymptotics of the natural L2-metric on the moduli space of rank-2 Higgs bundles over a Riemann surface as given by the set of solutions to the so-called self-duality equations for a unitary connection and a Higgs field.

Extended abstract (including formulas) can be found using the link above.