Course overview -- Wintersemester 2017/2018 (Archive)

Atiyah, Patodi and Singer constructed the relative K-theory class $[\alpha]$ associated with a flat unitary vector bundle over a closed manifold. This class is related to the spectral invariant rho of a Dirac operator by the so called index theorem for flat bundles, which computes the pairing between $[\alpha]$ and the K-homology class $[D]$ of the Dirac operator.

In this talk, after introducing the context and the needed tools, we show that $[\alpha]$ admits a canonical construction, using von Neumann algebras and that, as a secondary class, it results from Atiyah's $L^2$-index theorem for covering.

Taking an operator algebraic point of view, we show that Atiyah's property can be encoded using KK-theory with real coefficients (which will be introduced). This permits to generalise the constructions of secondary classes of rho-type in the noncommutative setting of a discrete group $\Gamma$ suitably acting on a $C^*$-algebra $A$. Based on joint work with Paolo Antonini and Georges Skandalis.

In my talk I will explain various results concerning Calabi-Yau threefolds and geometric objects on the Calabi-Yau variety, e. g. a divisor, a curve or a coherent sheaf. I will discuss their deformation theory and the connection to Picard-Fuchs equations and potential functions. These are special holomorphic functions describing the obstructions of a deformation problem.

The Crowley-Nordström ν invariant and its extension ν¯ is an invariant of compact G2 manifolds. It has been computed for several kinds of connected sums by Crowley, Nordström and Goette (see next talk), in this talk we discuss strategies of computations for Joyce manifolds.

There is nowadays a large supply of compact manifolds of special holonomy $G_2$. The Crowley-Nordström $\nu$ invariant and its extension $\bar\nu$ help to distinguish different connected components of the $G_2$-moduli space on a given manifold $M$. We already know some manifolds $M$ whose $G_2$-moduli space has at least 7 different connected components. However all examples of $G_2$-manifolds investigated so far turn out to be topologically $G_2$-nullbordant, that is, their $\nu$-invariant is divisible by 3.

In this talk, we will consider Nordström's extra twisted connected sum construction with at least one $\mathbb Z/3$- or $\mathbb Z/4$-block. Although $\bar\nu(M)$ is defined in terms of $\eta$-invariants, we will show that in the end it can be computed explicitly for these examples using elementary hyperbolic geometry.

I will talk about a generalisation of the Seiberg-Witten equations introduced by Taubes and Pidstrygach, in dimension 3 and 4 respectively, where the spinor representation is replaced by a hyperKahler manifold admitting certain symmetries. I will discuss the 4-dimensional equations and their relation with the almost-Kahler geometry of the underlying 4-manifold. In particular, I will show that the equations can be interpreted in terms of a PDE for an almost-complex structure on 4-manifold. This generalises a result of Donaldson.

Borcherds-Kac-Moody algebras are a generalization of finite dimensional semisimple Lie algebras obtained by weakening the requirements on Cartan matrices. In his proof of the Moonshine Conjectures, Borcherds related them to the vertex operator algebras from conformal field theory. At the same time, there appears to be a connection to automorphic forms via denominator functions. This can hopefully be leveraged, in particular, to investigate so-called Bogomol'nyi-Prasad-Sommerfield states in field theories with supersymmetry.

In this talk, we will study the Selberg zeta function and its relatives. We will recall the celebrated Selberg trace formula, and the geometric setting of our work, the Teichmüller space of Riemann surfaces of genus, g. As shown by Zograf and Takhtajan, the Selberg trace formula connects the Ricci curvature of the Hodge bundle $H^0 (K^m)$ over Teichmüller space together with the second variation of the Selberg zeta function at integer points. We will briefly explain this connection and the role of the Selberg trace formula in its derivation.

Further, we will investigate the behavior of the Selberg zeta function, $Z(s)$, as a function on Teichmüller space. We will deduce an explicit formula for the second variation of $\log( Z(s) )$ via a certain infinite sum involving lengths of closed geodesics of the underlying surface and their variations. We will then utilize this formula to study the asymptotics of the second variation of $\log( Z(s) )$ as $s \to \infty$. We shall see that the most prominent role is played by the systole geodesics. Moreover, the dimension of the kernel of the first variation of the latter appears in the signature of the Hessian of $\log Z(s)$ for large $s$. In conclusion, we will show how our variational formula and its asymptotics have interesting implications for the curvature of the Hodge bundle and its relationship to the Quillen curvature.

This is a joint work with Julie Rowlett and Genkai Zhang.

Shape analysis aims at a mathematical description and analysis of geometric data such as e.g. curves or surfaces. The key paradigm is to view these data as elements of an infinite-dimensional Riemannian manifold, which is called shape space. I will give an introductory talk to shape spaces and Riemannian metrics thereon. Some main results to be covered are (non-)degeneracy of the Riemannian path length functional and wellposedness of the geodesic equation.

We discuss recent results on the Ricci flow for spaces with incomplete edge singularities. In the special case of isolated cones we establish stability of the flow near Ricci flat metrics.

Characteristic classes of vector bundles provide an important tool to study these geometric objects using techniques from algebraic topology, i.e. cohomology. In my talk I will give an introduction to Chern classes, which are characteristic classes of complex vector bundles. I will present several points of view onto this topic, each emphasising a certain aspect of Chern classes. This will help to understand the significance of this machinery.