Oberseminare und sonstige Vorträge

The Mahler measure of polynomials was introduced by Mahler in 1962 as a tool to study transcendental number theory. In this talk, we discuss the relationship between Mahler measure and Beilinson's regulator. As an application, we show that, under some conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of special values of the L-function of elliptic curves and the Bloch?Wigner dilogarithm. In certain four-variable cases, the Mahler measure can be written as a Q-linear combination of special values of the L-function of K3 surfaces and the Riemann zeta function.

Abstract: In weighted Kähler geometry, we consider Kähler manifolds equipped with a torus action, and a fixed positive function on the moment polytope. I will introduce this setting, as well as the notions of canonical metrics in weighted Kähler geometry: weighted solitons and weighted cscK metrics. I will then review some results on existence of such metrics, and applications to the more classical Kähler-Einstein metrics, Kähler-Ricci solitons and Calabi's extremal Kähler metrics.

Dirac-Schrödinger operators (or Callias-type operators) are given by Dirac-type operators on a smooth manifold, together with a potential. I will describe a general setting with arbitrary signatures (with or without gradings), which allows us to study index pairings and spectral flow simultaneously. I will first describe a general Callias Theorem, which computes the index (or the spectral flow) of Dirac-Schrödinger operators in terms of K-theoretic index pairings on a compact hypersurface. Associated to each Dirac-Schrödinger operator is also a Toeplitz operator, which is obtained by compressing the potential to the kernel of the Dirac operator. I will then explain how the index or spectral flow of these Toeplitz operators is related to the index or spectral flow of Toeplitz operators on the compact hypersurface. These results generalise various known results from the literature, while presenting them in a common unified framework.

Abstract: The cohomology ring of moduli spaces of manifolds is an important object in geometric topology, as it classifies characteristic classes of manifold bundles. For even-dimensional manifolds, Galatius and Randal-Williams established a complete homotopy theoretic formula for this object after a certain stabilization procedure. In this talk, I will explain an odd-dimensional analog of Galatius and Randal-Williams' work in the context of odd-dimensional manifold triads. After stating this result, I will explain the strategy of proof and discuss some applications and examples.

TBA