Vorträge in der Woche 27.06.2016 bis 03.07.2016

Montag, 27.06.2016: Poles of maximal order of Igusa zeta functions

Dr. Johannes Nicaise (Imperial)

I will describe some surprising interactions between number theory, algebraic geometry and string theory that have appeared in my recent work with Mircea Mustata and Chenyang Xu and that have led to a solution of Veys’ 1999 conjecture on poles of maximal order of Igusa zeta functions. The talk will be aimed at a general audience and will emphasize some key ideas from each of the fields involved rather than the technical aspects of the proof.

Dienstag, 28.06.2016: Graphen und Klassenzahlen

Anton Deitmar

Mittwoch, 29.06.2016: Partial Differential Equations in Geometry

Prof. Dr. Simon Brendle (Stanford University und Columbia University)

A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In particular, we will focus on the geometry of hypersurfaces, and discuss the isoperimetric inequalities, Alexandrov’s theorem on embedded surfaces in R n of constant mean curvature, as well as our proof of Lawson’s 1969 conjecture concerning embedded minimal tori in S 3 . Time permitting, I will discuss some recent results on the classification of self-similar solutions to geometric flows.

Donnerstag, 30.06.2016: Singularity formation in Black hole interiors

Prof. Spyros Alexakis (University of Toronto)

The prediction that solutions of the Einstein equations in the interior of black holes must always terminate at a singularity was originally conceived by Penrose in 1969, under the name of “strong cosmic censorship hypothesis”. The nature of this break-down (i.e. the asymptotic properties of the space-time metric as one approaches the terminal singularity) is not predicted, and remains a very hotly debated question to this day. One key question is the causal nature of the singularity (space-like, vs null for example). Another is the rate of blow-up of natural physical/geometric quantities at the singularity. Mutually contradicting predictions abound in this topic. Much work has been done under the assumption of spherical symmetry (for various matter models). We present recent developments (partly due to the speaker and G. Fournodavlos) which go well beyond this restrictive class. A key role is played by the axial symmetry reduction of the Einstein equations, where a wave map structure appears.

Donnerstag, 30.06.2016: TOPOLOGY OF T-VARIETIES

Prof. Antonio Laface (Universidad de Concepción)

A T-variety is a normal complex variety X endowed with an effective action of an algebraic torus T. Such varieties generalize toric varieties and can be described by means of the language of divisorial fans developed by Altmann, Hausen and Süß. The topology of toric varieties has been widely studied. In this talk I will discuss some recent results about the topology of T-varieties focusing on the Chow ring of a rational projective Q-factorial T-variety of complexity one which admits an equivariant morphism to P1 .

Donnerstag, 30.06.2016: T. Krake

Der DMD-Algorithmus und seine Interpretation durch den Koopmanoperator

In meinem zweiten Vortrag stelle ich die Grundlagen von DMD nochmal zusammenfassend dar und führe weitere Operatoren ein, die einen wesentlichen Bezug zu DMD besitzen. Das Ziel ist eine Interpretation von DMD durch den Koopmanoperator zu erhalten.

Donnerstag, 30.06.2016: A Gutzwiller trace formula for large hermitian matrices

Jens Bolte (Royal Holloway, University of London)

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schrödinger equation. We construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantisation condition for eigenvalues as well as some simple examples.

Donnerstag, 30.06.2016: On the uniqueness of small surfaces minimizing the Willmore functional subject to a small area constraint

Prof. Dr. Jan Metzger (Universität Potsdam)

We consider the Willmore functional for surfaces immersed in a compact Riemannian manifold M and study minimizers subject to a small area constratint. We show that if the scalar curvature of M has a non-degenerate maximum then for small enough area these minimizers are unique. This is joint work with Tobias Lamm and Felix Schulze.