Vorträge in der Woche 15.12.2014 bis 21.12.2014

Dienstag, 16.12.2014: Grundzüge der Seibert-Witten-Floer-Homologie

Jonathan Walz (Tübingen)

Dienstag, 16.12.2014: Equivariant cohomology of isotropy actions

Michael Füllbier (Tübingen)

Mittwoch, 17.12.2014: Geometry in solid state physics: the case study of topological insulators

Domenico Monaco (Rom)

In this talk, I will present some new results regarding topological quantum phases of matter, in the context of time-reversal symmetric (TRS) topological insulators in 2 and 3 dimensions. I will first review how geometric information can be extracted from periodic gapped quantum systems, modeling e.g. electrons in crystalline solids. This information is encoded in integer-valued topological invariants, namely Chern numbers of the so-called Bloch bundle. In 2d systems, the Chern number is known to measure the obstruction to the existence of a smooth periodic Bloch frame for an isolated set of energy bands of the Hamiltonian, and is related to physical observables such as the transverse conductivity in the Quantum Hall Effect. I will then explain how the presence of further symmetries, such as time-reversal, leads to a "refinement" of the geometry. This is relevant in the framework of 2d and 3d TRS topological insulators, whose quantum phases are labelled by integers mod 2 and are distinguished by the presence of edge spin currents. Our result allows to identify these Z_2-valued invariants again as geometric obstructions. The results were obtained in collaboration with D. Fiorenza and G. Panati ("La Sapienza" Università di Roma).

Donnerstag, 18.12.2014: Toward a justification of variational asymtotics for multiscale systems with strong gyroscopic forcing

Marcel Oliver (Jacobs University, Bremen)

In systems with strong gyroscopic forces, approximate equations for the dynamics on a slow manifold can be found via variational asymptotics. The results generally differ from those obtained by classical Hamiltonian normal form theory. We explain the method, using the non-relativistic limit of the nonlinear Klein-Gordon equation as an example, and prove a shadowing result for this particular case.

Donnerstag, 18.12.2014: Mass-Angular momentum-Charge inequality in General Relativity

Dr. Ye Sle Cha (University of Oxford)

We consider the axisymmetric initial data of the Einstein-Maxwell equations, with a simply connected Riemanninan manifold having two ends, one asymptotically flat and the other either asymptotically flat or asymptotically cylindrical. Penrose's heuristic arguments relate the ADM mass, charge, and the angular momentum of the initial data via the mas-angular momentum-charge inequality. This has been proven when the initial data is maximal. Here we show how to reduce the general formulation for the non-maximal initial data, to the known maximal case, whenever a geometrically motivated system of two elliptic equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality for the scalar curvature. Each equation in the system is analyzed in detail individually, and it is shown that the appropriate existence/uniqueness of results holds with the solution satisfying the desired asymptotics. This is a joint work with Marcus Khuri.

Freitag, 19.12.2014: Surfaces minimizing the Willmore energy for small prescribed area

Prof. Dr. Jan Metzger (Universität Potsdam)

In this talk I will consider the Willmore functional for surfaces in three dimensional Riemannian manifolds subject to (small) prescribed area. The goal is to discuss old and new results on the existence of such surfaces and their interaction with the ambient geometry. This is joint work with Tobias Lamm and Felix Scchulze.