Vorträge in der Woche 25.06.2018 bis 01.07.2018

Dienstag, 26.06.2018: Quanteninvarianten von Knoten

Prof. Dr. Christoph Bohle (Universität Tübingen)

Der Plan meines (Doppel-)Vortrags ist, die geometrischen Ideen hinter den Knoteninvarianten von Reshetikhin und Turaev zu erklären und die Axiome von sogenannten Ribbon-Kategorien zu "malen". Die Theorie der Knoteninvarianten von Reshetikhin und Turaev kann man ausbauen zu einer (3,2,1)-topologischen Quantenfeldtheorie, die eine mathematische Präzisierung der Chern-Simons-Witten Quantenfeldtheorie ist. Der Fall von Knoten, auf den wir uns beschränken, entspricht dabei dem Fall von Invarianten der 3-Sphäre mit 1-dimensionalen "Defekten".

Mittwoch, 27.06.2018: On the classification of null hypersurfaces in Robertson Walker spacetimes

Dr. Didier Solis (Universidad Autónoma de Yucatán (UADY))

One of the most distinctive feature in Lorentzian geometry is the presence of null (degenerate) submanifolds. In this talk we address some recent developments in the classification of null hypersurfaces subject to geometric constraints immersed in Robertson-Walker spacetimes. In particular, we will focus on umbilical, isoparametric and Einstein null hypersurfaces. This is joint work with Oscar Palmas (UNAM) and Matias Navarro (UADY).

Donnerstag, 28.06.2018: ¼ Jahrtausend Eulers Calculus Integralis (nebst einer modernen Anwendung)

Prof. Gerhard Wanner (Universität Genf)

Donnerstag, 28.06.2018: Der spektrale Abbildungssatz für dynamische Banach-Moduln

S. Siewert (Tübingen)

Ziel des Vortrags ist es, das asymptotische Verhalten von Kozykeln über Flüssen mit Hilfe gewisser Eigenschaften des Spektrums des zugehörigen dynamischen Banach-Moduls zu untersuchen. Hierfür werden wir zunächst exponentielle Dichotomie eines dynamischen Banach-Moduls definieren und diese anhand des Spektrums charakterisieren. Anschließend werden wir zeigen, dass, unter geeigneten Voraussetzungen, der spektrale Abbildungssatz für dynamische Banach-Moduln gilt.

Donnerstag, 28.06.2018: A Direct Description of Nelson-Type Hamiltonians via Boundary Conditions

Julian Schmidt (Tübingen)

When quantum particles are linearly coupled to a real bosonic field, the standard approach is to apply a UV-renormalisation method in order to define a Hamiltonian for the system. The domain and the action of the Hamiltonian obtained by this procedure are however hard to describe. In this talk, I will present a new approach to the problem of defining such UV-divergent Hamiltonians on Fock space. In joint work with Jonas Lampart, a direct description of the Hamiltonian without UV-cutoff for a class of models -- including the so-called Nelson model -- has been obtained. After introducing the models and briefly explaining the problem and the standard method, I will construct the Hamiltonian for the model without cutoffs directly by using abstract interior-boundary conditions. These boundary conditions relate sectors with a different number of bosons but the techniques are similar to those employed in the theory of many-body point interactions. I will conclude by discussing more recent results and possible future projects.

Freitag, 29.06.2018: Consistency of relative neighborhood classification rules

Tobias Frangen (Uni Tuebingen)

In a classification task we are dealing with the problem of finding a function g, called a classifier, that for every point x of an input space X predicts a label y in a (usually finite) label space Y. Therefore we have access to training points (x_1,y_1),...,(x_n,y_n)\in X\times Y, which are realisations of i.i.d. random variables distributed according to some underlying probability distribution of (X,Y). The quality of a classifier can be measured by the expected 0-1-loss between the assigned label g(X) and the true label Y. A relative neighborhood classification rule uses a data dependent distance notion on X to estimate P[Y\in {y} | X] and to predict a label. This thesis shows that for a large class of distributions of the random variable (X,Y) on R^d \times R, the resulting classifier is nearly optimal with high probability if we sample enough training points.