Sommersemester 2015
Es werden Themen aus dem Grenzbereich zwischen Differentialgeometrie, mathematischer Physik und stochastischer Analysis behandelt.
Vorträge aus den Arbeitsgebieten der Teilnehmer
Montags, 16.00-17.30
Haus 8, Raum 0.50
| Datum | Vortrag | Referent | Inhalt |
| 27.04.2015 | String structures and connections | Christian Becker |
Let $X$ be a compact Riemannian $n$-manifold, with $n \geq 3$.
The bundle of orthonormal frames is a principal $O_n$-bundle.
Several geometric structures on $X$ can be described in terms of lifts of the structure group along a group homomorphism $G \to O_n$:
orientations are lifts to the connected component $SO_n \to O_n$, Spin structures are further lifts to the simply connected cover $\mathrm{Spin}_n \to SO_n$.
String structures are lifts to a $3$-connected cover of $\mathrm{Spin}_n$.
This group is usually called $\mathrm{String}_n$.
It is defined only up to homotopy and cannot be realized as a finite dimensional Lie group. String structures on $X$ can be described in terms of homotopy theory. The obstruction to the existence of a String structure is a certain characteristic cohomology class on $X$. Isomorphism classes of String structures correspond to certain cohomology classes on the Spin structure. However, there also exists a model of the group $\mathrm{String}_n$ as an infinite dimensional Fr\'echet Lie group. It is built as a subgroup of the gauge group of a certain bundle over $\mathrm{Spin}_n$. In this talk, based on joint work in progress with Christoph Wockel, we describe actual lifts of the structure group of a Spin structure to the group $\mathrm{String}_n$. These are built from gauge group bundles. We also discuss the construction of connections on our String bundles. These are built from connections on the underlying Spin bundles and additional data, so-called Higgs fields. |
| 04.05.2015 | The spectrum of the Dirac operator for generic metrics | Andreas Hermann |
Let $M$ be a closed spin manifold of dimension $n\geq 2$.
For every Riemannian metric on $M$ we define the spinor bundle on $M$, a complex vector bundle whose sections are called spinors.
We also define the Dirac operator, an elliptic differential operator of first order acting on spinors.
Explicit computations of the spectrum of the Dirac operator are only possible for particular metrics (e.g. round metrics on spheres, flat metrics on tori). In this talk I will give an overview on what is known about the spectrum of the Dirac operator for a generic choice of metric. I will describe some results by Dahl and Ammann-Dahl-Humbert about the eigenvalues of the Dirac operator and then explain my own results about zero sets of eigenspinors. |
| 18.05.15 | Applications of supergeometry in mathematics and physics | Florian Hanisch | Supergeometry aims at constructing spaces, whose algebras of functions contain fermionic observables, i.e. functions anticommuting with each other. We give a short introduction into the basic ideas of this subject, discussing the ringed space- as well as the functorial picture. The latter approach allows for a natural definition of mapping spaces and we will explain why these concepts are crucial for applications. Eventually, we will sketch some of these applications (e.g. construction of phase spaces for fermionic classical field theories) in more detail. |
| 15.06.15 | Renormalisation schemes in QFT, Number theory and combinatorics on cones | Sylvie Paycha |
The objects of study are multiple integrals and multiple sums with linear constraints arising from Feynman integrals, multizeta functions or exponential sums on cones, which we want to evaluate even though they might diverge.
We shall discuss various renormalisation methods to do so, which raises the question as to how to relate the different renormalised values they lead to. More precisely, starting from the observation that the divergences arise in the form of linear forms in the variables, our problem boils down to evaluating at the poles, meromorphic functions in several variables whose poles are linear. Assuming we can distinguish the pole part of the meromorphic function from its holomorphic part in an unambiguous manner, we can evaluate it at a pole by taking the value of its holomorphic part at the pole. We show how this can be carried out by means of an inner product on the underlying integration or summation space. Alternatively, the multiple integrals and multiple sums under consideration can be viewed as maps from a set of constraints--modelled by Feynman graphs or cones in our examples-- with values in a set of meromorphic functions. A complement map on the set of constraints can then be used to pick out subdivergences corresponding to a subset of constraints (a subdiagram, a face of a cone) which has the same effect on the meromorphic functions as the above procedure, since it separates the pole part from the holomorphic part in an unambigous manner, thus yielding a second renormalisation device. A third option is - as in the case of multizeta functions at poles--to fix the set of constraints (here a Chen cone) and to vary equip the set of integrands or the summands (here tensor products of symbols) with a complement map. Again this will have the effect of separating the pole part of the meromorphic functions from their holomorphic part in an unambigous manner, yielding a third renormalisation device. How these three renormalisation devices relate is yet unclear and requires identifying an underlying renormalisation group. |
| 29.06.15 | Anomalies in Quantum Field Theory: An Essay | Christoph Stephan |
Anomalies in a Quantum Field Theory describe the phenomenon that a symmetry of the classical theory is not necessarily a symmetry of the quantised theory.
Anomalies can be physically beneficial (e.g. for global symmetries) or disasterous (in the case of gauge symmetries). I will try to give an overview of the different types of anomalies and how they are related to topology and index theorems of certain Dirac operators. Unfortunately most of these considerations require the "space-time" manifold to be Riemannian. The distant aim of my considerations is to establish a rigorous connection between anomalies of Quantum Field Theories on Lorentzian manifols and the index theorem for Lorentzian manifolds recently published by Christian Bär and Alexander Strohmaier. |
| 06.07.15 | Koflächenformel für Lipschitz-Abbildungen zwischen Euklidischen Räumen | Christian Scharrer | nach Evans/Gariepy "Measure Theory and Fine Properties of Functions" Par. 3.4 und Federer "Geometric Measure Theory" 3.2.10 |
| 13.07.15 | Tunneling for a class of difference operators | Elke Rosenberger | I consider difference operators on an epsilon-scaled lattice with a multi-well potential. For the interaction between the wells - the tunneling - I give a general formula and show how, in special situations, it is possible to derive complete asymptotic expansions. |