Christian Bär
Sommersemester 2014
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Donnerstags, 16:15-17:45
Haus 8, Raum 0.53
| Datum | Vortrag | Referent | Inhalt |
| 10.04.14 | Bianchi type I solutions to Einstein's vacuum equations | Oliver Lindblad Petersen | {tex} \noindent In this talk we consider solutions to Einstein's vacuum equations starting with the Bianchi type I spacetimes. A Bianchi type I spacetime is given by the manifold $I \times \mathbb{R}^3$ equipped with a metric of the form $-dt^2 + \sum_{i=1}^3 a_i(t)^2 dx^i \otimes dx^i $, where $a_i:I \rightarrow \mathbb{R}_+$. Hence these spaces are spatially homogeneous. Nevertheless there is no assumption on spatial isotropy, and the solutions will indeed be anisotropic in general.\\ Apart from the trivial solution (Minkowski space), two different classes of metrics arise. The Flat and the Non-flat Kasner metrics. In both cases there will be a singularity in the past such that the geodesics cannot be extended beyond this point. In the case of Flat Kasner metric there is an isometric embedding into Minkowski space, but in the case of Non-flat Kasner metric the curvature blows up approaching this singularity and the spacetime is hence inextendable. In addition to this, considering the 3-torus as a submanifold of $\mathbb{R}^3$, we see that the volume of the 3-torus grows proportional to the time. {/tex} |
| 17.04.14 | Surgery and the Positive Mass Conjecture | Andreas Hermann | The Positive Mass Conjecture from General Relativity has been proved in some special cases (e.g. for manifolds of dimension at most 7 or for spin manifolds) but the general case is still subject to current research. In this talk we present a surgery result obtained with Emmanuel Humbert which might help to give a proof in the general case. |
| 24.04.14 | Path integrals on manifolds with boundary | Matthias Ludewig | We give path integral representations of the solution to the heat equation on various manifolds. To define our path integral, we use approximation of the path space by spaces of piecewise geodesics, as opposed to the more commonly used Wiener measure. This has the advantage of being closer to the original ideas of Feynman, in particular, our approach is applicable to the Schrödinger equation as well. As for the case of a manifold with boundary, we investigate the „reflected exponential map“ and highlight the difficulties coming from differently curved boundary. We give results in various special cases. |
| 08.05.14 | TQFT 1 | TBA | |
| 15.05.14 | TQFT 2 | Florian Hanisch | Having finished with 1-dimensional field theories in the previous talk, we now reduce the source dimension to zero, but add a super dimension. We will discuss the needed background on supermanifolds that are needed here. The reason to deal with super dimensions in the first place is that this gives the grading to the space of field theories needed to obtain a cohomology function. In our particular case, 0|1-dimensional field theories give rise to differential forms, while 0-dimensional field theories only yield smooth functions. |
| 27.05.14 | TQFT 3 | Christian Becker | 1-dimensional topological field theories essentially give rise to the 0-degree part of K-Theory, the stable isomorphism classes of vector bundles.
In this talk, we discuss how adding by a super dimension, we obtain K-theory of all degrees.
If time suffices, we might even discuss 2|1-dimensional field theories.
2|1 conformal field theories are conjectured to give rise to topological modular form theory (TMF), a certain generalized cohomology theory. Room 1.22.1.27 |
| 27.05.14 | Differential cohomology and locally covariant QFT | Alexander Schenkel | In this talk I will present ongoing joint work with Christian Becker and Richard J. Szabo on the quantization of differential cohomology theories.
Differential cohomology is a differential refinement of singular cohomology, which among other things has led to interesting refinements of the theory of characteristic classes, e.g. new invariants for flat connections.
In mathematical physics, differential cohomology can be used to give a nice and efficient description of the gauge orbit spaces of (higher) Abelian gauge theories.
In low degrees these are the sigma-model with target space being a circle, isomorphism classes of circle bundles with connections (Maxwell theory) and isomorphism classes of gerbes with connections.
We provide a construction of the quantum field theory functor for any degree k differential cohomology theory, therewith formulating the quantum field theory for all models listed above (and even much higher ones).
Our approach makes heavy use of the natural exact sequences and diagrams defining an abstract differential cohomology theory, which are going to reveal a rich subtheory structure of the quantum field theory (i.e. existence of subfunctors).
In particular, the (C*-algebras generated by the) Pontryagin duals of certain singular cohomology theories are subtheories and we prove that the infamous violation of the locality axiom is precisely due to these purely topological subtheories. Room 1.08.0.53 |
| 12.06.13 | Infinite-dimensional geometry of mapping spaces | Florian Hanisch | We give a brief introduction into the convenient setting by Kriegl and Michor and show how it can be used to equip spaces of smooth mappings with structures of infinite-dimensional manifolds. In case that the target is (the total space of) a vector bundle, we will show that the mapping space carries an induced bundle-structure. In this context, we will adress the role of parallel transport and its smoothness properties. If time allows, we will indicate how this can be used to define mapping spaces in the / a category of supermanifolds. |
| 26.06.14 | Dynamical Torsion within General Relativity and Dirac Operators of Simple Type | Jürgen Tolksdorf | I will discuss dynamical torsion within the geometrical setup of a distinguished class of Dirac operators. The Euler-Lagrange equations of the resulting variational problem are shown to look similar to the coupled Einstein-Dirac-Yang-Mills equations. I will comment on some special features in the case of two dimensional Riemannian manifolds. |
| 10.07.14 | Geometrische String-Strukturen und Chern-Simons-Theorie | Christian Becker | Unter String-Geometrie verstehen wir geometrische Strukturen auf einer Mannigfaltigkeit M, die sogenannten Spin-Strukturen auf dem Schleifenraum L(M) entsprechen.
Letztere wurden in den 1980er Jahren von Killingback eingeführt, motiviert durch zweidimensionale Feldtheorien (sog. nichtlineare Sigma-Modelle) und Stringtheorie.
In moderner Sprache werden geometrische String-Strukturen durch Trivialisierungen geeigneter Bündel-2-Gerben und deren Zusammenhänge beschrieben. In dem Vortrag werde ich zunächst die geometrische Begrifflichkeit (Gerben, Flächenholonomie etc.) einführen und die Transgression zum Schleifenraum erläutern. Die klassische Chern-Simons-Wirkung kann man als Holonomie längs 3-Mannigfaltigkeiten verstehen, analog liefert die Wess-Zumino-Witten-Wirkung eine Flächen-Holonomie. Geometrische String-Strukturen bzw. -Klassen setzen sich aus diesen Daten zusammen. |
| 17.07.14 | Loop group geometry and transgression | Konrad Waldorf | Brylinski's transgression functor takes abelian gerbes over a manifold M to line bundles over the free loop space LM. We will describe a characterization of the image of Brylinski's functor in terms of additional structure for line bundles over loop spaces, most prominently, a fusion product. We will explain that the universal central extension of a compact Lie group fits into this picture and so has interesting new features. As an application, we describe a refinement of Killingback's notion of spin structures on a loop space LM which is equivalent to string structures on the manifold M itself. |
| 04.09.14 | Generalized Killing Spinors on Riemannian Spin^c Manifolds | Roger Nakad | In a joint work with Nadine Große, we extend the study of generalized Killing spinors on Riemannian Spin^c manifolds started by Moroianu and Herzlich to complex Killing functions.
We prove that such spinor fields are always real Spin^c Killing spinors or imaginary generalized Spin^c Killing spinors, provided the dimension of the manifold is greater or equal to 4.
Moreover, we classify Riemannian Spin^c manifolds carrying imaginary and imaginary generalized Killing spinors. Haus 8, Raum 0.53 |
| 04.09.14 | The wave equation and redshift in Bianchi type I spacetimes | Oliver Lindblad Petersen | In the first part of the talk, we show how the solution to the scalar wave equation on the 3-torus-Bianchi type I spacetime can be written as a Fourier decomposition.
We present results on the behaviour of these Fourier modes and apply them to the case of Kasner spacetimes. In the second part of the talk, we consider the Cauchy problem for Maxwell's vacuum equations, with special initial data, in order to model light in Bianchi type I spacetimes. We calculate the redshift of the solution and show that it coincides with the cosmological redshift. Haus 8, Raum 0.53 |