Christian Bär
Wintersemester 2015/16
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Donnerstags, 16:15
Campus Golm, Haus 9, Raum 1.11
| Datum | Vortrag | Referent | Inhalt |
| 15.10.15 | Differential cohomology with compact support | Christian Becker | In this talk we introduce differential cohomology with compact support. There are several different models for differential cohomology. We use the model of differential characters which is originally due to J. Cheeger and J. Simons. |
| 22.10.15 | Locally covariant QFT and the gauge puzzle | Marco Benini | Using the simple example of a free scalar field, I will illustrate the axiomatic formulation of locally covariant quantum field theory (LCQFT). With this framework in mind, the case of certain Abelian gauge theories will be discussed. In particular, I will present a model for free electromagnetism (and higher analogues) and its variations in terms of doubled and self-dual fields. As it turns out, despite being among the easiest examples of a genuine gauge theory, these models do not fulfil the locality axiom of LCQFT, i.e. the requirement that causal embeddings between spacetimes induce injective morphisms between the associated (C*-)algebras of observables. This observation motivates us to look for a softer version of LCQFT that fits gauge theories too. For this purpose, we propose an homotopy theoretic approach to the problem. REFERENCES [1] R. Brunetti, K. Fredenhagen and R. Verch,The Generally covariant locality principle: A New paradigm for local quantum field theory,Commun. Math. Phys. 237 (2003) 31 [math-ph/0112041]. [2] C. Bär, N. Ginoux and F. Pfäffle,Wave equations on Lorenzian manifolds and quantization,Zürich, Switzerland: Eur. Math. Soc. (2007). [3] C. Becker, A. Schenkel and R. J. Szabo,Differential cohomology and locally covariant quantum field theory,arXiv:1406.1514 [hep-th]. |
| 29.10.15 | Asymptotic expansion of path integrals | Matthias Ludewig | Given a parameter-dependent integral of the form $\int_M e^{-\phi(x)/2t} a(x) dx$ on a Riemannian manifold, it has an asymptotic expansion for small times, which can be calculated using the Laplace method. We then discuss a heuristic, infinite-dimensional version of the Laplace-method that can be used to formally associate an asymptotic expansion to path integrals, i.e. integrals over infinite-dimensional domains. Finally, we show how parts of it can be made rigorous using finite-dimensional approximation methods. |
| 05.11.15 | The scalar wave equation in Schwarzschild coordinates | Ariane Beier | |
| 12.11.15 | Spectral functions of Dirac operators | Alexander Strohmaier | I will review some known general expansions of microlocal spectral counting functions of Dirac and Laplace operators. In special cases these relate directly to heat kernel coefficients. I will then explain some differences between expansions for Laplace and Dirac type operators and show that generalized Dirac operators can be characterized amongst Dirac type operators by the vanishing of a certain coefficient appearing in the expansion of the spectral counting function. (Joint work with Liangpan Li) |
| 19.11.15 | Intrinsische Metriken auf Graphen | Matthias Keller | |
| 26.11.15 | Path Integrals, Zeta-Determinants and the Gelfand-Yaglom Theorem | Matthias Ludewig | It is "well-known" in quantum field theories that the values of certain path integrals are given by associated zeta-determinants "up to a multiplicative constant". What is usually meant is that one can only calculate the ratio of path integrals by the ratio of zeta functions. The "relativity principle" for zeta functions in turn relates this to usual Hilbert space determinants. We give a rigorous meaning to all these statements and show how finite-dimensional approximation of path integrals can be used to prove them. |
| 03.12.15 | The Hamiltonian formulation of general relativity and the space of initial data | Oliver Lindblad Petersen | We recall the Hamiltonian formulation of Einstein's vacuum equation and explain the exact meaning of the lapse function and the shift vector. In this form, Einstein's equation can be seen as a flow on the space of initial data. We discuss the relation to the classical linearisation stability results and conclude with some remarks on the meaning of "gauge invariance" in this setting. |
| 10.12.15 | A new approach to physical solutions of scalar wave equations on globally hyperbolic Lorentzian manifolds | Max Lewandowski | After Radzikowski's celebrated equivalence theorem in the 1990's microlocal analysis and especially the wave front set of solutions of wave equations on globally hyperbolic Lorentzian manifolds experienced growing interest in the QFT on CST society. For solutions of the Klein Gordon equation he proved that their particular singularity structure on the light cone is equivalent to a certain condition on their wave front set. In my talk I will consider scalar solutions of d'Alembert's equation on Minkowski space which for microlocal reasons yield the same singularity structure as scalar solutions of general wave equations on Lorentzian manifolds. Instead of starting with Wightman's axioms I will present another approach finding appropriate solutions involving Riesz distributions and distributional regularization instead and in the end show that they coincide with those "physical solutions" Radzikowski's Theorem is aimed at. |
| 07.01.16 | Visualisierung von Lösungen der Wärme- und Wellengleichung mittels SAGE | Christian Bär | Mit Hilfe der Computeralgebra-Software SAGE berechnen und visualisieren wir Lösungen der Wärmeleitungsgleichung und der Wellengleichung auf flachen 2-dimensionalen Tori. Nach einer kurzen Zusammenfassung des mathematischen Hintergrunds wird auf programmiertechnische Aspekte eingegangen. |