Christian Bär
Wintersemester 2013/14
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Mo, 14:15-15:45
Haus 22, Raum 1.28
| Datum | Vortrag | Referent | Inhalt |
| 14.10.13 | Relative differential cohomology | Christian Becker | In this talk I will report on recent results concerning notions of relative differential cohomology.
Differential cohomology is a refinement of integral cohomology of a smooth manifold by differential forms.
In low degrees, differential cohomology classifies well-known geometric structures: U(1)-valued functions (in degree 1) and hermitian line bundles with connection (in degree 2). There exist several models for differential cohomology. In the talk I briefly recall the notion of differential characters, the original model, due to Cheeger and Simons. These characters are certain U(1)-valued homomorphisms on the group of smooth singular cycles. In degree 2 such a character is the same thing as the holonomy map of the corresponding hermitian line bundle with connection. From the perspective of differential characters, there arise two notions of relative differential cohomology (taking into account other models, there are even more). These are obtained by considering characters on two versions of cycles that represent relative homology classes. I will explain how these notions arise and derive long exact sequences for both. These relate absolute and relative differential cohomology groups. In degree 2, the characters again classify well-known objects: hermitian line bundles with connection and section (up to isomorphisms that preserve both) and hermitian line bundles with connection and parallel section. |
| 21.10.13 | Locally covariant charged Dirac fields | Jochen Zahn | The framework of locally covariant field theory, due to Brunetti, Fredenhagen & Verch and Hollands & Wald, proved extremely fruitful for the formulation of quantum field theories on curved spacetimes. In the talk, I present a recent generalization of the framework to accommodate fields charged under a gauge group. As an example, I discuss the charged Dirac field and possibly some applications. |
| 28.10.13 | Twisted Dirac structures and geometric quantization | Alexander Cardona | During this talk we will recall the definition of Dirac structures on manifolds and their use in Poisson geometry. Then we will discuss the definition of geometric quantization in the context of symplectic geometry and its generalization to the case of Dirac structures in the twisted case. |
| 04.11.13 | Green-hyperbolic operators I - Geometric foundations | Christian Bär | We introduce various compactness properties for closed subsets of globally hyperbolic manifolds and show their interrelation. These considerations will later be applied to the supports of sections. |
| 11.11.13 | Green-hyperbolic operators II - Function spaces | Christian Bär | We study various spaces of smooth sections of a vector bundle over a globally hyperbolic manifold. The crucial concept is that of a support system. This is a family of closed subsets of our manifold with certain properties making it suitable for defining a good space of sections by demanding that their supports be contained in the support system. We observe a duality principle; a distributional section has support in a support system if and only if it extends to a continuous linear functional on test sections with support in the dual support system. |
| 18.11.13 | Green-hyperbolic operators III - General properties | Christian Bär | We introduce Green’s operators and Green-hyperbolic differential operators. We give various examples and show that the class of Green-hyperbolic operators is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking “square roots”, and under the direct sum construction. This makes it a large and very flexible class of differential operators to consider. We show that the Green’s operators are unique and that they extend to several spaces of sections. We argue that Green-hyperbolic operators are not necessarily hyperbolic in any PDE-sense and that they cannot be characterized in general by well-posedness of a Cauchy problem. |
| 25.11.13 | Green-hyperbolic operators IV - Symmetric hyperbolic systems | Christian Bär | We study symmetric hyperbolic systems over globally hyperbolic manifolds. We provide detailed proofs of well-posedness of the Cauchy problem, finiteness of the speed of propagation and the existence of Green’s operators. The crucial step in these investigations is an energy estimate for the solution to such a symmetric hyperbolic system. We conclude by observing that a symmetric hyperbolic system can be quantized in two ways; one yields a bosonic and the other one a fermionic locally covariant quantum field theory. |
| 16.12.13 | Einige Aspekte des Lorentz'schen Yamabe-Problems | Nicolas Ginoux | Das Yamabe-Problem fragt nach der Existenz einer konformen Metrik mit konstanter Skalarkrümmung auf einer gegebenen pseudo-riemannschen Mannigfaltigkeit. Wir untersuchen dieses Problem auf Lorentz-Mannigfaltigkeiten, insbesondere globalhyperbolischen Raumzeiten und stellen einige Existenz- und Nichtexistenzergebnisse vor. |
| 13.01.14 | Elastische Membranen | Claudia Grabs | Im Vortrag werden die nötigen Gleichungen zur Berechnung der Dynamik einer elastischen Membran im umgebenden euklidischen Raum vorgestellt und teilweise hergeleitet.
Dabei soll zunächst überblicksartig Ordnung in den Zoo der Tensoren der Elastizitätstheorie (Verzerrungstensoren, Spannungstensoren, Verzerrungsratetensoren, ...) gebracht werden. Aus den klassischen Erhaltungssätzen der Kontinuumsmechanik ergeben sich dann die lokalen Bewegungsgleichungen, bei denen zwischen tangentialen und normalen Anteilen der Bewegung der Membran unterschieden werden kann. Aus energetischen Überlegungen ergibt sich weiterhin der Zusammenhang zwischen Spannungen und Verzerrungen für verschiedene Materialien. Kombiniert man die Bewegungsgleichungen mit einem konkreten Materialgesetz, so kann dann die Bewegung der Membran vollständig beschrieben werden. So lassen sich dann vermutlich verschiedene Anfangs- und Randwertprobleme für elastische Membranen lösen. |
| 20.01.14 | Globally non-trivial almost-commutative manifolds | Koen Van Den Dungen | Within the framework of Connes' noncommutative geometry, the special case of almost-commutative manifolds is of particular interest for the construction of models in particle physics (such as the full standard model or possible extensions thereof).
These almost-commutative manifolds can be described in terms of globally trivial vector bundles.
In this talk I will describe how to generalise this approach to the globally non-trivial case.
I will focus on those globally non-trivial almost-commutative manifolds which can be constructed from a principal fibre bundle and a finite spectral triple, and show that these describe a gauge theory on the underlying manifold.
I will also discuss till what extent these objects can be described in purely operator-algebraic terms. This talk is based on joint work with Jord Boeijink (Radboud University Nijmegen). |
| 27.01.14 | Symmetry Reduction and Invariant Connections in Loop Quantum Gravity | Maximilian Hanusch | In gauge field theories, symmetries usually are represented by Lie groups of automorphisms of the underlying principal fibre bundle. Quantization of the corresponding set of invariant connections provides a notion of a reduced quantum configuration space. Alternatively, one might aim at a corresponding symmetry reduction directly on the quantum level. In this talk we first discuss the former reduction concept for the case of loop quantum cosmology. Here we investigate some measure theoretical aspects of the symmetry reduced space in detail. Then we introduce an algebraic characterisation of invariant connections that often allows for their explicit calculation. In particular, we go beyond the respective classical result by Wang as we drop the assumption of fibre-transitivity. Finally, by means of C*-dynamical systems we lift the classical symmetry directly to the quantum level. It turns out that in general quantization and reduction do not commute. Again we give some details for the case of LQC. |
| 03.02.14 | On the characteristic initial value problem for nonlinear symmetric hyperbolic systems, Part I | Roger Tagne Wafo | We consider a characteristic initial value problem for a class of quasi-linear symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces.
We prove existence of solutions on a future neighborhood of the initial surfaces.
The general result is applied to general semilinear wave equations. The slides for the talk can be found here. |
| 10.02.14 | On the characteristic initial value problem for nonlinear symmetric hyperbolic systems, Part II | Roger Tagne Wafo | Second part of the talk. The slides for the talk can be found here. |