Christian Bär
Wintersemester 2016/17
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 0.12
| Datum | Vortrag | Referent | Inhalt |
| 19.10.16 | An analogue of the Coleman-Mandula theorem for QFT in curved spacetimes | Chris Fewster | The Coleman--Mandula (CM) theorem states that the Poincaré and internal symmetries of a Minkowski spacetime quantum field theory cannot combine nontrivially in an extended symmetry group. In this talk I will describe some of the background to the CM theorem and then establish an analogous result for a general class of quantum field theories in curved spacetimes. Unlike the CM theorem, our result is valid in dimensions n>=2 and for free or interacting theories. It makes use of a general analysis of symmetries induced by the action of a group G on the category of spacetimes. Such symmetries are shown to be canonically associated with a cohomology class in the second degree nonabelian cohomology of G with coefficients in the global gauge group of the theory. The main result proves that the cohomology class is trivial if G is the universal cover S of the restricted Lorentz group. Among other consequences, it follows that the extended symmetry group is a direct product of the global gauge group and S, all fields transform in multiplets of S, fields of different spin do not mix under the extended group, and the occurrence of noninteger spin is controlled by the centre of the global gauge group. |
| 27.10.16 | The Cauchy problem for the linearised Einstein equation | Oliver Lindblad Petersen | In this talk we discuss the well-posedness of the Cauchy problem for the linearised Einstein vacuum equation on arbitrary globally hyperbolic vacuum spacetimes.
The solution space of the linearised Einstein equation (graviational waves) modulo gauge solutions is shown to be in one-to-one correspondence with initial data modulo gauge.
This correspondence is given by an isomorphism of topological vector spaces.
One concludes global existence, uniqueness and continuous dependence on initial data.
This extends our results presented in an earlier research talk, where the special case of spatially compact spacetimes was considered. The statement is shown for smooth and distributional initial data of arbitrary Sobolev regularity. As an application, we give examples of spacetimes with arbitrarily irregular (non-gauge) solutions to the linearised Einstein equation. |
| 03.11.16 | On the construction of the Green operators and of the ground state for a massive scalar field theory in AdS | Claudio Dappiaggi | We consider a real, massive scalar field on the Poincaré domain of the (d+1)-dimensional AdS spacetime. Since the background is not globally hyperbolic, first we determine all admissible boundary conditions that can be applied on the conformal boundary and, subsequently, we address the problem of constructing the advanced and retarded Green operators as well as the two-point function for the ground state satisfying those boundary conditions. We unveil that, in some cases, bound states exist, while, when they are absent, the two-point function can be explicitly written in terms of special functions. In addition, we investigate the singularities of the resulting state, showing that they are consistent with the requirement of being of Hadamard form in every globally hyperbolic subregion of the Poincaré patch. |
| 17.11.16 | Zeta Functions and Mass of Conformal Differential Operators | Matthias Ludewig | I will explain the relation between the Green’s function of a Laplace type operator and its zeta function. In particular, we will see that the constant term in the asymptotic expansion (which is often called the mass of the operator) is given by a zeta value. If the operator is conformally invariant, it is well-known that certain zeta values give rise to conformal invariants. In particular, we show that the mass is such an invariant in odd dimensions. |
| 15.12.16 | The geometry of semiclassical limits on regular Dirichlet spaces | Batu Güneysu | In this talk, I will first explain how one can reformulate the known semiclassical limit results for the heat trace of Schrödinger operators on Riemannian manifolds and infinite weighted graphs in a form which makes sense for abstract Schrödinger type operators on locally compact spaces. Then I will give a probabilistic proof of this reformulation in case the "free operator" stems from a regular Dirichlet form which satisfies a principle of not feeling the boundary. This abstract result leads to completely new results for Schrödinger operators on arbitrary complete Riemannian manifolds, and allows to recover the known results for weighted infinite graphs. |
| 12.01.17 | First variation of the mass of a closed Riemannian manifold | Andreas Hermann | Let (M,g) be a closed Riemannian manifold such that all eigenvalues of the conformal Laplace operator L_g are strictly positive and such that g is flat on an open neighborhood of a point p.
The constant term in the expansion of the Green function of L_g at p is called the mass of (M,g) at p. In this talk we investigate the dependence of the mass on the Riemannian metric. We compute the first variation of the mass with respect to a change of metric and we discuss critical points of the mass. This is joint work with Emmanuel Humbert. |
| 19.01.17 | Index of the Dirac Operator on Lorentzian Spacetime | Sebastian Hannes | We will prove Fredholm property of the Dirac operator on a globally hyperbolic spacetime under generalized APS boundary conditions and their deformations. Then we can derive relative index formulas for the involved operators. |
| 26.01.17 | Spectral flow and the Riesz stability of the Atiyah-Singer Dirac operator under bounded perturbations of local boundary conditions | Lashi Bandara | We study the Atiyah-Singer Dirac operator on smooth Riemannian Spin manifolds with smooth compact boundary. Under lower bounds on injectivity radius and bounds on the Ricci curvature and its first derivatives, we demonstrate that this operator is stable in the Riesz topology under bounded perturbations of local boundary conditions. Our work is motivated by the spectral flow and its connection to the Riesz topology. These results are obtained by obtaining similar results for a more wider class of elliptic first-order differential operators on vector bundles satisfying certain general curvature conditions. At the heart of our proofs lie methods from Calderón-Zygmund harmonic analysis coupled with the modern operator theory point of view developed in proof of the Kato square root conjecture. |
| 02.02.17 | Konforme Geodäten im Cartankalkül | Daniel Platt | Ein Cartan-Zusammenhang ist eine 1-Form, die dieselben Invarianzeigenschaften wie ein Hauptfaserbündel-Zusammenhang hat und zusätzlich einen absoluten Parallelismus (in eine geeignete Lie-Algebra) in jedem Punkt definiert.
Es ist bekannt, dass viele G-Strukturen auf einer Mannigfaltigkeit kanonisch eine eindeutige Cartan-Geometrie induzieren, und dass dieses Verfahren umkehrbar ist.
Im Vortrag geben wir für den Fall einer CO(p,q)-Struktur eine explizite Konstruktion der induzierten Cartan-Geometrie mithilfe des Tractorbündels an. Auf einer konformen Mannigfaltigkeit werden durch eine Differentialgleichung die konformen Geodäten definiert. In einer allgemeinen Cartangeometrie werden mithilfe des Cartan-Zusammenhangs die kanonischen Kurven definiert. Wir geben einen neuen Beweis für die Tatsache, dass im Fall der konformen Cartan-Geometrie beide Definitionen übereinstimmen. Als Anwendung beweisen wir zum Schluss, dass die stereographische Einbettung R^n in S^n die eindeutige konforme Kompaktifizierung des R^n ist. |
| 09.02.17 | The curl operator on odd-dimensional manifolds | Christian Bär | We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres and spherical space forms. |