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27.06.2024, 16:15
– Raum 1.10

Forschungsseminar Differentialgeometrie

##### Fill-ins, Dirac eigenvalues and the hyperspherical radius

Christian Bär (UP)

Our group is working in differential geometry and related areas such as global analysis, topology and mathematical physics. In the following we give three examples of current research projects.

**1. Wave equations on curved spacetime **

In general relativity space and time are modelled by so-called Lorentzian manifolds. Many physical phenomena such as electro-magnetic waves or the recently detected gravitational waves can be described by fields satisfying a wave equation. We examine the underlying global analytic theory of such wave equations. Among other things, this contains the study of existence, uniqueness and stability of solutions. Moreover, we work on the construction of quantum field theories on curved spacetime and thereby support theoretical physicists in their search for a unified theory of general relativity and quantum physics. As a very recent development we consider index theory for Dirac operators on Lorentzian manifolds where we examine boundary value problems for Dirac fields.

**2. Analytic and spectral properties of geometric operators **

Many questions arising in physics are related to analytic or spectral properties of differential operators. For example the possible energies of a physical system are very often given by the eigenvalues of a differential operator. In this case one would like to know which geometric quantities of a system are determined by these eigenvalues. We examine some of the open questions arising in this context. An important example is given by the conformal Laplace operator. We examine under which circumstances one can conclude positivity of the so-called ADM mass of an asymptotically Euclidean manifold from the properties of this operator. Another important example is given by the fundamental solution of the heat equation. We examine its applications to the theory of path integrals.

**3. Supergeometry **

Supergeometry is a generalization of differential geometry in the sense that it includes spaces on which anticommuting functions exist. On the one hand it is based on ideas from algebraic geometry, on the other hand it allows constructions such as Lie theory, tensor calculus or calculus of variations as in classical differential geometry. The study of such structures is originally motivated by physics since both (pseudo)-classical Fermi fields as well as supersymmetric theories can be modelled geometrically within this framework.

We examine the geometric theory of partial differential equations on supermanifolds. Besides the solution theory we also examine its applications to field theory and geometry, for example the construction of supersymmetric/fermionic quantum field theories on curved spacetime and the discussion of geometric conditions for the existence of solutions. Another subject of our current research is the supersymmetric approach to index theory initiated by Witten.