Resonances of closed geodesics on real projective space and extensions of loop products
Montag, 2.6.25, 16:15-17:45, Seminarraum 404
Properties of closed geodesics on a Riemannian manifolds are a classical topic of mathematical research. The closed geodesics can be described as the critical points of the energy functional on the free loop space of a manifold. Since string topology studies algebraic structures on the homology of free loop spaces, there is hope that string topology operations give new insights into closed geodesics. In this talk we show one can adapt a result by Hingston and Rademacher for closed geodesics on the sphere to real projective space. Since even-dimensional real projective space are not orientable the classical string topology operations are not defined and some extra care is required. We will show that the loop product and coproduct can be defined on the universal covering space of the free loop space of real projective space and these new operations can be understood as extensions of the loop product and coproduct on the sphere.
Regularity of the Dirac Operator on corner domains
Montag, 16.6.25, 16:15-17:45, Seminarraum 404
This talk addresses the Dirac operator on polygonal domains in \(\mathbb{R}^2\) with local boundary conditions. While the theory is well-developed for smooth boundaries, much less is known in the presence of corners. We establish symmetry and regularity of the Dirac operator under these generalized conditions. Initial progress toward proving self-adjointness will also be discussed, including an explicit description of the adjoint operator and its decomposition into regular and singular components at the corners. These results provide a foundation for further study of spectral and boundary value problems for Dirac operators on non-smooth domains. The results are based on my own research.
t.b.a.
Montag, 23.6.25, 16:00-17:30, Seminarraum 404