The Frobenius relation in string topology
Monday, 21.10.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
String topology is the study of algebraic operations on the homology of the free loop space of a closed manifold. Two prominent operations are the Chas-Sullivan product and the Goresky-Hingston coproduct. It is an important question what structure these two operations form together. We show that under a transversality condition a Frobenius-type relation for the product and the coproduct holds. As an application this yields the behaviour of the coproduct on product manifolds. This talk is based on joint work with Nathalie Wahl.\n
Self-adjoint codimension 2 boundary conditions for Dirac operators
Monday, 28.10.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Joint work with Nadine Große.\n\nLet \(N\) be an oriented compact submanifold in an oriented complete Riemannian manifold \(M\). We assume that \(M\bsetminus N\) is spin and carries a unitary line bundle \(L\). We study the associated twisted Dirac operator, a priori defined on smooth section with compact support in the interior of \(M\bsetminus N\). We are interested in self-adjoint extensions of this operator.\n\nIf \(N\) has codimension~\(1\), then this is the well-studied subject of classical\nboundary values for Dirac operators. If \(N\) has codimension at least \(3\), or if \(N\) has codimension \(2\) and if \(L\) has trivial monodromy around \(N\), then we obtain a unique self-adjoint extension which coincides with the classical self-adjoint Dirac operator on \(M\). The submanifold \(N\) is thus ``invisible''.\n\nThe main topic of this talk is thus the case of codimension~\(2\) with non-trivial monodromy. We will classify all selfadjoint extensions.\n\nThis work is motivated by work of Portman, Sok and Solovej, who treated the special case of \(M=S^3\) with a link, a case important in mathematical physics.\nWe thank Boris Botvinnik and Nikolai Saveliev for stimulating discussions about this topic.