Course overview -- Wintersemester 2024/2025

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.

Joint work with Nadine Große.

Let $N$ be an oriented compact submanifold in an oriented complete Riemannian manifold $M$. We assume that $M\setminus N$ 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 $M\setminus N$. We are interested in self-adjoint extensions of this operator.

If $N$ has codimension~$1$, then this is the well-studied subject of classical boundary 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''.

The main topic of this talk is thus the case of codimension~$2$ with non-trivial monodromy. We will classify all selfadjoint extensions.

This 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. We thank Boris Botvinnik and Nikolai Saveliev for stimulating discussions about this topic.

We prove eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. This is joint work with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat.

The rectangular peg problem, an extension of the square peg problem which has a long history, is easy to outline but challenging to prove through elementary methods. I will report the recent progress on the existence and multiplicity results, utilizing advanced concepts from symplectic topology, e.g. J-holomorphic curves and Floer theory.

TBA

The sequential topological complexities (TCs) of a space are integer-valued homotopy invariants that are motivated by the motion planning problem from robotics and express the complexity of motion planning if the robots are supposed to make predetermined intermediate stops along their ways. After outlining their definitions, I will discuss the sequential TCs of aspherical spaces in the first part of my talk and describe how they can be investigated by purely algebraic means. We will also take a look at a generalization of this algebraic setting, namely sectional categories of subgroup inclusions. In the second part of my talk, I will present a general lower bound on their values and derive consequences for sequential TCs and parametrized topological complexities of epimorphisms. We will investigate the methodology of the proof of this lower bound, in which all key steps are carried out using elementary homological algebra. This is joint work with Arturo Espinosa Baro, Michael Farber and John Oprea.