Veranstaltungsübersicht -- 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.

Morse function on a manifold M is called strong if all its critical points have different critical values. Given a strong Morse function f and a field F we construct a bunch of elements of F, which we call Bruhat numbers (they're defined up to sign). More concretely, Bruhat number is written on each bar in the barcode of f. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f. We then construct the barcode and Bruhat numbers with twisted (a.k.a. local) coefficients and prove that mentioned product equals the Reidemeister torsion of M. In particular, it's again independent of f. This way we link Morse theory to the Reidemeister torsion via barcodes. Based on a joint work with Petya Pushkar.

From the physical perspective of relativistic wave mechanics, Dirac operators on Riemannian manifolds can be interpreted as infinitesimal generators of time translation, i.e. as Hamiltonians. Most mathematical research on Dirac operators focuses on what amounts to the study of free, massless fields. In this talk however, we will consider a Dirac Hamiltonian that is coupled to an electromagnetic field and a spatially non-constant mass term. After motivating and setting up the necessary notions, we will proceed to show that, given a suitably behaved "potential well", the spectrum of such a Dirac Hamiltonian must be discrete. The result being presented is a generalization of a theorem by N. Charalambous and N. Große in 2023.

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.

In joint work with Rachael Boyd and Corey Bregman we study the classifying space $B\mathrm{Diff}(M)$ of the diffeomorphism group of a connected, compact, orientable 3-manifold $M$. I will recall the construction of this $B \mathrm{Diff}(M)$, also known as the "moduli space of $M$", and explain how it parametrises smooth families of manifolds diffeomorphic to $M$.

Milnor's prime decomposition and Thurston's geometrisation conjecture allow us to cut $M$ into "geometric pieces", which each admit complete, locally homogeneous Riemannian metric. For such geometric manifolds $(N,g)$ recent work using Ricci flow shows that a certain space of metrics is contractible and thus that the generalised Smale conjecture (often) holds: the diffeomorphism group $\mathrm{Diff}(N)$ is homotopy equivalent to the isometry group $\mathrm{Isom}(N,g)$.

The purpose of this talk is to explain a technique for computing the moduli space $B\mathrm{Diff}(M)$ in terms of the moduli spaces of the pieces. We use this to prove that if $M$ has non-empty boundary, then $B \mathrm{Diff}(M\text{ rel boundary})$ has the homotopy type of a finite CW complex, as was conjectured by Kontsevich.

In this talk we will consider the $L^p$ spectrum of the Laplacian on differential forms. In particular, we will show that the resolvent set of the Laplacian on $L^p$ integrable $k$-forms lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. Moreover, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on $k$-forms, and we provide a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space. The above results are joint work with Zhiqin Lu.

In joint work with Cornelia Drutu, Panos Papasoglu, and Stephan Stadler, we study isoperimetric inequalities for null-homotopies of Lipschitz 2-spheres in Hadamard manifolds or, more generally, proper CAT(0) spaces. In one dimension less, for fillings of circles by discs, it is known that a  quadratic inequality with a constant smaller than the sharp threshold $1/(4\pi)$ implies that the  underlying space is Gromov hyperbolic and satisfies a linear inequality. Our main result is a first  analogous gap theorem in higher dimensions, yielding exponents arbitrarily close to 1. Towards  this we prove a Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently  missing in the literature, and introduce so-called minimal tetrahedra, which we demonstrate satisfy  a linear inequality.