Vorträge in der Woche 08.12.2025 bis 14.12.2025

Dienstag, 09.12.2025: Anosov representations and convex-cocompact groups

Giacomo Gavelli

Mittwoch, 10.12.2025: The dessins d'enfants stemming from boundary points of Teichmüller curves of origamis in H(2)

Sebastian Engelhardt (Universität des Saarlandes)

Origamis are translation surfaces that consist of finitely many Euclidean squares glued together along their edges. By deforming its translation structure, an origami defines a special kind of Teichmüller curve. These are complex algebraic curves in the moduli space M_g of closed complex regular curves of genus g. In this talk, we study the boundary points of Teichmüller curves stemming from origamis in the stratum H(2) on the Deligne–Mumford compactification of M_g. Stable reduction is a general method for finding boundary points of algebraic curves in M_g, represented by stable Riemann surfaces. For origami curves, the method can be described via contraction of the core curves of horizontal cylinders of the origami. Each irreducible component of the resulting stable surface naturally gives rise to a dessin d'enfant – a bipartite graph embedded into a Riemann surface. We give a graph-theoretical characterization of all occurring dessins for origamis in H(2).

Donnerstag, 11.12.2025: Stratifying moduli spaces of stable Higgs bundles

Aryaman Patel (Saarbrücken)

We study the behavior of slope-stability of reflexive twisted sheaves over a normal projective variety $X$ under pullback along a cover. Slope-stability is always preserved if the cover does not factor via a quasi-étale cover. Fixing the rank, there is one quasi-étale cover that checks whether a twisted sheaf remains slope-stable on all Galois covers yielding a stratification of the moduli space of slope-stable Higgs-bundles. [If time permits: As an application, we determine the image of the Hitchin morphism restricted to the smallest closed stratum of the Dolbeault moduli space if $X$ is smooth. This allows us to determine the image of the Hitchin morphism from the Dolbeault moduli space if $X$ is a hyperelliptic variety.]

Donnerstag, 11.12.2025: Rigidity aspects of a singularity theorem

Carl Rossdeutscher (Universität Wien)

In 2018 Galloway and Ling established the following cosmological singularity theorem: If a (3+1)-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface with a positive definite second fundamental form (i.e., it’s expanding in all directions), then the spacetime is past null geodesically incomplete unless the Cauchy surface is a spherical space. We present some rigidity results for this singularity theorem. In particular if the second fundamental form is only positive semidefinite and the spacetime is past null geodetically complete, we show that the Cauchy surface (or at least a finite cover thereof) is a surface bundle over the circle with totally geodesic fibers or a spherical space. Under certain additional assumptions on the Cauchy surface, we show that passing to a cover is unnecessary. Our results make in particular use of the positive resolution of the virtual positive first Betti number conjecture by Agol. If a spacetime admits a U(1) isometry group, we can relax the assumption on the second fundamental form further.

Donnerstag, 11.12.2025: Geometric Optimization in Scientific Machine Learning

Dr. Marius Zeinhofer (ETH Zürich)

We discuss an “optimize-then-project” approach for applications in scientific machine learning. The key idea is to design algorithms at the infinite-dimensional level and subsequently discretize them in the tangent space of the neural network ansatz. We illustrate this approach in the context of the variational Monte Carlo method for quantum many-body problems, where neural quantum states have recently emerged as powerful representations of high-dimensional wavefunctions. In this setting, we recover the celebrated stochastic reconfiguration algorithm, interpreting it as a projected Riemannian L2 gradient descent method. We further explore extensions to Riemannian Newton methods, and conclude with considerations related to the scalability of these schemes.

Donnerstag, 11.12.2025: Rigorous Schrödinger Quantum Mechanics of Countably Many Degrees of Freedom

Xabier Oianguren-Asua (Tübingen)

In this talk, we will provide a rigorous generalization of the quantum theories that employ square-integrable functions on "R^n" as their state vectors, to the case in which n is countably infinite. The resulting structure's configuration space ("R^\mathbb{N}") can parametrize, among others, the expansion coefficients of a field with respect to an orthonormal basis. Consequently, the talk will suggest a rigorous formulatation of quantum mechanics using wavefunctionals over field configuration spaces. For this purpose, we will employ von Neumann's infinite tensor product --which circumvents the absence of a well-behaved infinite-dimensional Lebesgue measure-- together with a joint spectral diagonalization theorem for arbitrarily many strongly commuting self-adjoint operators in non-separable Hilbert spaces. If time permits, we will also sketch the connection between the obvious CCR representation of our formulation and the usual Fock representation.