Vorträge

Donnerstag, 06.11.2025: The Conformal Method and Hamiltonian General Relativity

David Maxwell (Fairbanks, Alaska)

The conformal method is a workhorse tool for building solutions of the Einstein constraint equations, especially those with constant mean curvature. One frequent perspective on the technique is that it is simply a handy tool, a frequently effective mathematical artifice. In this talk we demonstrate instead that the conformal method is deeply rooted in the Hamiltonian approach to general relativity. We start with a careful analysis of the Gauss constraint $\mathrm{div} E = \rho$ of electromagnetism and describe an approach, grounded in Hamiltonian field theory, for generating its solutions. Then, after an unexpected detour into temporal wave gauge in general relativity, we exhibit a modern understanding of conformal method as an immediate generalization of the toy-model approach to the Gauss constraint. If time permits, we discuss how these ideas led to recent advances in formulating the conformal method in the non-vacuum setting, with applications to charged fluids.

Dienstag, 11.11.2025: Lattices, deformations and rigidity

Paul Vögele

Mittwoch, 12.11.2025: Enumerative Invariants of the Hirzebruch Surface F_n

Parisa Ebrahimian (Tübingen)

In the past two decades, tropical geometry has become a powerful modern tool for solving enumerative problems. In 2008, Hannah Markwig and Andreas Gathmann provided a tropical proof of the famous Kontsevich’s formula, making use of Mikhalkin’s correspondence theorem. In this talk, we present a further generalization of Kontsevich’s formula to the case of rational curves on the Hirzebruch surface F_n. Using Tyomkin’s correspondence theorem, which establishes the equality between algebraic and tropical counts of rational curves on toric surfaces satisfying point and cross-ratio conditions, we derive a recursive formula via tropical methods. This formula counts rational tropical curves of fixed degree in F_n that satisfy both point and multiple cross-ratio conditions, where the cross-ratio conditions here provide more flexibility than in previous works.

Donnerstag, 13.11.2025: Volume preserving mean curvature flow in asymptotically flat spaces

Jacopo Tena (Rome)

In this talk, I will discuss relations between curvature flows and foliations of asymptotically flat manifolds. Firstly, I will extend to the asymptotically flat setting a classical result by Huisken-Yau which allows one to construct a constant mean curvature foliation of the outer part of the manifold by an alternative approach to those by Nerz and Eichmair-Koerber. This is a joint work with C. Sinestrari. Secondly, I will provide an alternative construction of the constant spacetime-mean curvature foliation of an initial data set by Cederbaum-Sakovich by introducing a (volume preserving) fully nonlinear mean curvature flow whose speed takes into account the non-time-symmetric nature of the ambient.

Donnerstag, 13.11.2025: Boundary Superconductivity in BCS Theory

Dr. Barbara Roos (L'Aquila)

The connection between two important models of superconductivity, Bardeen-Cooper-Schrieffer (BCS) theory and Ginzburg-Landau (GL) theory is well-understood for systems without boundaries, where GL theory can be rigorously derived from BCS theory. GL theory is in particular used to model boundary superconductivity, however, a derivation of GL from BCS in the presence of boundaries is currently still out of reach. As a first step, we study BCS theory for systems with boundaries. In particular, I will discuss how the presence of a boundary influences the critical temperature.

Donnerstag, 13.11.2025: Large-scale response theory for gapless Fermi systems in low dimensions

Dr. Harman Preet Singh (Tübingen)

For a class of non-interacting, one-dimensional gapless systems of lattice fermions, we consider the density and current dynamical responses to a density-type perturbation with slow modulation in space and time, in the infinite volume and zero temperature limit. Furthermore, we consider the edge density and current responses for quantum Hall lattice systems on a cylinder. We prove the validity of the linear response for such quantities, thus justifying Kubo’s formula, which we evaluate explicitly solely in terms of the spectral data at the Fermi points, and of the profile of the perturbation. Finally, we present the extension to weakly interacting one-dimensional fermions. The proof of validity of linear response on the physically relevant time-scale relies on a precise analysis of the imaginary-time Duhamel series, and in particular on a cancellation for the scaling limit of the higher order correlation functions describing the nonlinear response, achieved by renormalisation group methods. Based on joint works with M. Porta and G. Scola.

Mittwoch, 19.11.2025: Quadratically enriched plane curve counting via tropical geometry

Felix Röhrle (Tübingen)

Consider the classical problem in enumerative geometry of counting rational plane curves through a fixed configuration of points. The problem may be considered over any base field and the point conditions might be scheme theoretic points. Recently, Kass--Levine--Solomon--Wickelgren have used techniques from $\mathbb{A}1^^1^$-homotopy theory to define an enumerative invariant for this problem which is defined over a large class of possible base fields. This new theory generalizes Gromov-Witten invariants (base field = complex numbers) and Welschinger invariants (base field = real numbers) simultaneously. In this talk I will present a tropical correspondence theorem, which allows to effectively compute these new invariants. This is joint work with Andrès Jaramillo-Puentes, Hannah Markwig, and Sabrina Pauli.

Mittwoch, 26.11.2025: The 1-fold tropical Abel-Prym map

Giusi Capobianco (Roma Tor Vergata)

The algebraic Abel-Prym map relates the geometry of a double cover of algebraic curves with their corresponding Prym varieties. Birkenhake and Lange proved that the map has degree 2 if and only if the source curve is hyperelliptic. In the talk I will present joint work with Yoav Len, in which we investigate the 1-fold Abel-Prym map in the tropical setting and prove similar results. I will show that the tropical Abel-Prym map is a harmonic morphism of degree 2 if and only if the source graph is hyperelliptic. Furthermore, I will prove that the Jacobian of the image of this map is isomorphic, as pptav, to the Prym variety of the cover and relate this result to the tropical bigonal construction.

Montag, 12.01.2026: tba

Stefan Friedl