Vorträge

Mittwoch, 09.07.2025: The tropical Donagi theorem

Thomas Saillez (Université Libre de Bruxelles)

In tropical geometry, we have a lot of analogues of theorems from algebraic geometry. The aim of this talk is to discuss the tropical analogy of the tetragonal construction, which is a combinatorial construction used in the study of abelian varieties. Under reasonable assumptions, the tetragonal construction associates to the data of a double cover of a quadruple cover of a curve two other pairs of such covers. A remarkable result is that indeed it forms a triality of pairs of covers (applying the construction to one pair of covers outputs the other two) and that the Prym varieties, a generalization of Jacobians arising from double covers, are isomorphic. We are going to introduce those elements in the context of tropical geometry and explore the results we can obtain there.

Mittwoch, 16.07.2025: What is (Mathematically) a Neural Network?

Dr. Armando Cabrera Pacheco (Universität Tübingen)

Given the hype surrounding machine learning and artificial intelligence, it is natural to be curious about what neural networks are and how they work. However, while there is a lot of information on the subject, it is not easy to find it in a form suitable for (some) mathematicians. When looking for a concrete explanation of what neural networks are, one often finds analogies related to neuroscience and not much of the actual mathematical framework behind them, which often leaves mathematicians unsatisfied. In this talk, we will look at neural networks from a purely mathematical point of view. We will discuss some of the straightforward challenges, both from a mathematical and computational viewpoint, that these powerful tools present.

Donnerstag, 17.07.2025: Cosmological quantum fields, Minkowski-like vacua and Dark Energy

Nicolai Rothe (TU Berlin)

We will discuss some newly found solutions to the full massless semiclassical Einstein equation (SCE) in a cosmological setting (with lambda=0). After a short introduction to the relevant notions, we present the SCE in a particular shape which allows for the construction of certain vacuum states. These states may be viewed as the least possible generalization of the Minkowski vacuum to generic cosmological space-times. In this setting, solving the SCE breaks down into solving a certain ODE which can be approached numerically and, at least generically, we obtain solutions that well fit physical expectations. Moreover, these solutions indicate dark energy as a quantum effect back-reacting on cosmological metrics and, since in our model m=lambda=0, this may not be traced back to the usual, obvious dark-energy/cosmological constant effect of a quantum field.

Freitag, 18.07.2025: Strenge Kommutativität von höheren Homotopiegruppen

Philipp Schmale (Uni Tübingen)

Montag, 21.07.2025: Asymptotic Stability of Solitons

Prof. Jonas Lührmann (Texas A&M)

Many wave phenomena in nature, from water waves to light pulses in optical fibers, can be described by nonlinear dispersive equations. Among their solutions, solitons are particularly remarkable: they are localized waves that retain their shape as they travel, behaving almost like particles despite being waves. In this talk, I will explain what solitons are and why they are important for understanding the long-time behavior of nonlinear dispersive equations. I will also discuss recent progress on the question of the asymptotic stability of solitons in lower-dimensional settings where the strong interactions between solitons and dispersive waves lead to especially rich dynamics.