Vorträge in der Woche 14.07.2025 bis 20.07.2025

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: Singularity formation in coupled Ricci and Yang-Mills flow

Andoni Royo Abrego (Universität Tübingen)

In this talk we will introduce a coupled geometric evolution equation describing the deformation of a Riemannian metric and a gauge field on closed manifolds. After recalling some basic properties of Ricci flow and Yang-Mills flow separately, we will motivate the coupled flow from a variational viewpoint and derive parabolic estimates. These will be used to characterize finite-time singularities, which are strongly dominated by the Ricci flow behavior. We will also describe an explicit non-trivial example of self-similarly shrinking solution over an SU(2)-bundle.

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: Einführung in die homotopie-kohärente Kategorientheorie

Philipp Schmale (Uni Tübingen)