Vorträge am Mathematischen Institut

Prof. Dr. Martin GROHE:

The Logic of Graph Neural Networks

Zeit und Ort

Donnerstag, 15.5.25, 15:00-16:30, Hörsaal 2

Zusammenfassung

Graph neural networks (GNNs) are deep learning models for graph data that play a key role in machine learning on graphs. A GNN describes a distributed algorithm carrying out local computations at the vertices of the input graph. Typically, the parameters governing this algorithm are acquired through data-driven learning processes.

After introducing the basic model, in this talk, I will focus on the expressiveness of GNNs: which functions on graphs or their vertices can be computed by GNNs? Understanding expressiveness will help us to understand the suitability of GNNs for various application tasks and guide our search for possible extensions.

Surprisingly, the expressiveness of GNNs has a clean and precise characterisation in terms of logic and Boolean circuits, that is, computation models of classical (descriptive) complexity theory.

Prof. Dr. Damien Gayet:

What does a random complex curve look like locally?

Zeit und Ort

Donnerstag, 26.6.25, 15:00-16:30, Hörsaal 2

Zusammenfassung

A complex curve of degree d in the complex projective plane is the vanishing locus of a homogeneous polynomial of degree d. These curves are real surfaces satisfying a spectacular collective feature: generically they are all topologically the same, that is connected compact Riemann surfaces with genus equal to (d-1)(d-2)/2. Now, if we fix a ball and we look at the intersection of the curve with this ball, this magical property disappears completely. For instance, the curve can simply miss the ball. But if the polynomial is chosen at random and has a high degree, we can expect intuitively that the ball will capture, in average, a fixed proportion of the global topology of the curve. I will explain that this is the case.

Prof. Dr. Barbara Niethammer:

Free boundary problems in models for cell polarization

Zeit und Ort

Donnerstag, 3.7.25, 15:00-16:30, Hörsaal 2

Zusammenfassung

Cell polarization denotes the rearrangement of certain substances on the membrane of a cell in response to an external chemical signal. It is a crucial ingredient in many biological processes, such as for example the motion of cells. Starting from a bulk-surface reaction-diffusion system for several protein densities describing this process, we rigorously derive nonlocal free boundary problems that allow for a relatively simple characterization of polarization. For these limit systems we prove global stability of steady states and characterize the parameter regime for the onset of polarization. We also discuss some aspects of regularity in time of the free boundary.

(joint work with A. Logioti (Stuttgart), M. Röger (Dortmund) and J. Velazquez (Bonn))