Colloquium SS 2026

Speaker: Piotr Bogusław Mucha, Uniwersytet Warszawski – Invited by E. Wiedemann

Abstract: The concept of weak solutions plays a central role in the analysis of nonlinear partial differential equations. Beyond reduced regularity requirements, weak formulations encode essential structural information through integral identities and energy inequalities, which are crucial for both existence theory and qualitative analysis. In this talk, I will discuss the relationship between weak and regular solutions, emphasizing how weak formulations are designed to capture physically and mathematically meaningful properties and, in favorable situations, allow for further regularity improvements. The Navier–Stokes equations, in both incompressible and compressible settings, will serve as the main example. I will highlight selected mathematical challenges related to existence, stability, and regularity of solutions, illustrating the delicate interplay between weak formulation and regularity theory in fluid mechanics.

Speaker: David Reutter, Universität Hamburg – Invited by C. Meusburger

Speaker: Gabriel Wittum, King Abdullah University of Science and Technology (Kingdom of Saudi Arabia) – Invited by N. Neuß

Abstract: Numerical simulation has become one of the major topics in Computational Science. To promote modelling and simulation of complex problems new strategies are needed allowing for the solution of large, complex model systems. Crucial issues for such strategies are reliability, efficiency, robustness, usability, and versatility.

After discussing the needs of large-scale simulation we point out basic simulation strategies such as adaptivity, parallelism and multi-grid solvers. To allow adaptive, parallel computations the load balancing problem for dynamically changing grids has to be solved efficiently by fast heuristics. These strategies are combined in the simulation system UG (“Unstructured Grids”) being presented in the following.

In the second part of the seminar we show the performance and efficiency of this strategy in various applications. In particular, the application and benefit of parallel adaptive multi-grid methods to modelling drug permeation through human skin is shown in detail.

Speaker: Lorenz Schwachhöfer, Technische Universität Dortmund – Invited by K.-H. Neeb

Speaker: Cristina Palmer-Anghel, Université Clermont Auvergne – Invited by C. Meusburger

Abstract: Quantum link invariants have their origin in representation theory and their geometry is a main open problem in quantum topology. Coloured Jones and coloured Alexander polynomials are two such sequences of invariants whose asymptotics are conjectured to capture deep geometric information. We will present a new topological perspective that unifies these invariants through the topology of configuration spaces. First, for a fixed level, we show that we can read off both coloured Jones and Alexander polynomials of a link from a fixed Lagrangian intersection in a configuration space. At the asymptotic level, Habiro defined his famous universal knot invariant globalising coloured Jones polynomials via representation theory, by introducing the Habiro ring. For the link case, this globalisation remained as an open problem for both sequences of invariants. We answer this open problem originating in representation theory using topological tools. On the representation theory side we develop extensions of Habiro type rings.On the topological side, we define geometrically a universal Jones link invariant and a universal Alexander link invariant via graded intersections in configuration spaces. Putting these together, our universal invariants of purely geometrical nature take values in the extended Habiro rings that we construct.

Speaker: Jussi Behrndt, Technische Universität Graz – Invited by H. Schulz-Baldes

Abstract: In this talk, we discuss qualitative spectral properties of self-adjoint Schrödinger and Dirac operators. We first briefly review some of the standard results for regular potentials from the literature and turn to more recent developments afterwards. Our main objective in this lecture is to discuss differential operators with singular potentials supported on curves or hyperplanes, where in the case of Dirac operators it is necessary to distinguish the so-called non-critical and critical cases for the strength of the singular perturbation. In particular, it turns out that Dirac operators with singular potentials in the critical case have some unexpected spectral properties.