Colloquium WS 2025/2026

Speaker: Giambattista Giacomin, Università degli Studi di Padova – Invited by H. Schulz-Baldes

Abstract: The transfer matrix method is a standard method to solve (in particular) one dimensional statistical mechanics models. In the case of spins taking only two values (Ising spins) the matrix involved is just a two by two matrix. In a somewhat surprising way and, for some cases, in highly non trivial way, the Ising transfer matrix is key to the solution of other models, notably dimer models and two dimensional Ising models. While for non disordered models everything boils down to computing the spectrum of the transfer matrix, in presence of disorder one needs to understand the much less explicit Lyapunov spectrum of the product of random Ising transfer matrices. In this field a number of mathematical challenges emerge from some remarkable research works in theoretical physics that, in mathematical terms, address the problem of singular behavior of Lyapunov exponents. The purpose of my presentation is to give an introduction to this research domain and present some rigorous results addressing precisely some of the issues tackled by physicists.

Speaker: Paul Wedrich, Universität Hamburg – Invited by C. Meusburger

Abstract: Quantum topology first revealed that the Jones polynomial—and many other knot and link invariants—originate from braided monoidal categories of quantum group representations, providing a foundation for associated 3- and 4-dimensional topological quantum field theories (TQFTs). Khovanov’s categorification of the Jones polynomial suggests an analogous higher categorical structure, hinting at connections to 4- and 5-dimensional TQFTs via braided monoidal 2-categories. In this talk, I will outline four types of TQFTs emerging from link homology—4d and 5d, linear and derived—and survey the current landscape of concrete examples.

Speaker: Jan Giesselmann, Technische Universität Darmstadt – Invited by M. Gugat

Abstract: A posteriori error estimates are computable bounds on the accuracy of numerical approximations to differential equations. Unlike classical error analyses, they rely only on the computed solution and the problem data, not on the unknown exact solution.

This talk will focus on the fundamental connection between such estimates and stability properties of the underlying continuous problems.

As a first example, we will discuss results for implicit Euler and Petrov–Galerkin schemes for ordinary differential equations. We then turn to systems of hyperbolic conservation laws. After reviewing error estimates based on relative entropy stability, we will outline more recent developments in the one-dimensional case, including approaches that exploit Bressan’s stability theory and a-contraction estimates.

Speaker: Aidan Sims, University of New South Wales – Invited by K. Li

Abstract: Operator algebras (specifically C*-algebras) are important because they underpin our understanding of quantum mechanics – for example, designing a quantum algorithm basically amounts to constructing just the right unitary operator. Mathematically, we use C*-algebras to “linearise” topological and dynamical data by representing them as collections of linear operators on Hilbert space. The problem is that while every C*-algebra can in principle be realised as a collection of linear operators on Hilbert space, this has, in general, the same sort of descriptive power as the observation that every group can, in principle, be realised as a group of permutations—and for the same reason: the set of which it is a group of permutations is…itself. Groupoids provide a means of “seeing” a coordinate system in a C*-algebra. I’ll try to outline how, and why this might be useful.

Speaker: Frank den Hollander, Universiteit Leiden – Invited by A. Greven

Abstract: Preferential attachment random graphs are dynamic random graphs that grow over time in such a way that vertices with large degrees are favoured over vertices with small degrees. Such graphs are used to model the power-law degree distributions that are found to be abundantly present in real-world networks.  Under standard preferential attachment, each time a new vertex comes in it attaches itself to an old vertex with a probability that is proportional to the degree of that old vertex. The resulting growing graph is a random tree whose vertices have degrees that grow polynomially fast in time, with a growth exponent that is computable. In the present lecture we consider a version with self-reinforcement, where the attachment probability is proportional to the sum (!) of the degrees over all prior times, which means that the growth mechanism has memory. We compute the growth exponent, show that it is strictly larger than the growth exponent in the absence of self-reinforcement, and develop insight into how the self-reinforcement affects the growth.

Speaker: Stefanie Rach, Otto-von-Guericke-Universität Magdeburg – Invited by A. Lindmeier

Abstract: Das Lernen von Mathematik an der Universität stellt viele Fach- und Lehramtsstudierende vor große Herausforderungen und führt häufig zu einem Studienabbruch. Welche individuellen Faktoren einen solchen Studienabbruch bedingen, wird in diesem Vortrag fokussiert. Als wichtige Faktoren werden sowohl das mathematische Vorwissen als auch die Motivation von Studierenden betrachtet. Basierend auf den Erkenntnissen mehrerer empirischer Studien werden Maßnahmen diskutiert, die Studierende in ihrem Lernprozess unterstützen könnten.

Speaker: Alain Joye, Université Grenoble Alpes – Invited by H. Schulz-Baldes

Abstract: The concept of quantum walks on graphs has attracted growing attention in recent years, due to its role at the intersection of theoretical computer science, physics, and mathematics. After giving an overview of several popular quantum walks to illustrate their significance in these fields, we will focus on the so-called coined quantum walks. From the perspective of discrete dynamical systems, we will examine some of their transport properties in various regimes, both random and deterministic.

This discussion will highlight similarities and differences between coined quantum walks and classical random walks.

Speaker: Andrea Bréard, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by K.-H. Neeb

Abstract: Early 20th century readers of Chinese mathematical texts rendered certain results directly into mathematical formulas. However, contemporary historians of science prefer to translate such texts into natural language to better reflect modalities of mathematical expression in the originals. While each option has its advantages and disadvantages and its own potential readers, I will show that it can be fruitful to follow both paths in order to understand the choice of style by the author himself. My argument is based on my translation of Li Shanlan’s _Comparable Categories of Discrete Accumulations_ 垛積比類 (1867) into French (Paris: Les Belles Lettres, 2023).

I will describe the kind of natural and visual language used in the original, where a large number of summation procedures are stated. Based on the arithmetic triangle, Li Shanlan constructed other triangular tables in his _Comparable Categories of Discrete Accumulations_ using operations known as “branching” (zhi 支) or “transformation” (bian 變). I will discuss how these operations on the tables enabled him to deduce the summation procedures for each of the diagonals of his triangles and to establish arithmetic links between the diagonal sums of different triangles. Even though Li does not give general procedures, but only the sums of the first diagonals from which the reader is invited to deduce the others, I will show that the tables revealed a plausible logic (or “patterns”) for their general calculation. The formulaic nature of the identified procedural and algorithmic code of the book not only incites the translator to equate the original text with algebraic formulas but it also reveals that Li Shanlan himself might have used algebra to derive his procedures.

Speaker: Igor Burban, Universität Paderborn – Invited by K.-H. Neeb