Termine

In this course I will introduce the Polchinski flow (or dynamics), a general framework to study asymptotic properties of statistical mechanics and field theory models, inspired by renormalisation group ideas. The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in apparently very different contexts (Markov chain mixing, optimal transport, functional inequalities...). I will motivate the construction of this object from a physics perspective, then consider concrete statistical mechanics models where it can be used to obtain functional inequalities. The connection with (optimal) transport and stochastic localisation will be discussed, together with some open questions. The course is based on a review paper with Roland Bauerschmidt and Thierry Bodineau, accessible here: https://projecteuclid.org/journals/probability-surveys/volume-21/issue-none/Stochastic-dynamics-and-the-Polchinski-equation-An-introduction/10.1214/24-PS27.full

tba If you are interested please contact Prof. Dr. Anna Siffert (asiffert@uni-muenster.de)

The modern theory of simple tracial C*-algebras is characterized by a rich collection of divisibility conditions, coming in numerous flavours and levels of strength. Z-stability and almost divisibility are primary examples of what might be called 'Cuntz-type' divisibility, while in the tracial setting we have counterparts like uniform property Gamma and tracial almost divisibility. On the dynamical side, analogous phenomena emerge through conditions like the small boundary property, the uniform Rokhlin property, and almost finiteness (in measure), often translating into 'relative' versions of the aforementioned properties for Cartan pairs. In this talk I will survey several of these regularity properties, alongside more recent notions introduced by Elliott and Niu, and explore some of the relationships between them. I will in furthermore try to relate some well-known instances of 'automatic centrality' which arises both in the setting of C*-algebras and in topological dynamics, by which I mean results that, under strong nuclearity assumptions, allow to upgrade non-central tracial divisibility conditions (e.g. tracial almost divisibility, small boundary property) to approximately central ones (uniform property Gamma, almost finiteness in measure).

Learning dynamical systems from data has become a vital interdisciplinary research area, uniting concepts from mathematics, engineering, and data science. These systems, which describe how states evolve over time according to underlying mathematical laws, are essential for modeling a wide array of time-dependent phenomena. For practical applications, achieving high accuracy, interpretability, and explainability are crucial properties that are inherently connected to the mathematical structure of the dynamical systems. For instance, in modeling mechanical or electro-mechanical processes, systems with second-order time derivatives typically arise, with data available in the frequency domain as samples of transfer functions. To effectively learn such structured systems from frequency domain data, we introduce a data-driven second-order balanced truncation method. This approach enables the construction of low-dimensional second-order models with generalized proportional damping by assembling appropriate Loewner-like matrices. Numerical experiments illustrate the effectiveness and potential of the proposed methodology.

Further information

Call a complex polynomial f(x,y) _expanding_ if there is e>0 such that for all sufficiently large finite sets A and B of complex numbers with |B| >= |A|, we have |f(A,B)| > |A|^{1+e}. A result of Elekes and Rónyai shows that the only non-expanding polynomials f(x,y) are those obtained from addition or multiplication by composing with unary polynomials. Thinking of B as parametrising a family of unary polynomials f_b(x) = f(x,b), we can see this conclusion as placing B in an algebraic group acting via f. Generalising in these terms, arbitrary nilpotent algebraic groups and their actions can arise. I will review some results indicating that this should be the most general situation, including work of Tingxiang Zou and myself which confirms this in certain cases, using methods from model theory, additive and incidence combinatorics, group theory, and number theory.

Get an insight into the research of five new postdoctoral researchers of Mathematics Münster. In short scientific presentations they will introduce their topics.
After the talks, there will be the opportunity to exchange ideas while enjoying tea, coffee and cake in the Common Room.

• Catrin Mair, Homotopy theory in a condensed world
• Stefan Schrott, Optimal transport of stochastic processes
• Mathias Sonnleitner, Connecting and separating dots
• Ferdinand Wagner, Habiro Cohomology & Refined THH
• Alexander Van Werde, On the spectral determinacy of random graphs

A smooth curve $\gamma: [0,1] \to \Ss^2$ is locally convex if its geodesic curvature is positive at every point. J.~A.~Little showed that the space of all locally positive curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components $\cL_{-1,c}$, $\cL_{+1}$, $\cL_{-1,n}$. These spaces and variants have been discussed, among others, by B.~Shapiro, M.~Shapiro and B.~Khesin. Our first aim is to describe the homotopy type of these spaces. The connected component $\cL_{-1,c}$ is known to be contractible. We construct maps from $\cL_{+1}$ and $\cL_{-1,n}$ to $\Omega\Ss^3 \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots$ and $\Omega\Ss^3 \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots$, respectively, and show that they are (weak) homotopy equivalences. More generally, a smooth curve $\gamma: [0,1] \to \Ss^n \subset \RR^{n+1}$ is locally convex if $\det(\gamma(t),\ldots,\gamma^{n}(t)) > 0$ (for all $t$). We would like to understand the homotopy type of the space $\cL$ of locally convex curves with $\gamma^{(j)}(0) = \gamma^{(j)}(1) = e_{j+1}$ (for all $j \ne n$). We describe a CW complex with the same homotopy type. The homotopy type of $\cL$ is described for $n = 3$.