Termine

It is well-established that Patterson-Sullivan measures on the boundary of a hyperbolic space, along with the associated Bowen-Margulis-Sullivan measure, provide valuable insights into the action of a group of isometries on the space's boundary through analysis of the geodesic flow. Given that paths of a random walk on a hyperbolic groups lie close to the group's quasi-geodesics, it is natural to ask whether similar behaviour can also be seen in the flow along bi-infinite random walk paths. In this talk, I will give an insight into classical Patterson-Sullivan theory, and show how studying bi-infinite random walks on a discrete hyperbolic group $G$ leads to an analogue of the Patterson-Sullivan measure on $\partial^2G$. This talk is based on joint work with Mahan Mj and Chiranjib Mukherjee.

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

Based on my master's thesis of the same name, this talk addresses the time-dependent Schrödinger equation, a fundamental governing equation in quantum mechanics. While established numerical methods typically follow a method-of-lines approach, full space-time variational formulations have gained growing interest. They elegantly capture low-regularity solutions, bypass restrictive CFL conditions, and, crucially for Model Order Reduction (MOR), treat time as an additional variable to inherently provide sharp, global-in-time error bounds.

The thesis builds upon the ultraweak space-time discontinuous Petrov-Galerkin (DPG) formulation introduced by Demkowicz et al. in 2017. While their foundational work was restricted to the free Schrödinger equation, we generalize this framework to incorporate bounded, time-independent, real-valued potentials, which dictate the underlying energy landscape in many physically relevant scenarios.

The presentation provides an overview of this work, beginning with an introduction to the DPG method and a summary of the underlying functional analytic framework. We then detail the generalization to include a potential, highlighting continuous well-posedness alongside stationary theoretical derivations regarding Fortin operator stability and the identification of an additional data oscillation term. Following a discussion on implementation and full-order model numerical experiments, we motivate the need for MOR and the Reduced Basis Method (RBM) in parametric many-query scenarios. Finally, marking the first explicit realization of an RBM-DPG coupling, we present theoretical guarantees, including uniform well-posedness and exponential Kolmogorov $N$-width decay, and conclude with numerical validations of the reduced-order model.

The term Berry?s random wave describes a canonical model for a Gaussian Laplace eigenfunction on the plane. This universal object naturally arises in connection with several questions in stochastic geometry, especially those concerning the local behaviour of Laplace eigenfunctions?random or deterministic?on chaotic surfaces. After a brief introduction to this model, I will highlight two striking (and still quite mysterious) structures that emerge in high-frequency regimes: * Randomly scattered scars * Total disorder processes (if time permits) The second part of the talk will mainly focus on open problems and conjectures, and will include some personal reflections and anecdotes about doing mathematical research in the era of AI.

In 1902, William Burnside asked whether a finitely generated group in which every element has finite order must be finite. This question, now known as the Burnside Problem, was answered negatively by Novikov and Adian in 1968, who proved that the free Burnside group B(m,n) is infinite for sufficiently large odd exponents n. More recently, in 2023, Agatha Atkarskaya, Eliyahu Rips, and Katrin Tent proved that B(m,n) is infinite for all odd n?557, the best currently known bound, using methods from iterated small cancellation theory. After outlining the ideas behind this result, we discuss connections between free Burnside groups, the space of amenable subgroups, and C*-simplicity, including work in progress with Katrin Tent and Sam Shepherd on constructing new classes of C*-simple groups via Burnside quotients.

For any prime $p$, a finite group has as many irreducible complex characters of degree prime to $p$ as the normalizer of a Sylow $p$-subgroup. This equality, conjectured by John McKay in 1971, was reduced in 2007 by Isaacs--Malle--Navarro to a conjecture on representations of finite simple groups. Thanks to their classification we know that the latter are essentially finite groups of Lie type. Deligne--Lusztig theory helps to prove the McKay conjecture by this approach. For groups of characteristic different from $p$, the normalizers of Sylow $p$-subgroups belong to a larger class of subgroups related to parabolic subgroups of the ambient algebraic group for which Deligne-Lusztig varieties and induction functors have been used in the 1990s to provide a substitute to parabolic induction. This works well for unipotent characters. An important step is the construction of a Jordan decomposition of characters which is equivariant with respect to automorphisms of the simple group.

The Bayesian approach to inverse problems provides a principled and flexible methodology for estimating unknown parameters in partial differential equations (PDEs), making it an important frontier for statistical inference from the sparse, noise-corrupted data characteristic of physics-informed settings. Recent algorithmic advances and growing computational capacity have brought a new class of high-dimensional PDE inference problems into focus, with rich mathematical challenges and impactful applications. This talk will survey this emerging field and describe some of our recent and ongoing work in this domain. We consider some model PDE inference problems related to the measurement of fluid flows. We will then review some recent developments in Markov Chain Monte Carlo (MCMC) sampling. In particular, we will introduce several of our new algorithms, multiproposal preconditioned Crank-Nicolson (mpCN) - with local and global variants - as well as surrogate trajectory Hamiltonian methods. These methods are carefully tailored for resolving high-dimensional, strongly anisotropic posteriors with nonlinear correlation geometry while taking advantage of parallel computing architectures. These new MCMC algorithms have broad scope, but we are using our fluids model problems as a physically motivated challenge test bed. We describe how our new methods have been derived on the basis of an "involutive theory" that we developed. This theory provides a unified view of reversible, Metropolized MCMC while serving as an oracle to derive and validate new methods. We will also outline some rigorous bounds on mixing rates for mpCN that we derived on the basis of the so-called Weak Harris theory. These bounds demonstrate that our methods partially beat the "curse of dimensionality". This is joint work with Jeff Borggaard (Virginia Tech), Giulia Carigi (Indiana), Andrew Holbrook (UCLA), Justin Krometis (Virginia Tech), Cecilia Mondaini (Drexel), and Guillermina Senn (Indiana, NTNU).