Termine

In their seminal work, Xiao-Zhu have constructed exotic Hecke correspondences between the special fibers of different Shimura varieties, which do not necessarily lift to the generic fibers. As a consequence, they have constructed geometric realizations of the geometric Jacquet-Langlands correspondence, and verified generic instances of the Tate conjectures for certain Shimura varieties. Crucially, they only address Shimura varieties with good reduction, i.e., with hyperspecial level at a fixed prime. In these two talks, I will explain how to generalize their methods when allowing bad reduction (more precisely, when the level at a fixed prime is very special, which exists when the local group is quasi-split, but not necessarily unramified). In the first talk, I will explain how to construct exotic Hecke correspondences in these ramified situations between PEL type Shimura varieties. Combined with the "spherical" part of the categorical local Langlands correspondence, these yield new instances of the geometric Jacquet-Langlands correspondence. The key is to resolve the integral models by the splitting models of Pappas-Rapoport. In the second talk, I will define splitting versions of affine Deligne-Lustzig varieties, and explain how to study their irreducible components. Combining this with the aforementioned geometric Jacquet-Langlands correspondence, this can be used to verify generic instances of the Tate conjecture for certain Shimura varieties at very special level.

The seminar takes place online via Zoom. The Zoom links and announcements are sent out through our mailing list. You can find the link for the mailing list here: https://www.uni-muenster.de/MathematicsMuenster/ai-math/ Abstract: Mathematics is entering a period of remarkable promise and significant unease. AI systems are beginning to contribute to mathematical discovery, research-level reasoning, and formal verification, while also raising serious questions about reliability, attribution, access, education, and the future role of mathematicians. This talk surveys these developments through recent examples, including AI-guided search, advances in longstanding problems, and Lean formalization. The central theme is that AI is most powerful when it complements rather than replaces mathematical judgment: broadening discovery, accelerating exploration, and strengthening trust through machine-checkable proof.

In joint work with Macpherson, Steinhorn and Wolf on multidimensional asymptotic classes, we introduced ring-measurable structures. In these, every definable set is assigned a "size" in an ordered ring, subject to additivity and Fubini-type conditions on definable families. This generalises both the measurable structures of Macpherson and Steinhorn and the asymptotic counting in pseudofinite structures. We asked there whether ring-measurability implies simplicity. In ongoing work with Macpherson and Steinhorn, we show that the generic triangle-free graph (the Henson graph H_3), which is not simple, is not ring-measurable: counting configurations in two different ways yields polynomial identities with no admissible common solution. I will explain the framework and sketch the argument. I will then discuss the generic bowtie-free graph, which defines copies of H_3 that are not fully embedded, so that the argument does not transfer directly.

I will present an overview of the results on existence and asymptotic properties of magnetic skyrmions defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.

Abstract

Abstract: Embedding group rings into division rings provides methods for tackling problems which superficially seem unrelated. One way of constructing such embeddings (in characteristic 0) is by proving that they satisfy the Atiyah Conjecture. I will give a survey of the status of the Atiyah Conjecture, as well as applications to other problems, and discuss joint work with Sam Fisher on proving the conjecture for congruence subgroups of mapping class groups and outer automorphism groups of RAAGs.