Oberseminare und sonstige Vorträge

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.

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.

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.

Abstract: It is now well-known that two birationally equivalent hyperkähler manifolds must belong to the same deformation family, as complex manifolds. This property is known to fail for other classes of varieties with trivial first Chern class, such as Calabi?Yau threefolds. In this talk, I will report on joint work in progress with Isabel Stenger and Long Wang, where we investigate a rather dramatic failure of that property. I?ll introduce a specific Calabi?Yau threefold introduced by Schoen and studied by Namikawa and Grassi?Morrison in the early 90s. I will then recount how this threefold has many non-isomorphic birational models with interesting geometric features. At last, my goal is to explain how these birational geometric features can be used to distinguish between many deformation types.

Real rank was introduced by Brown and Pedersen in 1991 as a variant of stable rank, generalizing a characterization of covering dimension in terms of the instability of tuples of continuous functions. Nuclear dimension, introduced by Winter and Zacharias in 2010, generalizes a different characterization of covering dimension, based on the colorability of refinements of open covers. The two notions agree for commutative C*-algebras, where they coincide with the covering dimension of the spectrum, but differ already for homogeneous C*-algebras. In this talk, I will recall both notions of dimension, discuss examples illustrating their differences, and sketch the proof of the main result: the real rank of a C*-algebra is bounded above by its nuclear dimension.

Abstract. Interesting invariants of manifolds with compact, effective, smooth torus actions (with nonempty and finite fixed points) are recorded in the respective equivariant cohomology ring. A graph-theoretical approach to the cohomology description in a wide class of GKM manifolds is given by the Chang-Skjelbred sequence. However, the resulting answer is not very convenient, because it lacks generators and relations description. This was resolved with a face ring isomorphism by M. Masuda and T. Panov in complexity n-k=0 case. The case n-k=1 was partially resolved by A. Ayzenberg, M. Masuda and myself in 2022. In particular, for GKM T^k-actions on M^{2n} with n-k>0 the problem is open, in general. In this talk I will describe a new solution to the generators and relations problem in Hamiltonian, as well as in n-k<4 cases (under some additional technical assumptions). If time permits, I will outline a construction of new GKM 8-manifolds of complexity 0 whose orbit spaces bound Brieskorn homology 3-spheres (and do not satisfy the above assumptions). This follows by producing a smooth manifold with corners structure on the respective collars (the "disks, beams, plates" construction for the smooth conical stratification on the boundary), then using the recent realization theorem of Y. Karshon and S. Kuroki. Based on arXiv 2509.00392, 2604.13629, 2608.30513.