Veranstaltungen am Mathematischen Institut

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.

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.

By a connection with a formal parameter, we mean a connection on a scheme $X$ that is relatively smooth over the ring $R = k[[t]]$, where $k$ is a field of characteristic $0$. The monodromy of such a connection is a flat affine group scheme over $R$. We introduce the notion of prudent group schemes in terms of their representations; this property turns out to be equivalent to the projectivity of the coordinate ring (a result that holds over any complete DVR). In this talk, we discuss the question: When is the monodromy group of a connection prudent? Our main tool is the concept of hyper-prudence, which involves the $t$-adic completion of the representation category. This is a joint work with J.P. dos Santos (Montpellier) and Dao Van Thinh, Pham Thanh Tam (Hanoi).