Geometry Seminar / Graduate Lectures

Andreas Thom (TU Dresden)
Lie groups are countably representable

Can a (second countable) Lie group act faithfully on a countable set? For continuous actions on discrete sets, a connected Lie group can only act trivially. The question becomes very different when we regard Lie groups as abstract groups. In joint work with Nicolas Monod, we show that every second-countable Lie group admits a faithful action on a countable set, answering a classical question of Ulam. As a consequence, the same holds for every locally compact second-countable group. I will explain the main difficulties beyond the linear case and the role of central extensions. The key new ingredient is a deformation of a subring of the real numbers of countable additive index, which allows us to overcome the relevant cohomological obstruction.

Preprint: https://arxiv.org/abs/2610.18464

27.10.2026 (Tue)
Room: WIL C103
13:30

Alexander Ziegler (Bergische Universität Wuppertal)
Computing Chow Rings of Classifying Spaces from Representation Rings

Given a (smooth) variety X, one can learn a lot about X by understanding how any two subvarieties of X intersect. By considering formal sums of subvarieties up to rational equivalence, intersection theory gives rise to the so-called Chow ring of X, whose multiplication is intersection. Computing these Chow rings is notoriously difficult.

The goal of this talk is to gently introduce the setting of Chow rings of étale classifying spaces of finite groups, which is the algebro-geometric (or rather motivic) analog of the cohomology of finite groups. We will introduce the relevant ideas from intersection theory and group cohomology. Finally, we give a rough outline of a novel method to compute Chow rings of classifying spaces via representation theory. This method is part of the speaker's Ph.D. project and extends previous work by Burt Totaro from 2014. If time permits, we will sketch the implications of this work for the intersection theory of varieties and group cohomology.

13.10.2026 (Tue)
Room: WIL C103
13:30

Rizeng Chen (MPI-MiS Leipzig)
A Geometric Approach to Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this talk, we will revisit this problem from a more geometric perspective, and report a new algorithm that performs the projection variety-by-variety. Experiments suggest that the new algorithm is very competitive with the existing methods.

Rigidity for metric reduced products under forcing axioms

Forcing axioms are powerful set theoretic statements that decide many questions left open by the usual axioms of set theory. Certain mild consequences of these forcing axioms imply general lifting theorems for homomorphisms between reduced products of countable structures. These allow to characterise precisely those homomorphisms, between such quotients of products, which have a lift that factorises into homomorphisms between the factor structures. We will discuss extensions of such lifting theorems to the metric setting and ensuing rigidity phenomena for particular (metric) reduced product structures of interest.
This is based on joint work with Ilijas Farah and Alessandro Vignati, with Alessandro Vignati, and with Andreas Thom.

There is a well-known hierarchy of regularity for continuous Z-periodic functions on Rⁿ,

C⁰⊃C¹⊃. . .⊃C^∞⊃analytic⊃trigonometric polynomials,

and the decay of Fourier coefficients precisely reflects this hierarchy. In particular, the Fourier transform ^f has finite support if and only if f is a trigonometric polynomial.

Periodic hypersurfaces in Rⁿ exhibit a similar hierarchy of regularity, yet no direct analogue of this Fourier characterization is known.

In this talk, I will present joint work with Mario Kummer in which we give a surprising sufficient Fourier criterion ensuring that a C^1+ε periodic hypersurface Σ⊂Rⁿ is the zero set of a trigonometric polynomial of the form

p(e^2πix₁, . . . , e^2πix_n),

where p is a Lee--Yang polynomial.

The criterion is formulated using a notion recently introduced by Yves Meyer: a periodic positive Radon measure m on Rⁿ is called a lighthouse measure if supp(m) has zero Lebesgue measure and supp(^m) is contained in a proper double cone.

Our proof produces explicit examples of such lighthouse measures and establishes an inverse theorem. It relies on recent breakthroughs concerning the existence and classification of one-dimensional Fourier quasicrystals. No field-specific background is assumed. This work builds on collaborations with Alex Cohen, Pavel Kurasov, and Cynthia Vinzant.