Seminar Geometrie / Graduate Lectures

Geometric Representation of Structured Extensions in Ergodic Theory

Abstract: One of the early results of mathematical ergodic theory is the Halmos-von Neumann representation theorem from 1942: Every compact (i.e., “structured”) and ergodic (i.e., “irreducible”) measure-preserving transformation can be represented as a rotation on a compact abelian group. The result has later been generalized to actions of groups and to a “relative situation”: The so-called Mackey-Zimmer theorem allows to represent “structured” factor maps between ergodic measure-preserving systems as skew-products by homogeneous spaces of a compact group. This has become a crucial tool  featuring, e.g., in Furstenberg's famous ergodic theoretic proof of Szemerédi's theorem from additive combinatorics, and the powerful Host-Kra-Ziegler structure theory.
In the talk, we present a new topological approach to the Mackey-Zimmer theorem based on groupoids. This allows to prove a very general version of the result which should be useful to extend known ergodic structure theory to actions of general abelian groups. The talk is based on joint work with Nikolai Edeko (Zurich) and Asgar Jamneshan (Dresden).

Ulrike Tillmann
Mathematics and Biology

Der genaue Ort wird vor Ort bekanntgegeben.

Andreas Thom (TU Dresden)
Polynomial maps between non-abelian groups

Abstract: Alexander Leibman introduced a notion of discrete derivative for maps between groups. This leads to a notion of polynomial, where a polynomial of degree $n$ is defined to be a map such that all discrete derivatives have degree $n-1$. Maps of degree zero are defined to be constant. It turns out that every polynomial map of degree $1$ is essentially a homomorphism. We study polynomial maps of higher degree and obtain partial results. One of our results says that any polynomial map defined on a perfect group is essentially a homomorphism. This nicely complements results of Leibman which apply mostly to nilpotent groups. (This is joint work with Asgar Jamneshan and Jakob Schneider.)

15.10.2024 (Di)
WIL B 321
14:00

Georgy Scholten (MPI-CBG Dresden)
Truncated Moment Cone and Connections to the Coalescence Manifold

Abstract: In this talk, we will look at the univariate moment problem of piecewise-constant density functions on the interval [0,1] and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of n moments is achieved by a step function with at most n−1 breakpoints and that this bound is tight. We use this to show that any point in the nth coalescence manifold in population genetics can be attained by a piecewise constant population history with at most n−2 changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program.