# Geometry Seminar / Graduate Lectures

The Geometry seminar encompasses invited talks, Graduate Lectures of the Department of Mathematics as well as talks by Ph. D. students and theses defences. Upcoming events will be announced through the seminar's mailing list.

Everyone interested is welcome to attend.

Due to the institute getting a new address a standard room for the seminar does not exist yet.

12.11.2024 (Tue) WIL C/103 13:30 |
Dmitrii Pavlov (TU Dresden)Santaló geometry of convex polytopesAbstract: The Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set. In my talk I will describe this geometry and dive into connections with statistics, optimization and physics. This is joint work with Simon Telen. |

05.11.2024 (Tue) WIL C/103 13:30 |
Henrik Kreidler (Wuppertal)Geometric Representation of Structured Extensions in Ergodic TheoryAbstract: 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). |

29.10.2024 (Tue) Place: MPI-CBG 14:00 |
Ulrike TillmannMathematics and BiologyExact directions will be available in situ. |

22.10.2024 (Tue) WIL C 104 13:30 |
Andreas Thom (TU Dresden) 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 (Tue) |
Georgy Scholten (MPI-CBG Dresden) 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. |

For current lectures one can also refer to the Events calendar - Faculty of Mathematics.

The list of talks from past semesters can be found in the archive.