Vorträge in der Woche 11.11.2024 bis 17.11.2024

Montag, 11.11.2024: Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

Christopher Voll (Bielefeld)

In this talk, I would like you to meet Hall-Littlewood-Schubert series, a new class of multivariate generating functions. Their definition features semistandard Young tableaux and polynomials resembling the classical Hall-Littlewood polynomials. Their intrinsic beauty notwithstanding, Hall-Littlewood-Schubert series have many applications to counting problems in algebra, geometry, and number theory. In my talk the spotlight will be on applications to affine Schubert series. These may be seen as an integral analogue of the Poincare polynomials enumerating the rational points over finite fields of classical Schubert varieties. The latter parametrize subspaces of a given vector space by the intersection dimensions with a fixed flag of reference. This is joint work with Joshua Maglione. I will explain things from scratch, assuming no familiarity with the advanced technical vocabulary used in this abstract.

Dienstag, 12.11.2024: Representation Theory of semisimple Lie groups

Anton Deitmar

Mittwoch, 13.11.2024: Brill--Noether theory of special curves

Hannah Larson (Berkeley)

Brill--Noether theory studies the maps of curves C to projective spaces. The classical Brill--Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz--Brill--Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. I'll also present results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane.

Donnerstag, 14.11.2024: Asymptotic properties at early times of solutions to Einstein's equations via scattering constructions

Dr. Leonhard Kehrberger (MPI Leipzig)

In the standard approaches to tackling the evolution of the Einstein equations in general relativity, one typically has to make an ad hoc assumption on the asymptotic behaviour on either the Cauchy data towards spacelike infinity, or the characteristic data towards future null infinity. By the nature of these asymptotic regimes, it is difficult to directly give any physical meaning to such assumptions. In this talk, I will explain why the scattering problem, with data posed at the infinite past, is a more natural starting point of dynamics from this perspective. In particular, I will explain how data posed at the infinite past can be given physical motivation in a very straight-forward way, and how one then can dynamically derive the asymptotic properties towards spacelike infinity, future null infinity, and future timelike infinity. As a result of this analysis, we will prove that the typical assumptions placed on Cauchy data (Schwarzschild plus faster decay of ) and on characteristic data (smoothness of null infinity) are too strong for capturing very basic physical situations, and that this slower initial decay also leads to -in principle observable- very slow decay at late times. Finally, in view of the audience, I want to formulate an open problem concerning the possibility of detecting physically relevant assumptions purely at the level of Cauchy data.

Donnerstag, 14.11.2024: Derivation of the Effective Dynamics for the Bose Polaron

Siegfried Spruck (Tübingen)

We consider the dynamics of a dense quantum gas consisting of $N$ bosons evolving in $\mathbb{R}^3$ in the presence of an impurity particle in the mean-field scaling with initially high density $\rho$ and large volume $\Lambda$ of the gas. In the initial state of the system almost all bosons are in the Bose-Einstein condensate, with a few excitations. For this system we derive from the microscopic dynamics in the limit of large densities and volumes the effective description by a quantum field theory modelled by the Bogoliubov-Fröhlich Hamiltonian and thus prove the existence of a quasi-particle, the Bose polaron.

Donnerstag, 14.11.2024: Singularity structure of FLRW spacetimes at low regularities

Dr. Jan Sbierski (University of Edinburgh)

This talk investigates the structure of the Big Bang singularity in a variety of FLRW spacetimes. It is straightforward to compute scalar curvature invariants to determine whether a curvature singularity is present which excludes a continuation as a strong solution to the Einstein equations. In this talk the focus is on capturing the singularity structure at the level of the connection and the metric itself, determining which geometric quantities blow up and in which regularity class the solution breaks down.

Freitag, 15.11.2024: Tagung zur Hochschuldidaktik der Mathematik

Diverse Vortragende

Im Rahmen der Herbsttagung des Arbeitskreises HochschulMathematikDidaktik gibt es diverse Vorträge, die insbesondere auch für Dozentinnen und Dozenten am Fachbereich interessant sein könnten. Die Tagung findet vom 15.11.2024 ca. 14 Uhr bis in den frühen Nachmittag des 16.11. statt. Die Vorträge finden in N14 und C4H33 statt. Am Samstag um 9 Uhr trägt im Plenum Prof. Anselm Knebusch vor, der diese Jahr den Ars-legendi-Fakultätenpreis für Mathematik erhalten hat. Mehr Einzelheiten finden Sie auf der Seite der Arbeitsgruppe Mathematik und ihre Didaktik.

Freitag, 15.11.2024: Projectivity and beyond: Foundations for scalable statistical relational artifical intelligence

Dr. Felix Weitkämper (German University of Digital Science Potsdam)

Abstract: Statistical relational artificial intelligence, sometimes abbreviated as StarAI, combines the expressivity of first-order logic with a probabilistic approach to uncertainty. Typically, StarAI approaches either extend relational logical for- malisms such as logic programming with probabilities (resulting in probabilistic logic programming) or lift probabilistic graphical models such as Bayesian networks or Markov random fields to a relational setting (leading to relational Bayesian net- works or Markov logic networks). While the synthesis of logical and probabilistic modelling allows for very expressive, domain-independent representations, it also creates new challenges for learning and reasoning. In particular, available learning algorithms scale poorly with the underlying data set, exacerbating issues already felt in non-probabilistic relational learning. The scaling behaviour of models after learning is often very brittle, and with increasing domain size, derived probabilities can converge to certainty or impossibility regardless of the parameters encoded in the model. This hampers the domain-independent representation, one of StarAI’s hallmark features, and prevents learning models on smaller training sets to alleviate scalability issues. This talk aims to give an overview over my recent work on this question, identifying the restrictive fragments of commonly used statistical relational languages that behave particularly well when used across domain size (projective fragments) and presenting more expressive statistical relational frameworks that uniformly approximate projective distributions in the limit. The talk will conclude with a discussion of the implications of those approximation results for scalable statistical relational learning.

Samstag, 16.11.2024: Tagung zur Hochschuldidaktik der Mathematik

Diverse Vortragende

Im Rahmen der Herbsttagung des Arbeitskreises HochschulMathematikDidaktik gibt es diverse Vorträge, die insbesondere auch für Dozentinnen und Dozenten am Fachbereich interessant sein könnten. Die Tagung findet vom 15.11.2024 ca. 14 Uhr bis in den frühen Nachmittag des 16.11. statt. Die Vorträge finden in N14 und C4H33 statt. Am Samstag um 9 Uhr trägt im Plenum Prof. Anselm Knebusch vor, der diese Jahr den Ars-legendi-Fakultätenpreis für Mathematik erhalten hat. Mehr Einzelheiten finden Sie auf der Seite der Arbeitsgruppe Mathematik und ihre Didaktik.