Vorträge am Mathematischen Institut

Rosa Winter:

Many rational points on del Pezzo surfaces of low degree

Zeit und Ort

Freitag, 17.10.25, 10:30-11:30, Seminarraum 404

Zusammenfassung

Let X be an algebraic variety over a number field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. Questions one might ask are, is X(k) empty or not? And if it is not empty, how ‘large’ is X(k)? Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d at least 3, these are the smooth surfaces of degree d in P^d). The lower the degree, the more complex del Pezzo surfaces are. I will give an overview of different notions of ‘many’ rational points, and go over several results for rational points on del Pezzo surfaces of degree 1 and 2. I will then focus on work in progress joint with Julian Demeio and Sam Streeter on the so-called Hilbert property for del Pezzo surfaces of degree 1.

Francesco Cattafi:

An overview on Lie pseudgroups and geometric structures

Zeit und Ort

Montag, 20.10.25, 16:15-17:45, Seminarraum 404

Zusammenfassung

The space of (local) symmetries of a given geometric structure has the natural structure of a Lie (pseudo)group. Conversely, geometric structures admitting a local model can be described via the pseudogroup of symmetries of such local model.

The main goal of this talk is to provide several examples and give an intuitive understanding of the slogan above, which can be made precise at various levels of generality (depending on the definition of "geometric structure") and using different tools/methods.

Moreover, I will sketch a new framework, which include previous formalisms (e.g. G-structures or Cartan geometries) and allows us to prove integrability theorems. In particular, I will provide intuition on the relevant objects which make this approach work, namely Lie groupoids endowed with a multiplicative "PDE-structure" and their principal actions. Poisson geometry will give us the guiding principles to understand those objects, which are directly inspired from, respectively, symplectic groupoids and principal Hamiltonian bundles.

This is based on a forthcoming book written jointly with Luca Accornero, Marius Crainic and María Amelia Salazar.

Dr. Henrike Allmendinger :

Mathematische Orientierung: Fachliche Überhöhungen und ihr Einfluss auf den Unterricht

Zeit und Ort

Dienstag, 21.10.25, 18:30-20:00, Hörsaal 2

Zusammenfassung

Lehrkräfte sollten mehr Mathematik verstehen, als sie im Unterricht vermitteln. Wie können fachliche Überhöhungen in der Lehramtsausbildung eine „Mathematical Orientation“ fördern (vgl. Allmendinger, Aslaksen & Buchholtz, ZDM 2023), die wissenschaftlich fundiert und schulpraktisch relevant ist? Der Vortrag skizziert ein Rahmenkonzept und zeigt anhand eines konkreten Beispiels aus der Sekundarstufenmathematik, wie Vertiefungen jenseits des Curriculums zentrale Zusammenhänge zwischen Geometrie, Arithmetik und Algebra sichtbar machen. Die Analyse von Studierendenreflexionen verdeutlicht, wie diese Einsichten in Unterrichtsimpulse übersetzt werden. So schlagen fachliche Überhöhungen Brücken zwischen Hochschulmathematik und Praxis.

Alessandro Vegnuti:

Archimedean classes of hypernaturals, and their use in ART

Zeit und Ort

Dienstag, 28.10.25, 14:30-16:00, Seminarraum 404

Zusammenfassung

Arithmetic Ramsey Theory (ART) studies what kind of arithmetic configurations we cannot avoid taking a finite partition of the naturals: arithmetic progressions, large sets with all possible sums of their elements, solutions to certain polynomials are just some examples of these configurations (usually called Partition Regular, PR). How to deal with such problems? In the last years, ideas coming from nonstandard analysis - and linked with ultrafilter algebra - have provided a natural framework to study Ramsey-theoretic questions. In this talk, we will present a new tool to prove that certain polynomials are not PR: by adopting the nonstandard point of view, we will show how the notion of Archimedean classes of hypernaturals can easily produce negative results in ART. This is a joint work with Lorenzo Luperi Baglini.

Mingwei Zhang:

The curl operator and Sobolev inequalities for differential forms

Zeit und Ort

Dienstag, 28.10.25, 16:15-17:45, Seminarraum 125

Zusammenfassung

The curl operator for vectors in R^3 is of special importance and gives rise to various Sobolev inequalities. In this talk we will introduce the generalized curl operator for differential forms in higher dimensions and discuss the spectral analysis. As an application, we prove that fundamental bubbles (Killing forms) are local minimizers of one Sobolev inequality, but not local minimizers of another Sobolev inequality. This is a joint work with Prof. Guofang Wang.