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.

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.

Xuwen Zhang:

Anisotropic minimal graphs with free boundary

Zeit und Ort

Dienstag, 4.11.25, 16:15-17:45, Seminarraum 125

Zusammenfassung

Minimal surface equation is a classical topic in Geometric Analysis and PDEs. In this talk, we discuss recent progress on anisotropic minimal surface equation, and prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-side linear growth. This is a joint work with Guofang Wang, Wei Wei, and Chao Xia.

Maxwell Levine:

Recent Developments in Namba Forcing

Zeit und Ort

Donnerstag, 6.11.25, 15:00-16:30, Hörsaal 2

Zusammenfassung

I will discuss work done in Freiburg on a technique called Namba forcing. This technique was originally used by Namba and Bukovsky to, in essence, demonstrate certain differences between the cardinals \(\aleph_0\), \(\aleph_1\), and \(\aleph_2\). I found an argument for what is called ``the weak approximation property,'' which, in the context of forcing, means that certain functions are not added in the extension. In joint work with Heike Mildenberger and with Hannes Jakob, this led to the resolution of some longstanding open questions in PCF theory, which concerns the study of singular cardinals. With a similar argument I solved an old question about the minimality of forcing extensions. The talk is not meant to be technical, but rather an overview of what is happening in the area.

Bernd Rummler:

Remarks to exact Poincaré Constants in n-dimensional Annuli and Balls

Zeit und Ort

Dienstag, 11.11.25, 14:15-15:15, Seminarraum 226, HH10

Zusammenfassung

We study n-dimensional annuli and n-dimensional balls, where we suppose n ∈ {2,..,N} with N < ∞. We investigate in our non-dimensional setting each annulus ΩA- defined via two concentrical balls with radii A/2 and A/2 + 1 in Rn - and n-dimensional open unit balls as ”limits” of ΩAfor A → 0. We provide calculated (precise) Poincar´ e constants for scalar functions (with vanishing Dirichlet traces on the boundary) in dependence of the inner diameter A and the dimension nof the space Rn for these geometries. Addi- tionally we lay open the direct match of the Poincar´ e constants for solenoidal vector fields and the Poincaré constants for scalar functions (both with vanishing Dirichlet traces on the boundary) for solenoidal vector in space R2 resp. R3 with the Poincar´ e constants for scalar functions in R4 resp. R5. Generally we use the first eigenvalues of the scalar Laplacian (or the first eigenvalues the Stokes operator) for the calculation of the Poincar´ e constants. Supplementary, corresponding problems in domains Ω∗ σ (cf. e.g. the 3d-annuli from [12]) are investigated - for comparison but also to provide the limits for A → 0. These domains Ω∗ σ enable us to use the Green’s function of the Laplacian on Ω∗ σ with vanishing Dirichlet traces on ∂Ω∗ σ to show that for σ → 0 the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in Rn . On the other hand, we take advantage of the so-called small-gap limit for A → ∞ like in our papers to Poincar´ e constants in annuli (cf. [10] and [11]).

Bohan Wang:

The stability of Sobolev inequality on the Heisenberg group

Zeit und Ort

Dienstag, 11.11.25, 16:15-17:45, Seminarraum 125

Zusammenfassung

In this talk, we are concerned with the optimal asymptotic lower bound for the stability of Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of Sobolev inequality on the CR sphere through bispherical harmonics and complicated orthogonality technique (see Lemma 3.1). The loss of the Polya-Szego inequality and the Riesz rearrangement inequality on the Heisenberg group makes it impossible to use any rearrangement flow technique to derive the optimal stability of Sobolev inequality on the CR sphere from corresponding optimal local stability. To circumvent this, we will use the CR Yamabe flow to pass from the local stability to the global stability and thus establish the optimal stability of Sobolev inequality on the Heisenberg group with the dimension-dependent constants (see Theorem 1.1). This work was accomplished together with Lu Chen, Guozhen Lu and Hanli Tang.

Nils Kober:

Transfer learning for maximum likelihood estimation

Zeit und Ort

Mittwoch, 12.11.25, 14:15-15:45, SR 127/128

Zusammenfassung

Niklas Müller:

Singularities of base spaces of Lagrangian fibrations

Zeit und Ort

Freitag, 14.11.25, 10:30-12:00, Seminarraum 404

Zusammenfassung

Irreducible holomorphic symplectic varieties (or IHS for short) are a special class of projective algebraic varieties that can be studied from various angles; they are interesting because they are expected to satisfy many special geometric properties. Yet, at this time, they remain largely illusive. One promising way to understand the geometry of IHS varieties is through the study of so-called Lagrangian fibrations. A folklore conjecture attributed to Matsushita claims that the base \(X\) of such a fibration is necessarily isomorphic to the complex projective space. In this talk, we will survey several aspects of the geometry of IHS varieties. Finally, we present a new and short proof of Matsushita's conjecture in case \(\dim X = 2\). This talk is based on joint work with Zheng Xu.

Simon Buchwald:

A greedy reconstruction algorithm for minimal neural network architectures

Zeit und Ort

Dienstag, 18.11.25, 14:15-15:15, Seminarraum 226, HH10

Zusammenfassung

n many machine learning applications the choice of an appropriate/optimal neural network architecture is based purely on heuristic experience, or determined by trial and error. Moreover, even when a “good” network is found, it is a common issue that the training data is not distributed evenly, leading to bias in the networks.

To address these issues, we introduce a new greedy algorithm that selects simultaneously a subset of optimal training data points and the smallest neural network that is able to learn the selected data, while also representing well the non-selected data. By this approach, we are able to keep a perfect balance between under- and overfitting. Additionally, the non-selected training data is turned into validation data, which is especially useful in settings where only limited data is available.

We demonstrate the effectiveness of our new method by numerical experiments for function approximation and classification problems. This talk is based on a joint work with Gabriele Ciaramella and Marco Verani.

Yuchen Bi:

Stability of the Clifford Torus as a Willmore Minimizer

Zeit und Ort

Dienstag, 18.11.25, 16:15-17:45, Seminarraum 125

Zusammenfassung

This is joint work with Jie Zhou (Capital Normal University). We prove that surfaces in \(\mathbb{S}^3\) with genus \(\geq 1\) and Willmore energy \(\leq 2\pi^2 + \delta^2\) are quantitatively close to the Clifford torus after a conformal transformation. The closeness is measured in three aspects: \(W^{2,2}\) parametrization, \(L^\infty\) conformal factor, and conformal structure, with linear dependence on \(\delta\).

Amador Martin-Pizarro:

Model theory, differential algebra and functional transcendence

Zeit und Ort

Freitag, 21.11.25, 10:30-12:00, Seminarraum 404

Zusammenfassung

A fundamental problem in the study of algebraic differential equations is determining the possible algebraic relations among different solutions of a given differential equation. Freitag, Jaoui, and Moosa have isolated an essential property, called property D2, in order to show that if a differential equation given by an irreducible differential polynomial of order n is defined over the constants and has property D2, then any number of pairwise distinct solutions together with their derivatives up to order n-1 are algebraically independent. The property D2 requires that, given two distinct solutions, there is no non-trivial algebraic dependence between the solutions and their first n-1 derivatives.

The proof of Freitag, Jaoui and Moosa is extremely elegant and short, yet it uses in a clever way fundamental results of the model theory of differentially closed fields of characteristic 0. The goal of this talk is to introduce the model-theoretic tools at the core of their proof, without assuming a deep knowledge in (geometric) model theory (but some familiarity with basic notions in algebraic geometry).

Fateme Sajadi:

A Unified Finiteness Theorem For Curves

Zeit und Ort

Freitag, 21.11.25, 14:00-15:30, Seminarraum 404

Zusammenfassung

This talk presents a unified framework for finiteness results concerning arithmetic points on algebraic curves, exploring the analogy between number fields and function fields. The number field setting, joint work with F. Janbazi, generalizes and extends classical results of Birch–Merriman, Siegel, and Faltings. We prove that the set of Galois-conjugate points on a smooth projective curve with good reduction outside a fixed finite set of places is finite, when considered up to the action of the automorphism group of a proper integral model. Motivated by this, we consider the function field analogue, involving a smooth and proper family of curves over an affine curve defined over a finite field. In this setting, we show that for a fixed degree, there are only finitely many étale relative divisors over the base, up to the action of the family's automorphism group (and including the Frobenius in the isotrivial case). Together, these results illustrate both the parallels and distinctions between the two arithmetic settings, contributing to a broader unifying perspective on finiteness.

Dr. Lucía Martín Merchán:

Topological aspects of compact holonomy and closed G₂ manifolds

Zeit und Ort

Montag, 24.11.25, 16:15-17:45, Seminarraum 404

Zusammenfassung

Within Berger’s classification of holonomy groups, G₂ is the distinguished case in dimension seven, and a G₂-holonomy metric determines a parallel 3-form ϕ. As in other special geometries, the existence of such metrics imposes topological constraints on compact manifolds; analogues in Kähler geometry include the hard Lefschetz property, the Hodge decomposition, and formality. Formality, first discovered as a property of compact Kähler manifolds by Deligne, Griffiths, Morgan, and Sullivan in 1975, depends on the rational homotopy type of a manifold.

We review recent developments in the topology of compact holonomy G₂ manifolds by focusing on two results: one showing that compact holonomy G₂ manifolds need not be formal (arXiv:2409.04362), and another presenting examples of compact closed G₂ manifolds (dϕ=0) that satisfy all known topological obstructions to admitting holonomy G₂ metrics, for which the existence of such metrics cannot be confirmed or excluded with current techniques.

Wilfried Kuissi-Kamdem:

Asset-liability management with Epstein-Zin utility under stochastic interest rate and unknown market price of risk

Zeit und Ort

Mittwoch, 26.11.25, 16:00-17:30, Seminarraum 226 (HH10)

Zusammenfassung

In this talk we present a stochastic control problem with Epstein-Zin recursive utility under partial information (unknown market price of risk), in which an investor is constrained to a liability at the end of the investment period. Introducing liabilities is the main novelty of the model and appears for the first time in the literature of recursive utilities. Such constraint leads to a coupled forward-backward stochastic differential equation (FBSDE), which well-posedness has not been addressed in the literature. We derive an explicit solution to the FBSDE, contrasting with the existence and uniqueness results with no explicit expression of the solutions typically found in most related literature. Moreover, under minimal additional assumptions, we obtain the Malliavin differentiability of the solution of the FBSDE. We solve the problem completely and find the expression of the controls and the value function. Finally, we determine the utility loss that investors suffer from ignoring the fact that they can learn about the market price of risk.

Norbert Schappacher:

How to grasp Emmy Noether’s approach to mathematics (and physics)

Zeit und Ort

Donnerstag, 27.11.25, 15:00-16:30, Hörsaal 2

Zusammenfassung

Paul Schuh:

Tangentialer Sp(1)-Bordismus

Zeit und Ort

Montag, 1.12.25, 16:15-17:45, Seminarraum 404

Zusammenfassung

Alen Kushova:

tba

Zeit und Ort

Dienstag, 2.12.25, 14:15-15:15, Seminarraum 226, HH10

Zusammenfassung

Dr. Raja Herold-Blasius:

Mathematisches Problemlösen mit Strategieschlüsseln für alle

Zeit und Ort

Dienstag, 2.12.25, 18:30-20:00, Hörsaal 2

Zusammenfassung

Das mathematische Problemlösen ist eine der Kompetenzen, die großen Einfluss auf Lernerfolge hat. Trotzdem wird das Problemlösen immer wieder vernachlässigt. Das liegt nicht zuletzt daran, dass Schüler:innen das Bearbeiten und Lösen mathematischer Probleme als herausfordernd erleben. Zu deren Unterstützung entwickle und untersuche ich Konzepte und Materialien, die Schüler:innen verschiedenen Alters und unterschiedlicher Leistungsstände erlauben, Hürden beim Problemlösen zu überwinden und so Problemlöseprozesse erfolgreich zu bewältigen.

Im Vortrag werden Einblicke in die Arbeit mit einem digitalen Training zum Strategieerwerb mit Strategieschlüsseln gegeben und zur Diskussion gestellt.

Benjamin Gess:

Fluctuations in Continuum

Zeit und Ort

Donnerstag, 4.12.25, 15:00-16:30, Hörsaal 2

Zusammenfassung

Himanshu Chaudary:

tba

Zeit und Ort

Montag, 15.12.25, 16:15-17:45, Seminarraum 404

Zusammenfassung

Dr. Kristin Litteck:

Die Bedeutung individuell verfügbaren Vorwissens für den Erwerb von Wissen zum Ableitungsbegriff

Zeit und Ort

Dienstag, 16.12.25, 18:30-20:00, Hörsaal 2

Zusammenfassung

Der Ableitungsbegriff bildet einen zentralen Zugang zur Analysis in der gymnasialen Oberstufe. Zahlreiche Studien zeigen jedoch, dass viele Schülerinnen und Schüler erhebliche Schwierigkeiten beim Verständnis dieses Begriffs haben. Ein möglicher Erklärungsfaktor liegt im unzureichend entwickelten Vorwissen aus der Sekundarstufe I.

Im Vortrag werden typische Verständnishürden des Ableitungsbegriffs sowie Ergebnisse quantitativer Studien zum Einfluss spezifischen Vorwissens vorgestellt und diskutiert. Darüber hinaus wird eine Kurzintervention präsentiert, die sich als praxistauglicher Ansatz zur Kompensation von Defiziten im Vorwissen bewährt hat.

Sara Mazzonetto:

tba

Zeit und Ort

Dienstag, 13.1.26, 14:15-15:15, Seminarraum 226, HH10

Zusammenfassung

Henrik Ossadnik:

Kernideen als Leitlinien: Vom Vorstellungsaufbau in der Sek I zum Testen von Hypothesen in der Sek II

Zeit und Ort

Dienstag, 20.1.26, 18:30-20:00, Hörsaal 2

Zusammenfassung

Anschlussfähiges Unterrichten in der Oberstufe bedeutet, auf Vorerfahrungen und Vorstellungen der Mittelstufe aufzubauen und diese über Jahrgangsstufen hinweg zu entwickeln. Fehlt ein solcher Vorstellungsaufbau, bleibt die Stochastik in der Oberstufe – insbesondere die beurteilende Statistik und das Testen von Hypothesen – für Schülerinnen und Schüler oft unzugänglich. Interpretations- und Deutungsfragen treten dann in den Hintergrund.

Der Vortrag diskutiert auf Grundlage aktueller Forschungsergebnisse, welche Vorstellungen in der Sek I angebahnt werden sollten und wie in der Sek II beim Testen von Hypothesen darauf aufgebaut werden kann. Anhand von Kernideen (der Stochastik) und ausgewählten Beispielen wird verdeutlicht, wie solche Brücken in die Oberstufe gelingen können sowie welche Potenziale und Herausforderungen damit verbunden sind.

Grey Suchan:

tba

Zeit und Ort

Montag, 2.2.26, 16:15-17:45, Seminarraum 404

Zusammenfassung

Carla Cederbaum:

Combining potential theory with general relativity: a divergence theorem-based approach to proving geometric inequalities

Zeit und Ort

Donnerstag, 5.2.26, 15:00-16:30, Hörsaal 2

Zusammenfassung