Vorträge am Mathematischen Institut

Ferdinand Suchanek:

,,Varianzapproximationen des ML-Schätzers im Admixture-Modell"

Zeit und Ort

Donnerstag, 5.3.26, 10:00-11:30, Seminarraum 232

Zusammenfassung

Jannek Link:

Differential Forms on Valuation Rings

Zeit und Ort

Freitag, 13.2.26, 10:30-12:00, Seminarraum 404

Zusammenfassung

Liang Yu:

When A+xA=\mathbb{R}

Zeit und Ort

Donnerstag, 5.2.26, 16:00-18:00, Seminarraum 125

Zusammenfassung

We investigate which algebra substructures A of reals are such that there is a real x for which A+xA=\mathbb{R}.

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

Daniel Junginger:

Das Wortproblem für Halbgruppen

Zeit und Ort

Dienstag, 3.2.26, 14:30-16:00, SR 125

Zusammenfassung

Grey Suchan:

tba

Zeit und Ort

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

Zusammenfassung

Andrea Agazzi:

Transformers as mean-field interacting particle systems

Zeit und Ort

Freitag, 30.1.26, 12:00-13:30, Seminarraum 404

Zusammenfassung

Transformers are a central architecture in modern deep learning, forming the backbone of large language models such as ChatGPT. In this talk, I will present a mathematical framework for studying how information propagates through the layers of such neural networks. Specifically, we model the network’s internal representations as a large system of weakly interacting particles and study the associated mean-field PDE governing their depth evolution. Numerical experiments reveal that, under certain conditions, these dynamics exhibit a metastable clustering phenomenon, where tokens group into well-separated clusters that evolve slowly over time. A rigorous analysis of this behavior uncovers a range of open questions and unexpected connections to analysis and geometry.

Damian Sercombe:

Maximal toroids and Cartan subgroups of algebraic groups.

Zeit und Ort

Freitag, 30.1.26, 10:30-12:00, Seminarraum 404

Zusammenfassung

The theory of Cartan subgroups and maximal tori for smooth affine algebraic groups is well-established, and is crucial to the structure theory. An analogous, but more complicated, theory holds for restricted Lie algebras. We generalise and unify both theories into one which holds for all affine algebraic groups over a field, without any smoothness assumptions. We find that many results which hold in both special cases generalise remarkably well. We conclude with some applications to, and a discussion of, some generation problems for algebraic groups.

Lukas Riepl:

Sharp nonparametric testing for constant volatility

Zeit und Ort

Mittwoch, 28.1.26, 14:15-15:45, SR 127/128 (E1)

Zusammenfassung

Liuluan Yang:

Spacelike spherical constant Gauss curvature and k-Hessian curvature hypersurfaces in Schwarzschild spacetimes

Zeit und Ort

Dienstag, 27.1.26, 16:15-17:45, Seminarraum 125

Zusammenfassung

Nils Kober:

Master-Vortrag "Convergence rate analysis of the nonparametric maximum likelihood estimator with external data"

Zeit und Ort

Dienstag, 27.1.26, 15:15-16:45, Seminarraum 232

Zusammenfassung

Lukas Riepl:

Master-Vortrag "Sharp nonparametric testing for constant volatility"

Zeit und Ort

Dienstag, 27.1.26, 14:15-15:45, Seminarraum 232

Zusammenfassung

Eric Trébuchon:

The Dirac operator on conical domains

Zeit und Ort

Montag, 26.1.26, 16:15-17:45, Seminarraum 404

Zusammenfassung

In this talk, I will give an overview of my current research on Dirac operators on conical domains. I will outline the main ideas behind the proof that the Dirac operator is regular and self-adjoint for a large class of local boundary conditions. Previously, such results were only available for two-dimensional corner domains under very restrictive boundary conditions, or for rotationally symmetric convex three-dimensional cones equipped with MIT bag boundary conditions.

Recently, Pankrashkin proved self-adjointness of the Dirac operator with MIT bag boundary conditions on all (possibly non-smooth) convex domains. Combining this result with my own, the class of admissible domains extends to a huge family of domains with convex edges and arbitrary (non-convex, non-Lipschitz, non-connected, ...) "smooth" cones. That said, non-convex polyhedra are not yet understood.

David Belius:

A survey of novel optimizations algorithms for deep neural networks

Zeit und Ort

Freitag, 23.1.26, 12:00-13:30, Seminarraum 404

Zusammenfassung

The Adam/AdamW optimization algorithms have long been the workhorses for training deep neural networks. Recently, the alternative optimization algorithm Muon (Keller, Berstein et al '24) has been gaining traction. As well as outperforming Adam in many circumstances (in terms of test error), it has a fascinating theoretical motivation as an algorithm that favors feature learning in wide networks, thus avoiding the Neural Tangent Kernel infinite-width limit, which is incapable of feature learning. This survey talk will explain Muon, as well as the related optimization algorithms MuP and Scion. If time permits I will also discuss the Shampoo and SOAP training algorithms, which also outperform Adam in certain situations, while having a more traditional theoretical motivation as a pseudo-second order methods, and their possible relation to Muon & friends.

Xinrui You:

Kac polynomials

Zeit und Ort

Freitag, 23.1.26, 10:30-12:00, Seminarraum 404

Zusammenfassung

This is a presentation of a master thesis, written under the supervision of Prof. Letellier and Prof. Hennecart with local supervisor Prof. Soergel. Kac studied the counting of absolutely indecomposable quiver representations of fixed dimensions over finite fields in 1982. This counting yields a polynomial over the cardinal of finite fields, named Kac polynomial. We will discuss the polynomiality and integrality of Kac polynomials, after Kac.

Tatjana Schreiber:

Digital Twins in Research and Industry: An Overview

Zeit und Ort

Mittwoch, 21.1.26, 14:15-15:45, SR 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.

Jorge Carrasco Coquillat:

Minimal types and independence in differentially closed fields

Zeit und Ort

Dienstag, 20.1.26, 14:30-16:00, SR 125

Zusammenfassung

Working in the theory of differentially closed fields of characteristic 0, Fre- itag and Moosa introduced the degree of nonminimality d of a type as a measure of how many parameters are needed to witness that the type is not minimal. Together with Jaoui, they showed that if a stationary type has nonminimality degree \(d\ge 2\), weakly orthogonal and internal to the field of constants, its binding group acts generically \(d\)-transitively on the set of realizations of p. This can be used to show that if every triple of different realizations of a nonalgebraic stationary type is independent over its set of parameters, then the type is minimal.

Rupert Klein:

Thoughts on Machine Learning

Zeit und Ort

Dienstag, 20.1.26, 14:15-15:15, Seminarraum 226, HH10

Zusammenfassung

Techniques of machine learning (ML) find a rapidly increasing range of applications touching upon social, economic, and technological aspects of everyday life. They are also being used with great enthusiasm to fill in gaps in our scientific knowledge by data-based modelling approaches. I have followed these developments for a while with interest, concern, and mounting disappointment. When these technologies are employed to take over decisive functionality in safety-critical applications, we would like to exactly know how to guarantee their compliance with pre-defined guardrails and limitations. Moreover, when they are utilized as building blocks in scientific research, it would violate scientific standards -in my opinion- if these building blocks were used without a throrough understanding of their functionality, including inaccuracies, uncertainties, and other pitfalls. In this context, I will juxtapose (a subset of) deep neural network methods with the family of entropy-optimal Sparse Probabilistic Approximation (eSPA) and entropy-optimal network (EON) techniques developed recently by Illia Horenko (RPTU Kaiserslautern-Landau) and colleagues.

Nadine Große:

On local boundary conditions for Dirac-type operators

Zeit und Ort

Montag, 19.1.26, 16:15-17:45, Seminarraum 404

Zusammenfassung

We discuss smooth local boundary conditions for Dirac-type operators, giving existence and non-existence results for local self-adjoint boundary conditions and discuss concrete parametrizations of the space of all those conditions in low dimensions. We also give concrete conditions when these boundary conditions are elliptic/regular/Shapiro-Lopatinski (i.e. in particular giving rise to self-adjoint Dirac operators with domain in H^1). This is joint work with Hanne van den Bosch (Universidad de Chile) and Alejandro Uribe (University of Michigan).

Kang Zuo:

TBA

Zeit und Ort

Freitag, 16.1.26, 16:00-18:00, Seminarraum 125

Zusammenfassung

Qun Chen:

TBA

Zeit und Ort

Freitag, 16.1.26, 14:00-16:00, Seminarraum 125

Zusammenfassung

Sara Mazzonetto:

On some results for Threshold Diffusions

Zeit und Ort

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

Zusammenfassung

Threshold diffusions are solutions to stochastic differential equations whose coefficients change discontinuously depending on the position of the process relative to certain barriers (thresholds). They are used to model a variety of phenomena in finance/economics, engineering, and other sciences. Many questions remain about existence and uniqueness, numerical approximation, and statistical inference for these processes. This seminar will primarily offer a mathematical introduction to some of these processes. We will also consider parameter inference problems, presenting asymptotic results under different assumptions and highlighting technical challenges arising, in particular, from the interplay between discontinuities and local times.

Ernst Eberlein:

The term structure of implied correlations between S&P and VIX markets

Zeit und Ort

Freitag, 9.1.26, 12:00-13:30, Seminarraum 404

Zusammenfassung

We develop a joint model for the S&P500 and the VIX indices with the aim of extracting forward looking information on the correlation between the two markets. We achieve this by building the model on time changed Lévy processes, deriving closed analytical expressions for relevant quantities directly from the joint characteristic function, and exploiting the market quotes of options on both indices. We perform a piecewise joint calibration to the option prices to ensure the highest level of precision within the limits of the availability of quotes in the dataset and their liquidity. Using the calibrated parameters, we are able to quantify the leverage effect along the term structure of the VIX options and corresponding VIX futures. We illustrate the model using market data on S&P500 options and both futures and options on the VIX. This is joint work with Laura Ballotta and Grégory Rayée.

Laurent Mazliak:

Emile Borel and the probabilistic turn of a worried Cantorian

Zeit und Ort

Donnerstag, 8.1.26, 15:00-16:30, Hörsaal 2

Zusammenfassung

In this talk, I shall present the singular way in which Émile Borel, from his studies on the structure of real numbers and a certain rejection of Cantor's abstract vision, found in the calculus of probabilities an adequate tool to formulate a new approach to problems. At the same time, he became aware of its usefulness for the study of phenomena of physics and society and developed a singular viewpoint on the concept of probability, merging subjectivist and objectivist aspects under an idiosyncratic formulation of the so-called Cournot principle.

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.

Wei Wei:

Brendle' s non-compactness of the Yamabe equation in the manifolds with non-vanishing Weyl tensor

Zeit und Ort

Dienstag, 16.12.25, 16:15-17:45, Seminarraum 125

Zusammenfassung

Marius Zeinhofer:

Geometric Optimization in Scientific Machine Learning

Zeit und Ort

Dienstag, 16.12.25, 14:15-15:15, Seminarraum 226, HH10

Zusammenfassung

We discuss an “optimize-then-project” approach for applications in scientific machine learning. The key idea is to design algorithms at the infinite-dimensional level and subsequently discretize them in the tangent space of the neural network ansatz. We illustrate this approach in the context of the variational Monte Carlo method for quantum many-body problems, where neural quantum states have recently emerged as powerful representations of high-dimensional wavefunctions. In this setting, we recover the celebrated stochastic reconfiguration algorithm, interpreting it as a projected Riemannian L2 gradient descent method. We further explore extensions to Riemannian Newton methods, and conclude with considerations related to the scalability of these schemes.

Himanshu Chaudary:

tba

Zeit und Ort

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

Zusammenfassung

Ben Deitmar:

The Kolmogorov-Smirnov test

Zeit und Ort

Mittwoch, 10.12.25, 14:15-15:45, SR 226 (HH10)

Zusammenfassung

Benjamin Gess:

Fluctuations in Continuum

Zeit und Ort

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

Zusammenfassung

Fluctuations are ubiquitous in real world contexts and in key technological challenges, ranging from thermal fluctuations in physical systems to algorithmic stochasticity in machine learning and fluctuations driven by small-scale weather patterns in climate dynamics. At the same time, such complex systems are influenced by a multitude of factors, relying on a wide range of parameters and interactions. This motivates the exploration of scaling limits and continuum dynamics. A systematic understanding of such interplay of stochasticity, complex dynamical behaviour, and continuum limits seeks to reveal universal properties, irrespective of the specific details of the systems in question. In this presentation we will examine the importance of modelling fluctuations in large systems through several examples, ranging from non-equilibrium thermodynamics to stochastic fluid dynamics and machine learning. We will demonstrate how their analysis uncovers deep connections between probabilistic features, partial differential equations with irregular coefficients, and geometric structures in the form of gradient flows on infinite dimensional manifolds.

Simone Pavarana:

Bayesian estimation with set-valued observations

Zeit und Ort

Mittwoch, 3.12.25, 16:00-17:30, Seminarraum 226 (HH10)

Zusammenfassung

In this work, we develop a Bayesian estimation framework in which the observable quantity is a random set rather than a classical random vector. The motivation comes from expert-opinion modelling in highly uncertain environments—such as robotic risk management—where experts typically provide confidence intervals or other set-valued assessments for key risk factors. We first show that, under suitable assumptions, the standard Bayesian update formula remains valid and does not require substantial modification compared to the classical setting where both signal and observation take values in Polish spaces. However, whereas these assumptions (such as absolute continuity of the conditional distribution of the observation given the signal) are straightforward to verify for vector-valued observations, the set-valued case requires a more delicate, case-by-case analysis.

We then establish that the required regularity conditions indeed hold for a broad class of random sets, including random intervals, balls, finite sets, and more generally convex compact sets. Particular attention is devoted to finite-valued random sets, which are more challenging and also appear in Bayesian filtering within the FISST framework introduced by Ronald P. S. Mahler in the mid-1990s.

Work together with Prof. Dr. Cagin Ararat

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.

Mingyang Han:

The super-Liouville equation on the sphere

Zeit und Ort

Dienstag, 2.12.25, 16:15-17:45, Seminarraum 125

Zusammenfassung

In this work, we study the super-Liouville equation on the sphere with positive coefficient functions. We begin by deriving estimates for the spinor component of the equation, thereby finding that the energy of the spinor part of solutions is uniformly bounded. We then analyze the compactness of the solution space in two aspects: the compactness for solutions with small energy and the compactness with respect to the conformal transformation group of the sphere. Finally, by introducing a new natural constraint, the Nehari manifold, and employing variational methods, we obtain the existence of the least energy solutions when the coefficient functions are even.

Alen Kushova:

Space-time least squares approximation for Schrödinger equation and efficient solver

Zeit und Ort

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

Zusammenfassung

We propose a space-time least-squares Galerkin formulation for the numerical solution of the Schrödinger equation, which overcomes the numerical instability of the plain Galerkin formulation in space-time and provides a well-posed method with quasi-optimal convergence. When discretized on a tensor product grid, the arising linear system is a sum of Kronecker product matrices. We propose a preconditioning strategy following a variant of the Fast Diagonalization approach. We also derive an explicit bound for the eigenvalues of the preconditioned system when a tensor product spline space is considered for the discretization. This bound is independent of the mesh size and depends on the quasi-uniformity parameter of the mesh and the spline degree p. Numerical results validate the theoretical convergence and demonstrate the computational efficiency of the approach.

Paul Schuh:

Tangentialer Sp(1)-Bordismus

Zeit und Ort

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

Zusammenfassung

Rolf Backofen:

Machine-Learning in the Context of CRISPR Research

Zeit und Ort

Freitag, 28.11.25, 12:00-13:30, Seminarraum 404

Zusammenfassung

As microbial genomes become available at an increasing rate, searching for CRISPR-systems is a essential task for determining both evolution of CRISPR system and new CRISPR-related functions. Bioinformatics has always been a driving force in this respect. We also had realized, however, that the annotation of CRISPR system is a difficult and labor-intensive work.

In recent years, the situation has improved by applying state-of-the-art machine learning approach to detect and annotated CRISPR systems. In this talk, we will discuss various annotation task that have been solved in our group using advanced machine learning. One lesson to be learned is that we do not have a swiss knife in machine learning that can be used for all task. Instead, many problems require specific ML-solutions, most likely due to the fact that we still have limited data available.

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

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.

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.

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.

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).

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 \(\delta\).

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.

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.

Nils Kober:

Transfer learning for maximum likelihood estimation

Zeit und Ort

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

Zusammenfassung

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.

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]).

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.

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.

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.

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.

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.

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.

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.