Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces - Glöckner, Helge (Axioms)
We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

Birkhoff decompositions for loop groups with coefficient algebras - Glöckner, Helge (arXiv:2206.11711)
Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with valued in complex Banach-Lie groups.

Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization - Gebken, Bennet
Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind.

Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients - Berkemeier, Manuel Bastian; Peitz, Sebastian (arXiv:2208.12094)
In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem.

On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation - Bieker, Katharina; Gebken, Bennet; Peitz, Sebastian (IEEE Transactions on Pattern Analysis and Machine Intelligence)
We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.

Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature - Kolb, Martin; Weich, Tobias; Wolf, Lasse Lennart (Annales Henri Poincaré)
Abstract The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$ M is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$ M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$ L 2 -spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Ruelle–Pollicott resonances for manifolds with hyperbolic cusps - Guedes Bonthonneau, Yannick; Weich, Tobias (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data - Hesse, Kerstin; Le Gia, Quoc Thong (Journal of Computational and Applied Mathematics)
Keywords: Applied Mathematics; Computational Mathematics

Regularity of Lie groups - Hanusch, Maximilian (Communications in Analysis and Geometry)
Keywords: regularity of Lie groups

Analysis 1 und 2 Skript/Buch - Hanusch, Maximilian

Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary - Kolb, Martin; Klump, Alexander (Theory of Probability and its Applications)

On non-extinction in a Fleming-Viot-type particle model with Bessel drift - Kolb, Martin; Liesenfeld, Matthias (Electronic Journal of Probability)
Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles.

Persistence of autoregressive sequences with logarithmic tails - Denisov, Denis; Hinrichs, Günter; Kolb, Martin; Wachtel, Vitali (Electronic Journal of Probability)
We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in- dependent and identically distributed innovations {ξk }. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains Xn and Mn are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for Xn and Mn conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of Xn and Mn in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential.

Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik - Aufgabenbeispiele und Ergebnisse einer Evaluationsstudie. - Hoffmann, Max; Biehler, Rolf (Bedarfsgerechte fachmathematische Lehramtsausbildung. Analyse, Zielsetzungen und Konzepte unter heterogenen Voraussetzungen)

Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium - Biehler, Rolf; Hoffmann, Max (Professionsorientierte Fachwissenschaft. Kohärenzstiftende Lerngelegenheiten für das Lehramtsstudium Mathematik)

First-year university students' self-regulated learning during the COVID-19 pandemic: a qualitative longitudinal study - Liebendörfer, Michael; Kempen, Leander; Schukajlow, Stanislaw (ZDM – Mathematics Education)
Abstract When the COVID-19 pandemic began, many universities switched to fully online teaching. This unexpected switching to online teaching was challenging for both teachers and students, and restrictions that were put in place because of pandemic made this challenge even greater. However, new ways of teaching might also open new opportunities for students’ learning. The research question driving our study was as follows: how do students regulate their learning and specifically their choice of resources and peer learning in university mathematics classes that are fully taught online as offered during the COVID-19 pandemic? We report on a longitudinal, qualitative study in which students recorded a brief audio diary twice a week over one whole semester (14 weeks). We focused on three students who completed 70 interviews in total and finished the semester with varying degrees of success. The results show how the students structured their studying (e.g., the roles that deadlines or synchronous teaching events played). They illustrate the strengths and limitations of digital materials provided by the lecturer and the use of complementary media. Further, the pandemic uncovered the double-edged role of simple, often anonymous exchanges (e.g., via Discord servers), with few binding forces for either side, and the significance of stable learning partnerships for students’ success. Our research highlights aspects that should be focal points when comparing traditional instruction and online instruction during the pandemic from a self-regulatory perspective. Practical implications refer to how these aspects can be combined sensibly in fully online courses, but also in blended learning contexts.
Keywords: General Mathematics; Education

Exploring the Perceived Relevance of University Mathematics Studies by First-Semester Teaching Students - Büdenbender-Kuklinski, Christiane; Hochmuth, Reinhard; Liebendörfer, Michael (International Journal of Research in Undergraduate Mathematics Education)
Abstract As earlier research results suggest that many mathematics teaching students criticize a missing relevance in their studies, we explore explanations and interrelationships of their relevance assessments. We aim at finding out how one could support the students in attributing relevance to their study programs. A two-fold model for relevance assessments in mathematics teacher education is proposed, consisting of relevance content and relevance reasons. We investigate students' relevance perceptions of mathematical topics and of topics’ complexities, as well as their rating of individual and societal/ vocational relevance reasons, all in relation to their perception of the relevance of their overall program of study. Contrary to earlier research findings, our results suggest that mathematics teaching students already do attribute relevance to many content areas and that a preparation for the teaching profession is not the only reason for them to assign relevance. There also seem to be many students who would attribute relevance if they could develop as individuals and pursue their interests. We suggest that giving students opportunities to set individual priorities in their studies could hence support their relevance assessments. As low relevance assessments seem to be connected to students’ motivational problems, students might profit from motivational support, as well.
Keywords: Education; Mathematics (miscellaneous)

Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. - Winkler, Michael (Calculus of Variations and Partial Differential Equations)

Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. - Winkler, Michael (Advanced Nonlinear Studies)

Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in a nutrient-poor environments. - Winkler, Michael (Nonlinear Analysis: Real World Applications)