Dunkl operators are commuting differential-reflection operators on $\mathbb R^n$ which generalize the usual partial derivatives by additional reflection terms. They are associated with some underlying finite reflection group and its root system. Dunkl operators allow a rich harmonic analysis, including a generalization of the Fourier transform with many similar properties. In this talk, we give a basic introduction to Dunkl theory and explain some uncertainty principles related to them.

Large random matrices tend to exhibit universal fluctuations. Beyond the well-known Wigner-Dyson and Tracy-Widom eigenvalue distributions, we overview other universality results for Hermitian and non-Hermitian matrices. We discuss the emergence of normal distribution involving eigenvectors, especially the random matrix version of quantum unique ergodicity. We also explain why results on non-Hermitian random matrices are much harder than their Hermitian counterparts and highlight our new methods to tackle them.

Valued difference fields are fields equipped with a valuation and a compatible automorphism. I will talk about joint work with Jan Dobrowolski and Francesco Gallinaro on the ``universal domains`` for this class, that is, those where the automorphism is ``generic``. We prove that such structures admit an equivariant section of the valuation, show that the automorphism is also generic as an automorphism of fields, characterise which amalgamation problems in the category of ac-valued difference fields of residue characteristic 0 admit solutions, and deduce that valued difference fields of residue characteristic 0 satisfy a certain notion of combinatorial tameness, namely they are NTP2 in the sense of positive logic.

In this talk I would like to discuss some numerical algorithms for mean curvature flow of two-dimensional surfaces through singularities. It is well known that geometric surfaces flows often undergo topological changes. For mean curvature flow this has been heavily investigated by, e.g., Hamilton, Huisken and Sinestrari, and Brendle and Huisken. Based on these works we will discuss an algorithm for mean curvature flow with numerical surgery, a process which automatically handles these topological changes, by cutting away some surface-portions and closing the remaining parts by spherical caps. In view of this, the following ethical statement seems appropriate: \emph{None} of the test subjects were permanently harmed during the experiments.

Die Art und Weise, wie Lehrkräfte Schüler:innen in Diskussionen anleiten, wirkt sich auf das Lernen der Schüler:innen aus (Webb et al., 2014). Dies setzt eine angemessene Förderung durch Lehrkräfte voraus (Franke et al., 2015). Damit Mathematiklehrkräfte Gespräche zur Förderung des Verständnisses der Schüler:innen führen können, benötigen sie ihrerseits die dafür notwendige Expertise (Prediger, 2019). In den letzten Jahren wurden simulationsbasierte Aktivitäten verwendet, die als Script-Writing Aktivitäten (SWA) bezeichnet werden (Zazkis et al., 2013; Zazkis & Marmur, 2018; Shure et al., 2022), um (angehenden) Lehrkräften fiktive Situationen zu bieten und dabei ihre Expertise bei der Gesprächsführung untersuchen zu können. Konkret werden fiktive Dialoge zwischen Schüler:innen, die ihre Lösungen für Aufgaben diskutieren, eingebettet und Lehrkräfte werden gebeten, die Dialoge zu vervollständigen. Auf diese Weise kann untersucht werden, wie sie das Verständnis bei den Schüler:innen fördern würden. Im Vortrag wird diese Methode zur Untersuchung der Expertise bezüglich Diskussionsführungsansätze von (angehenden) Lehrkräften dargestellt und anhand von Beispielen konkretisiert. Diskutiert wird, inwieweit SWA ein ökonomisches Mittel darstellen, um Gesprächsführungspraktiken und zugrundeliegende Orientierungen von Lehrkräften zu untersuchen.

Im Vortrag wird ein umfassender Überblick über die verschiedenen Formen des Ganztags gegeben, wobei die Beteiligung unterschiedlicher Träger, Institutionen und Akteur:innen die Komplexität des Themas verdeutlicht. Neben dem Unterricht existiert eine Vielzahl an außerunterrichtlichen Angeboten, deren pädagogische Settings näher beleuchtet werden. Dabei wird der Zusammenhang von Unterricht und Ganztagsangeboten differenziert betrachtet, etwa im Hinblick auf komplementäre versus kompensatorische Ansätze. Im Weiteren werden empirische Befunde zu den Wirkungen dieser Angebote vorgestellt. Ein Schwerpunkt der Darstellung liegt auf der Förderung fachlicher Kompetenzen, insbesondere im Fach Mathematik. Hierbei werden gezielt Hausaufgabenbetreuung, Lernzeiten und spezielle Matheangebote zur Kompetenzförderung und Potenzialentwicklung thematisiert.

Computer-assisted assessment tasks enable elementary school students to benefit from using the digital geometry environment for the exploration and construction, and also automatically assess the characteristics of students’ work in school geometry topics. To achieve this we designed assessment tasks in the STEP system on triangle altitudes. Using these assessment tasks, we explored what characteristics of elementary school students' work can be noticed from their submission of digital formative assessment tasks in geometry? And which tasks are better than others in diagnosing each work characteristic? Using Psychometric methods, we discovered distinct work patterns that either led, hindered, or did not affect students’ success. This leads us to conclude that it is possible to digitally assess a sequence of rich, student-centered, authentic geometry construction tasks to assess not only the level of proficiency but also students’ work characteristics.

https://wwwold.mathematik.tu-dortmund.de/~dplauman//squares2025.html

The new wave of artificial intelligence is impacting industry, public life, and the sciences in an unprecedented manner. However, one current major drawback is the lack of reliability as well as the enormous energy consumption of AI systems.
In this talk we will take a mathematical viewpoint towards this problem, showing the power of such approaches in the field of AI. We will first provide an introduction into this vibrant research area. We will then discuss key mathematical research directions toward reliability of AI and survey some results on, in particular, performance guarantees as well as explainability. This is followed by a discussion of fundamental limitations also in terms of sustainability. Our mathematical viewpoint will lead us naturally to the necessity of novel (analog) hardware such as neuromorphic computing and the related model of spiking neural networks. We will finish with some very recent mathematical results for spiking neural networks.

In this talk we show how to apply the framework developed by Sam and Snowden to study structural properties (eg. bound on rank and order of torsion) of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. In particular, we focus on magnitude homology for graphs, which were introduced by Hepworth and Willerton. The talk is organised as follows; we start with a short introduction to modules over categories and to the theory of Groebner categories. Then, we introduce magnitude homology and see some examples. Finally, we will see how to use the theory of Groebner categories to obtain information on magnitude (co)homology. Time permitting I will also describe how to extend some of the results presented to other graph homology theories. This talk is based on joint work with Luigi Caputi and Eric Ramos.

Implicit-explicit (IMEX) stepping is a popular technique in flow simulation, as it can improve computational efficiency and simplify implementation, while still maintaining good numerical stability. For standard, homogenous fluid models, IMEX usually means keeping viscous terms implicit and treating convection (semi-)explicitly. Yet, for fluids obeying more complex rheologies or balance equations, it can be attractive to treat also other terms, even diffusive ones, in an explicit or semi-implicit manner. An important advantage is avoiding certain couplings that would otherwise arise in implicit discretisations. In this talk, we will see different IMEX formulations for non-homogeneous flow problems and discuss important implications of those approaches. We will see that while some variants are unconditionally stable, others can suffer from numerical instabilities. Advantages and trade-offs will be discussed for different models, such as viscoelastic and multiphase flows.

Der Einsatz von Künstlicher Intelligenz (KI) im Mathematikunterricht der Primarstufe eröffnet vielversprechende Möglichkeiten zur Unterstützung individueller Lernprozesse. KI-gestützte Apps bieten Potentiale für die Unterstützung und Entlastung von Lehrkräften, indem sie beispielsweise bei der Diagnostik unterstützen, Aufgaben passend auswählen, individuell anpassen, und Schüler: innen beim mathematischen Problemlösen mit lernförderlichem Feedback begleiten. Zugleich ergeben sich jedoch erhebliche Herausforderungen, insbesondere in Bezug auf ethische Fragestellungen, Datenschutz und die Sicherstellung der fachdidaktischen Qualität KI-generierter Rückmeldungen und Interaktionen. Dieser Vortrag reflektiert den aktuellen Forschungsstand zum Einsatz von KI in Anwendungen für den Mathematikunterricht der Grundschule und präsentiert erste Erfahrungen mit selbstentwickelten Prototypen die im Rahmen fachdidaktischer Entwicklungsforschung weiter untersucht und optimiert werden sollen. Anhand konkreter Beispiele werden Potenziale sowie Grenzen und notwendige Rahmenbedingungen für den erfolgreichen Einsatz und die Ausgestaltung von KI-gestützten Apps im Mathematikunterricht diskutiert. Ein besonderer Fokus liegt dabei auf der Frage, wie eine fachdidaktisch sinn- volle und verantwortungsvolle Integration von KI gestaltet werden kann.

Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic, and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability. In particular, I will explain its crucial interplay with the heat flow and its link to the curvature-dimension condition formulated in functional-analytic terms by Bakry-Émery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions.

In recent years, data science, machine learning, and artificial intelligence have emerged both as entirely new majors and as electives within many mathematics, physics, or computer science curricula. While sometimes perceived as competitors to traditional areas of mathematics, the core of many data science or machine learning methods is of course classical linear algebra, analysis, probability theory, and, especially in the context of neural networks, functional analysis. In my talk I will outline the latter connection. In particular, I will show that Cybenko's 1990s theorem on the so-called expressivity of neural networks fits perfectly into a classical functional analysis course---with the advantage that all the prerequisites for a full proof will be available.

Are there really primitive tribes whose system of counting goes: 'One, Two, Many,. . .' indicating that from three on it's more or less a blur?
Maybe we modern humans are such a tribe. Despite the sophistication we see in ourselves compared with our less advanced ancestors from times long past, it's surprising how little progress we've made in addressing some basic problems in 3D or beyond, or when solving seemingly simple equations in ≥ 3 variables. While our accomplishments include astonishing mastery of some ABCs, such as Air (flight, weather prediction), Biology (medical breakthroughs, DNA) and Communications (phone, video, email), we often struggle to get past 1, 2, 3, in other domains. Or sometimes even to get to 3.
We'll survey a dozen fun topics in shapes and numbers and patterns whose basics and generalisations can be explored with little mathematical background, and which speedily lead to 'what if?' questions ranging from easy to tricky to 'we just don't know.' Fruit, cakes, doughnuts, bagels, coins, boxes, cubes, primes, squares, and sums involving powers will all make appearances. Occasionally, we will stray into deeper waters and touch on more sophisticated topics.
(At the end we mention some advanced forays into modern maths.)

We discuss the algebraic geometry of low rank symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories.

In the Middle School Mathematics and the Institutional Settting of Teaching project (2007 – 2015), colleagues and I partnered with four, large school districts who were engaged in ambitious instructional improvement efforts in middle-grades mathematics (youth ages 12 – 14). Our intent was to generate an empircially-grounded theory of action for instructional improvement at scale. In this presentation, I will share about the background of the project, the various lines of inquiry and key findings, and how those findings informed a theory of action. In addition, I will share key learnings about how our team organized coherent and complementary lines of inquiry, and developed measures to assess critical aspects of improving instruction at scale.

We study ordinary differential equations defined over Hardy fields using polynomials and real powers. Building on work of Aschenbrenner, van den Dries, and van der Hoeven, we show that these equations have Hardy field solutions if and only if they have formal solutions in the field of LE-transseries. This is done via a suitable algebraic/model-theoretic framework, which shows that solvability for a family of these equations depends ``tamely`` on real parameters. This framework also allows us to prove a relative decidability result.

Spätestens seit der Corona-Pandemie stehen digitale Weiterentwicklungen der Hochschullehre verstärkt im Fokus: Beim Blended Learning werden digitale und Präsenzformate sinnvoll miteinander kombiniert. Im Inverted Classroom dient das Selbststudium der Vorbereitung, während die Präsenzzeit für gemeinsames Arbeiten genutzt wird. Hybride Formate ermöglichen es Studierenden, entweder online oder vor Ort an synchronen Sitzungen teilzunehmen. Durch die Integration digitaler Plattformen in diese Methoden können auch Personen außerhalb der Hochschule Zugang zu den Lehrveranstaltungen erhalten. In diesem Vortrag wird das Lehrkonzept eines hybriden Inverted Classrooms in einer Mathematikveranstaltung für das Grundschullehramt vorgestellt und sowohl fachdidaktisch als auch lernpsychologisch motiviert. Die Verknüpfung von Lehre und Wissenschaftskommunikation im Social Web wird zur Diskussion gestellt.

In this talk, we present a regularity result for a degenerate elliptic PDE. We prove the continuity of the stress field for a solution of a minimization problem with linear growth arising from Hencky plasticity. The proof is based on an approximation argument, a Sobolev estimate, and a maximum principle. Finally, we give an application to the regularity of the level sets of the solution.

Abstract: In dimension 3 and higher, it is known that minimal resolutions of singularities of a complex projective variety need not be unique. Typically, there arise small modifications which toggle back and forth between different minimal models of the same variety. This framework is best understood for Calabi—Yau varieties, whose minimal models are connected by finite sequences of so-called flops. Some finite sequences of flops loop, and thereby define non-trivial automorphisms on one minimal model. Up to this automorphism group, it is however conjectured that any given Calabi—Yau variety has finitely many minimal models. As the class of Calabi-Yau varieties is closed under taking quotients by finite group actions in a natural way, it is natural to expect finiteness properties of minimal models to descend under finite quotients. This talk presents an equivariant descent result in that spirit. We will highlight the role of convex geometry, in particular of self-dual homogeneous cones and hyperbolic reflections, in that result.

Algebraisches Denken ist in der Grundschule untrennbar mit der Leitidee Muster, Strukturen und funktionaler Zusammenhang verwoben, die ja bekanntlich einen wesentlichen Kern des Mathematikunterrichts ausmacht und alle anderen Leitideen durchzieht. Algebraisches Denken bedeutet, Muster zu erkunden und die ihnen zugrunde liegenden Strukturen, d. h. mathematischen Eigenschaften und Relationen, zu verstehen. Im Vortrag wird zunächst eine Charakterisierung des algebraischen Denkens und eine damit verbundene Differenzierung der Begriffe Muster und Strukturen grundgelegt. Anschließend werden Einblicke in Studien vorgestellt, in der Verallgemeinerungsprozesse mit Fokus auf Strukturen in den arithmetischen Themenfeldern Operationen und Funktionen aus epistemologischer Perspektive betrachtet werden.

Beweisassistenten werden mehr und mehr zu einem hilfreichen Tool für Mathematiker:innen und Informatiker:innen. Wir werden die Stunde nutzen, um die Grundlagen des Beweisassistenten Lean kennen zu lernen. Wer möchte, kann sich Lean gerne unter https://lean-lang.org/download/ installieren und den Vortrag interaktiv verfolgen. Den verwendeten Code werde ich einen Tag vor dem Vortrag unter https://github.com/StamesJames/lean-introduction-code veröffentlichen.

Nach meinem Mathematik-Studium an der TU Dortmund von 2000 bis 2005 und weiteren Stationen an der Universität arbeite ich seit 2012 in der IT-Beratung. Meine Kollegen und ich beantworten dort Kundenfragen der Form 'Sollen wir die Software einkaufen oder selbst programmieren? Was sollen wir programmieren? Wie können wir das Programmierte testen?'. Die im Studium gelernten Ergebnisse wie der Satz von Stokes, der Hilbertsche Nullstellensatz oder der Simplex-Algorithmus helfen uns hierbei nicht, sehr wohl aber die gelernten Fähigkeiten wie das Verstehen schwieriger Texte, das Argumentieren und das technische Schreiben. Im Vortrag möchte ich meinen Werdegang, die in unseren Projekten eingesetzten mathematischen Fähigkeiten und die Firma Consileon vorstellen.

We consider the dimension reduction and homogenization of a Stokes-system with no-slip boundary conditions posed in thin periodically perforated layers whose thickness and hetero- geneity scales are governed by a small parameter ε > 0. The periodicity of the microstructure is of order ε, while the thickness of the layer is given by εα, with α ∈ (0, ∞). In this talk, we aim to prove that the limit solution as ε → 0 satisfies a two-pressure Stokes-system. In addition, we develop the analytical tools necessary for this process. Specifically, we extend the concept of two-scale convergence to such thin, heterogeneous domains and establish a priori estimates for the microscopic solution that are uniform with respect to both the periodicity and the thickness of the layer.

Motivated by engineering and Photonics research on resonators in structured deterministic or stochastic environments, the talk plans to introduce rigorous randomizations of boundary conditions for acoustic wave equations in Lipschitz domains. We give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra. Based on these results, examples of mathematically convenient randomizations are constructed in terms of eigenfunctions of Laplace-Beltrami operators on boundaries. It will be shown that, for these special randomizations, the resolvent compactness is connected with the Weyl law for Laplace-Beltrami eigenvalues. The analytical techniques of Dirichlet-to-Neumann (DtN) maps and generalized Krein resolvent formulae are used substantially, and, as a by-product, an approach to DtN maps via abstract Weyl functions of Derkach-Malamud is studied. The talk is based on the paper

Machine Learning has emerged as one of the most transformative forces in science and technology. Beneath its powerful algorithms lie mathematical foundations deeply rooted in classical disciplines such as Applied Mathematics and Systems Control. This lecture adopts a mathematician’s perspective to examine why Machine Learning works so effectively and how its data-driven paradigms can be rigorously integrated into traditional analytical frameworks.
We will revisit the historical and conceptual links between Machine Learning and Systems Control - also known as Cybernetics - a field shaped by the pioneering ideas of Ampère and Wiener. Their parallel evolution reveals a deep mathematical unity and highlights the power of mathematics to model complex systems and drive innovation.
This dual perspective is mutually enriching. On one hand, Machine Learning raises fundamental mathematical questions that challenge and inspire the mathematical community. On the other, it offers opportunities to expand the scope of classical applied mathematics by developing hybrid methodologies that integrate data-driven insights.
The lecture will conclude by outlining promising directions for future research at the intersection of Machine Learning, Applied Mathematics, and Control Theory.

Under the name practical rationality of mathematics teaching, my colleagues and I have sought to describe elements of tacit knowledge of mathematics teaching that regulate teachers’ action and serve as sources of justification for action in the mathematics class-room. I will describe instruments developed to do this research and results that connect constructs of practical rationality (e.g., norms and obligations) with measures of teacher knowledge.

On the full, flat space, Egidi-Veselic and Wang-Wang-Zhang-Zhang established that given a measurable set $\omega$, the following three properties are equivalent: (i) thickness: $\omega$ has equidistributed measure in every ball at some scale, (ii) a spectral inequality bounding the $L^2$ norm of functions with compact spectral support by their $L^2$ norm on $\omega$, and (iii) observability of the heat semigroup from $\omega$. The aim of the talk will be to discuss how these properties interact in the manifold setting and how to generalize the equivalence to non-compact manifolds, under relevant bounds on curvature. I will focus on ongoing work with Alix Deleporte and Jean Lagacé, in which we aim to prove that (i) implies (ii) on manifolds with bounded curvature, with a constant that depends only on curvature bounds rather than the metric itself. The proof crucially relies on elliptic estimates by Logunov and Malinnikova, combined with tools from geometric analysis.

Totally positive functions in sampling theory and time-frequency analysis:
Totally positive functions play an important role in approximation theory, statistics, and nowadays also in time-frequency analysis. In this talk I will present sampling theorems for spaces that are spanned by the integer shifts of a totally positive function. These are often used as a substitute for the bandlimited functions. A related problem are the spanning properties of time-frequency shifts. These questions go back to the mathematical foundations of quantum mechanics as formulated by von Neumann.

I will discuss the large-scale/ long-time behavior of a Brownian particle in a random, incompressible (divergence-free) vector field. If this vector field has correlations which decay sufficiently slowly, then the behavior of the particle is superdiffusive instead of diffusive. In particular, after time $t$, the particle will typically be a much larger distance than $\sqrt{t}$ from the origin. This phenomenon was explained in the physics literature in the late 1980s, using heuristic renormalization group arguments, as a divergence of effective diffusivity as we zoom out to large length scales. I will discuss recent mathematical innovations based on some analytic/PDE methods that allow us to make some of these renormalization group arguments rigorous.
​​​​​​​(This is joint work with A. Bou-Rabee and T. Kuusi.)

Ein konzeptuelles Verständnis des Stellenwertsystems bildet eine zentrale Grundlage für den Aufbau arithmetischer Kompetenzen. Besonders am Übergang von der Primar- zur Sekundarstufe ist es bedeutsam, da der Umgang mit erweiterten Zahlenräumen zunehmend abstrakt erfolgt. Dennoch zeigen viele Schülerinnen und Schüler am Ende der Grundschulzeit deutliche Schwierigkeiten im Stellenwertverständnis. Daher erscheint es notwendig, dieses Verständnis und damit verbundene Fehlermuster differenziert zu untersuchen: Welche Muster lassen sich identifizieren, und wie konsistent treten sie individuell auf? Wie hängt das Stellenwertverständnis mit anderen arithmetischen Kompetenzen wie Zahlverständnis, Zahlenfolgen oder schriftlicher Subtraktion zusammen? Der Vortrag stellt Ergebnisse eines Forschungsprojekts vor, das ein Erhebungsinstrument entwickelte und in mehreren Untersuchungen anwendete.

We consider a class of dynamical systems that are described by time and space dependent partial differential equations. This class fits perfectly the port-Hamiltonian framework. We cover the wave equation, Maxwell's equations, the Kirchhoff-Love plate model, piezo-electromagnetic systems and many more. Our goal is to characterize boundary conditions that make the systems passive (the energy of solutions decays). This is done by constructing a boundary triple for the underlying differential operator. As a by-product we develop the theory of quasi Gelfand triples, which enables us to regard L2 boundary conditions even though the ``natural`` boundary spaces are neither included nor covering L2.

Redundancy plays a role in the life of everybody. We repeat sentences (sometimes with a slight change of the wording) in order to make sure that a message is received correctly, we double-check that the door is locked and the oven switched off, and so on. The principle of redundancy is also incorporated in mathematical methods for signal transmission, in order to guarantee that a transmitted signal can be recognized correctly, even if errors occur during transmission. Such methods are often based on so-called frames, which are basic building blocks that can be used to represent arbitrary elements in the underlying (Hilbert) space. Intuitively, a frame can be considered as a collection of vectors that contain ``more elements than a basis``. The talk will give an introduction to frame theory and discuss a class of frames, for which surprisingly sparse subfamilies keep the frame property, due to a very particular ``distribution`` of the redundancy.