A spectral result for massive electromagnetic Dirac Hamiltonians
Monday, 2.12.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
From the physical perspective of relativistic wave mechanics, Dirac operators on Riemannian manifolds can be interpreted as infinitesimal generators of time translation, i.e. as Hamiltonians. Most mathematical research on Dirac operators focuses on what amounts to the study of free, massless fields. In this talk however, we will consider a Dirac Hamiltonian that is coupled to an electromagnetic field and a spatially non-constant mass term. After motivating and setting up the necessary notions, we will proceed to show that, given a suitably behaved "potential well", the spectrum of such a Dirac Hamiltonian must be discrete. The result being presented is a generalization of a theorem by N. Charalambous and N. Große in 2023.
Model theory of difference fields with an additive character on the fixed field
Tuesday, 3.12.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field working in (a mild version of) continuous logic. Building on results by Hrushovski we can recover it as the characteristic 0 asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+ and obtain that ACFA+ is simple as well as a description of the connected component of the Kim-Pillay group. If time permits we present some results on higher amalgamation.\n
Apps, Projekte und KI-Tools für den digitalen Mathematikunterricht
Tuesday, 3.12.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Im Zeitalter der Künstlichen Intelligenz sollten digitale Medien im Mathematikunterricht nicht nur zur Reproduktion von Wissen, sondern vor allem zur Förderung und Stärkung von Kompetenzen eingesetzt werden. Dazu ist die Verknüpfung von mobilen Endgeräten mit individuellen, forschenden, kreativen und projektartigen Arbeitsaufträgen unerlässlich. Im Vortrag werden zunächst praktische Hinweise zum effektiven Einsatz digitaler Medien im Mathematikunterricht gegeben und einzelne Apps vorgestellt, die den Unterricht bereichern können. Anschließend wird auf die Bedeutung einer projektorientierten Lernkultur und den gezielten Einsatz von künstlicher Intelligenz eingegangen. Langfristiges Ziel eines digital angereicherten Mathematikunterrichts ist die\nEtablierung einer neuen Lern- und Prüfungskultur, die es den Schülerinnen und Schülern ermöglicht, ihre Fähigkeiten in einer dynamischen und vernetzten Welt optimal zu entfalten.
TBA
Thursday, 5.12.24, 10:15-11:15, Raum 414, Ernst-Zermelo-Str. 1
On a generalization of indiscernible sequences
Tuesday, 10.12.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Guingona and Hill introduced and studied a new hierarchy of dividing lines for first-order structures, denoted by (NCK)K , where K ranges in the theorie of ultrahomogeneous omega-categorical Ramsey structures. In a subsequent paper, Guigonna, Hill and Scow give a characterisation in terms of (generalised) K-indiscernible sequences.\nIn this talk, I will present a joint work with Nadav Meir and Aris Papadopoulos, in which we develop around these notions of K-indiscernibility. In particular, we will answer (negatively) a question posed by Guingona and Hill about the linearity of the NC_K hierarchy. As an application, we will also see that the ordered random graph admits a unique proper Ramsey reduct, namely the linear order.
Sequential topological complexities and sectional categories of subgroup inclusions
Thursday, 12.12.24, 10:00-11:00, Raum 414, Ernst-Zermelo-Str. 1
The sequential topological complexities (TCs) of a space are integer-valued homotopy invariants that are motivated by the motion planning problem from robotics and express the complexity of motion planning if the robots are supposed to make predetermined intermediate stops along their ways. After outlining their definitions, I will discuss the sequential TCs of aspherical spaces in the first part of my talk and describe how they can be investigated by purely algebraic means. We will also take a look at a generalization of this algebraic setting, namely sectional categories of subgroup inclusions. In the second part of my talk, I will present a general lower bound on their values and derive consequences for sequential TCs and parametrized topological complexities of epimorphisms. We will investigate the methodology of the proof of this lower bound, in which all key steps are carried out using elementary homological algebra. This is joint work with Arturo Espinosa Baro, Michael Farber and John Oprea.
Moduli spaces of 3-manifolds with boundary are finite
Monday, 16.12.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In joint work with Rachael Boyd and Corey Bregman we study the classifying\nspace \(B\bmathrm{Diff}(M)\) of the diffeomorphism group of a connected, compact, orientable\n3-manifold \(M\). I will recall the construction of this \(B \bmathrm{Diff}(M)\), also known as\nthe "moduli space of \(M\)", and explain how it parametrises smooth families of\nmanifolds diffeomorphic to \(M\).\n\nMilnor's prime decomposition and Thurston's geometrisation conjecture allow us\nto cut \(M\) into "geometric pieces", which each admit complete, locally\nhomogeneous Riemannian metric. For such geometric manifolds \((N,g)\) recent work\nusing Ricci flow shows that a certain space of metrics is contractible and thus\nthat the generalised Smale conjecture (often) holds: the diffeomorphism group\n\(\bmathrm{Diff}(N)\) is homotopy equivalent to the isometry group \(\bmathrm{Isom}(N,g)\).\n\nThe purpose of this talk is to explain a technique for computing the moduli\nspace \(B\bmathrm{Diff}(M)\) in terms of the moduli spaces of the pieces. We use this to\nprove that if \(M\) has non-empty boundary, then \(B \bmathrm{Diff}(M\btext{ rel boundary})\) has the\nhomotopy type of a finite CW complex, as was conjectured by Kontsevich.\n\n\n
The 1d inelastic Boltzmann equation for moderately hard potentials
Tuesday, 17.12.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Inelastic interaction of granular matter is a common phenomenon in natural processes. A mathematical description of such behaviour is given by a modification of the Boltzmann equation, where the dissipation of kinetic energy during collisions characterises the inelasticity at the particle level.\n\nIn this talk, we consider the occurrence of self-similar behaviour in the long-time limit for the one-dimensional inelastic Boltzmann equation. More precisely, we prove that self-similar profiles are unique in the regime of moderately hard potentials. The proof relies on a perturbation argument from the Maxwell model, together with a spectral gap for the corresponding linearised operator.
Ein roter Faden durch die Stochastik
Tuesday, 17.12.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Wenn man darauf verzichtet, Wahrscheinlichkeiten als objektiv - unabhängig vom Menschen - existierende Größen zu begreifen, sie stattdessen konsequent als unsichere und revidierbare Festlegungen deutet, die als Modelle dadurch entstehen, dass aus Erfahrungen Erwartungen werden, verschwinden seit Jahrzehnten ungelöste didaktische Probleme.\n\nUnter dem Hut eines neu entwickelten Wahrscheinlichkeits-begriffs wachsen der klassische (LAPLACE), der frequentistische (V. MISES), der subjektivistische (BAYES) und der axiomatische (KOLMOGOROFF) Wahrscheinlichkeitsbegriff zu einer Einheit zusammen. Mit seiner Hilfe entspinnt sich ein roter Faden durch die Stochastik, der Wahrscheinlichkeitsrechnung mit beschreibender und beurteilender Statistik von der Grundschule bis zum Abitur … und auch danach … zu einer Einheit verschmelzen lässt.\n\nMöglicherweise wird der in diesem Experimentalvortrag, vorgenommene Perspektivwechsel Ihr stochastisches Weltbild erweitern, vielleicht sogar konstruktiv erschüttern, wenigstens ein bisschen!
Pattern retrieval in the Hopfield model
Friday, 20.12.24, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
Hopfield proposed in 1982 as a simple model capable to store and\nsuccessively retrieve a number of high dimensional patterns, which\nrepresents nowadays a cornerstone of artificially intelligence (he was\nawarded the Nobel prize in Physics 2024 for this work). I will present some\nold and new mathematical results about pattern retrieval in the Hopfield\nmodel along with some open problems.\n