Felix Klein benannte vor mehr als hundert Jahren die doppelte Diskontinuität in der Mathematiklehrerausbildung – ein Problem, das bis heute ungelöst ist. Dieser Beitrag zeichnet Kleins Werk als Reformer und Lehrerbildner nach und skizziert die Forschungsdiskussion, die seine Diagnose ausgelöst hat. Dann stellt er das Framework der mathematischen Orientierung vor: ein philosophisch begründetes Begriffsgerüst, das präziser benennt, was Dozierende von Fachveranstaltungen im Lehramtsstudium anstreben können. Eine Relektüre von Kleins Vorlesungsreihe mit dieser Terminologie führt zu einer aufschlussreichen Einsicht: Die Vorlesungsreihe leistet bemerkenswert viel – und macht zugleich sichtbar, wo die Grenzen eines einzelnen Formats liegen: Das Angestrebte entfaltet sich nicht in einer Vorlesungsreihe allein.
Eine zentraler Aspekt im Werk von Felix Klein war die Entwicklung der Theorie der Modulformen. Der Vortrag im Rahmen der „Topic Days Felix Klein“ gibt eine kurze, informelle Einführung in Modulformen und beschreibt einige Anwendungen.
The escaping set of an entire function consists of the points in the complex plane that tend to infinity under iteration. This set plays a central role in the dynamics of transcendental entire functions. The goal of this survey is to explain this role, to summarise some of the main results in the area, and to identify a number of open questions.
Understanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics.
We review the relationship between discrete groups of symmetries of Euclidean three-space, constructions in algebraic geometry around Kleinian singularities including versions of Hilbert and Quot schemes, and their relationship to finite-dimensional and affine Lie algebras via the McKay correspondence. We focus on combinatorial aspects, such as the enumeration of certain types of partition-like objects, reviewing in particular a recently developed root-of-unity-substitution calculus. While the most complete results are in type A , we also develop aspects of the theory in type D , and end with some questions about the exceptional type E cases.
The discriminant of a number field is an important invariant measuring its size and ramification behavior. We will first recall classical results by Minkowski, Odlyzko, Serre, and Golod and Šafarevič about discriminants. We then discuss the relevance of number fields with small root discriminants to cryptography.
