Splendid and perverse equivalences

Inspired by the works of Rickard on splendid equivalences and of Chuang and Rouquier on perverse equivalences, we are here interested in the combination of both, a splendid perverse equivalence. This is naturally the right framework to understand the relations between global and local perverse equivalences between blocks of finite groups, as a splendid equivalence induces local derived equivalences via the Brauer functor. We prove that under certain conditions, we have an equivalence between a perverse equivalence between the homotopy category of p-permutation modules and local derived perverse equivalences, in the case of abelian defect group.Comment: 13 pages, 4 figure

Model Theory and Groups

The aim of the workshop was to discuss the connections between model theory and group theory. Main topics have been the interaction between geometric group theory and model theory, the study of the asymptotic behaviour of geometric properties on groups, and the model theoretic investigations of groups of finite Morley rank around the Cherlin-Zilber Conjecture

The McKay conjecture and Brauer's induction theorem

Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p'$-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p'$-degree of $G$ and $N_G (P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Brou\'e abelian defect group conjecture. We verify the proposed conjectures in several special cases.Comment: Minor changes made throughout the pape

A cohomological approach to the classification of pp-groups

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by \F_pG. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$

Connecting Mathematics and Mathematics Education

This open access book features a selection of articles written by Erich Ch. Wittmann between 1984 to 2019, which shows how the “design science conception” has been continuously developed over a number of decades. The articles not only describe this conception in general terms, but also demonstrate various substantial learning environments that serve as typical examples. In terms of teacher education, the book provides clear information on how to combine (well-understood) mathematics and methods courses to benefit of teachers. The role of mathematics in mathematics education is often explicitly and implicitly reduced to the delivery of subject matter that then has to be selected and made palpable for students using methods imported from psychology, sociology, educational research and related disciplines. While these fields have made significant contributions to mathematics education in recent decades, it cannot be ignored that mathematics itself, if well understood, provides essential knowledge for teaching mathematics beyond the pure delivery of subject matter. For this purpose, mathematics has to be conceived of as an organism that is deeply rooted in elementary operations of the human mind, which can be seamlessly developed to higher and higher levels so that the full richness of problems of various degrees of difficulty, and different means of representation, problem-solving strategies, and forms of proof can be used in ways that are appropriate for the respective level. This view of mathematics is essential for designing learning environments and curricula, for conducting empirical studies on truly mathematical processes and also for implementing the findings of mathematics education in teacher education, where it is crucial to take systemic constraints into account

Connecting Mathematics and Mathematics Education

This open access book features a selection of articles written by Erich Ch. Wittmann between 1984 to 2019, which shows how the “design science conception” has been continuously developed over a number of decades. The articles not only describe this conception in general terms, but also demonstrate various substantial learning environments that serve as typical examples. In terms of teacher education, the book provides clear information on how to combine (well-understood) mathematics and methods courses to benefit of teachers. The role of mathematics in mathematics education is often explicitly and implicitly reduced to the delivery of subject matter that then has to be selected and made palpable for students using methods imported from psychology, sociology, educational research and related disciplines. While these fields have made significant contributions to mathematics education in recent decades, it cannot be ignored that mathematics itself, if well understood, provides essential knowledge for teaching mathematics beyond the pure delivery of subject matter. For this purpose, mathematics has to be conceived of as an organism that is deeply rooted in elementary operations of the human mind, which can be seamlessly developed to higher and higher levels so that the full richness of problems of various degrees of difficulty, and different means of representation, problem-solving strategies, and forms of proof can be used in ways that are appropriate for the respective level. This view of mathematics is essential for designing learning environments and curricula, for conducting empirical studies on truly mathematical processes and also for implementing the findings of mathematics education in teacher education, where it is crucial to take systemic constraints into account