FusionSystems · Simple fusion systems and linking systems
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2016-08-03 → 2018-08-02
- EU contribution
- €195,455
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Simple fusion systems and linking systems
"The Classification of the Finite Simple Groups (CFSG) constitutes one of the greatest acheivements of twentieth-century mathematics. It describes the fundamental building blocks of finite symmetry, in the same way the discovery and classification of the elements describe the fundamental building blocks of molecules. The original proof of the CFSG spans somewhere between ten and fifteen thousand pages of journal articles, with contributions by over one hundred mathematicians. The proof of the CFSG is considered to be the longest proof in history of a single mathematical statement. It is thus of fundamental mathematical and philosophical interest to investigate ways of simplifying it. The theory of fusion systems provides one context for such a simplification. A fusion system is considered a snapshot of a finite group at a prime number p, although certain ""exotic"" fusion systems provide snapshots of finite simple groups that do not actually exist. Fusion systems were first considered in the study of representations of groups, namely the study of the ways in which groups can realize their symmetries in nature. They were later found to be important in certain segments of topology, or ""rubber-sheet geometry"". More recently, and because certain technical difficulties in the CFSG do not arise in fusion systems, the study of fusion systems has been taken up as an avenue to simplify the CFSG. The Dichotomy Theorem says that fusion systems can be partitioned into those of component type and those of characteristic p-type. The objective of the FusionSystems project was to make significant contributions to the classification of simple fusion systems of component type at the prime two, as well as the classification of simple fusion systems of characteristic p-type for an arbitrary prime p. The action will result in the publication of three peer-reviewed articles in international journals and/or conference proceedings, including public access to final-draft post-refereed versions of the articles. In addition, the action results in thirteen conference and seminar talks communicating the results of the action and of closely related research, one talk to the general public, and the support of four international visitors. "
Data: CORDIS, © European Union
Project objective
The classification of finite simple groups is often regarded as one of the major mathematical achievements of the 20th century. Its importance lies not only in the fundamental result, but also in the methodology and conceptual framework developed for its proof. Most intriguingly, it turned out that the structure of a finite group is closely connected to the structure of the p-local subgroups, i.e. the normalizers of non-trivial p-subgroups, for a suitably chosen prime p. Of particular importance is the prime 2. Even though the proof of the classification of finite simple groups is insightful in its major conceptual approach, it is extremely long and difficult in its details. Thus, it would be of great interest to obtain a simplified proof. Moreover, to gain the maximum benefit from the methods of the proof of the classification, it is highly desirable to work in a more general context which in particular allows also for applications in the modular representation theory of finite groups. Saturated fusion systems provide a conceptual framework for this and connect to important questions in homotopy theory. A program to find a new and better proof of the classification of finite simple groups through a classification of simple fusion systems at the prime 2 has been recently outlined by Aschbacher. Two parts of our proposal concern classification problems for fusion systems. Their significance lies not only in completing important cases in Aschbacher's program, but also in giving new insight into the relative abundance of exotic examples, i.e. fusion systems not induced by any finite group. In the third part we attack a major problem in the algebraic theory of fusion systems by defining an analogue in fusion systems of centralizers of subgroups of finite groups. This will simultaneously facilitate a classification of fusion systems in the spirit of Aschbacher's program, and lead to a combinatorial understanding of maps between classifying spaces of fusion systems.
Original text from CORDIS.
Participants
- THE UNIVERSITY COURT OF THE UNIVERSITY OF ABERDEEN · AberdeenCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union