## Periodic Reporting for period 1 - FusionSystems (Simple fusion systems and linking systems)

Reporting period: 2016-08-03 to 2018-08-02

"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.

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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.

"

The FusionSystems project began with conference talk at the University of British Columbia discussing, in part, some of the background of the project in the context of automorphisms of linking systems and its connection with the construction of centralizers in fusion systems. Initial effort in Aberdeen was dedicated to the preparation and deliverance of a talk to the general public on the subject of the research. Work then commenced on an investigation of the automorphisms and extensions of the Benson-Solomon fusion systems, the only currently known family of exotic systems at the prime two. The first publication, giving a complete description of such automorphisms and extensions, was widely disseminated and submitted for publication in January 2017, and was communicated formally in the Bristol-Leicester-Oxford Colloquium at City University, London. Also in December 2016-January 2017, the project also made progress on the problem of constructing centralizers of fusion subsystems, resulting in the development of an obstruction theory for rigid actions on linking systems. The research surrounding this portion of the project remains in progress. Effort subsequently turned to a solution of the Benson-Solomon component problem in the subintrinsic case, the background of which was communicated in the Algebra Seminar at the University of Birmingham near the beginning of the effort. A manuscript containing the solution is now completed, and is expected to be prepared for dissemination and submission by the end of November 2017. Concurrently, the project finished the determination of the complete list of the centric and radical subgroups and of their outer automorphism groups in these systems, a result cited in the solution to the Benson-Solomon component problem. This action resulted in a third completed manuscript, which is expected to be disseminated and submitted for publication by the end of November 2017. Further formal communication of the results of the project will continue in a deliberate fashion for one to two years.

Determining the automorphisms and extensions of fusion systems, the subject of the first publication, is known to be a delicate endeavour even in the case of the fusion system of a finite group. Several recent important research papers are dedicated to the determination of automorphisms and extensions under this condition. The FusionSystems project's article on the Benson-Solomon systems is, to its knowledge, the first to explicitly carry out such a description for an exotic fusion system. It should serve as a model for other researchers working with exotic systems, especially in the context of classification problems.

A classification of fusion systems with involution centralizer having a Benson-Solomon component, the subject of the second manuscript, is broadly awaited by researchers in fusion systems. Because a Benson-Solomon system is not the fusion system of any finite group, any simple fusion system having such an involution centralizer would necessarily be a new exotic system at the prime 2, which has been sought by many authors. The FusionSystems project's results provide additional evidence there is no such exotic system, strengthening the case for a conjecture of R. Solomon. The results also form part of one of the major steps in the classification of simple fusion systems of component type, namely the proof of an analogue of a theorem of J. Walter within the CFSG.

The description of the centric and radical subgroups in the Benson-Solomon systems builds on previous research on these systems. It allowed FusionSystems in the third manuscript to compute the number of simple representations that a Benson-Solomon finite simple group would have if it existed, thus answering a question posed by one of the leading figures in the representation theory of finite groups.

A classification of fusion systems with involution centralizer having a Benson-Solomon component, the subject of the second manuscript, is broadly awaited by researchers in fusion systems. Because a Benson-Solomon system is not the fusion system of any finite group, any simple fusion system having such an involution centralizer would necessarily be a new exotic system at the prime 2, which has been sought by many authors. The FusionSystems project's results provide additional evidence there is no such exotic system, strengthening the case for a conjecture of R. Solomon. The results also form part of one of the major steps in the classification of simple fusion systems of component type, namely the proof of an analogue of a theorem of J. Walter within the CFSG.

The description of the centric and radical subgroups in the Benson-Solomon systems builds on previous research on these systems. It allowed FusionSystems in the third manuscript to compute the number of simple representations that a Benson-Solomon finite simple group would have if it existed, thus answering a question posed by one of the leading figures in the representation theory of finite groups.