# Algebra

## In 'Pure Mathematics'

Back to 'Mathematics Research'Algebra research at the University of Birmingham covers a wide range of topics across group and representation theory. These areas lie at the heart of algebra and give a mathematical abstraction for the study of symmetry. Although symmetry has been studied for millennia as represented in the art and architecture of of many different cultures, it was not until the work of Galois in the early 19th century on the symmetries of roots of polynomials that the definition of a symmetry group was introduced. Subsequently group theory and representation theory have found profound applications across the physical sciences, for example in the chemistry of molecular vibrations and in particle physics, and have a close relationship with geometry, number theory and combinatorics. The research of algebraists in Birmingham involves an interaction of algebraic, geometric and combinatorial methods, collaboration with mathematicians worldwide and is supported by the Royal Society and the European Union.

The research interests of the members of the group are given below.

## Professor Paul Flavell

**Head of School****Professor in Algebra**

Paul's main area of research is Group Theory, which is an area of Abstract Algebra. In very general terms, groups are used to measure the abstract notion of symmetry. As a consequence, they appear in very many areas of science as well as being fascinating objects of study from the pure mathematical point of view.

There are two themes to Paul's research. Firstly, the further development of the abstract theory of finite groups. Secondly, participation on the ongoing international project to produce a new and simplified proof of the Classification Theorem for the Finite Simple Groups.

In the abstract theory of finite groups, the theory of Automorphisms of Finite Groups presents many formidable challenges and opportunities to considerably extend existing theory. For example Paul's Hall-Higman-Shult type theorem for arbitrary finite groups is a generalization of classical theorems regarding representations of solvable groups to nonsolvable groups. As an application, Paul has obtained a local version of Thompson's Thesis, which relates the structure of the fixed point subgroup of an automorphism to that of the whole group.

Paul has developed a substantial theory concerning automorphisms. In particular, relating local structure to global structure. Further high points of this work to date are new proofs of the Solvable and Nonsolvable Signalizer Functor Theorems. These results are two of the pillars on which both the first generation and second generation Gorenstein-Lyons-Solomon proofs of the Classification Theorem of the Finite Simple Groups are built.

Another long standing interest is Generation Properties of Finite Groups. A highlight of this work is Paul's short and direct proof that a finite group is solvable if and only if every pair of its elements generate a solvable subgroups. This result had previously been obtained by Thompson as a corollary of his monumental classification of the Minimal Simple Groups.

## Professor Chris Parker

**Professor in Group Theory **

Chris's main research interests are in group theory. Recently his research has focussed on theorems designed to recognise simple groups from some fragment of their *p*-local subgroup structure. These theorems are used in the projects aimed at understanding the classification of the finite simple groups. For example work together with Stroth (Halle) applies the results to give a classification of the finite simplegroups of local characteristic *p* which have a simple group of Lie type containing a Sylow *p*-subgroup. Chris and his PhD students are working on problems in the theory of saturated p-fusion systems. Particularly theyare interested in discovering patterns in the appearance of exotic fusion systems for odd primes *p*.

## Professor Sergey Shpectorov

**Professor in Group Theory and Geometry**

Sergey's research interests are in algebra, geometry, and combinatorics. The unifying theme is actions of groups: on geometries, graphs, Riemann surfaces, and, recently, axial algebras, which form a new interesting class of non-associative structures having applications in physics. In core group theory, Sergey is interested in finite simple groups, their classification, origin and properties of sporadic simple groups, and fusion systems. He is also interested in purely geometric and purely combinatorial questions, in particular, in the theory of diagram geometries and buildings, algebraic graph theory, and metric graph theory, including partial cubes and similar graphs with good metric properties.

## Dr Simon Goodwin

**Senior Lecturer**

Simon's research interests cover a range of topics in representation theory and Lie theory. His interests are centered around questions in the representation theory of Lie algebras and algebraic groups. Recently he has been interested in the representation theory of modular Lie algebras and of Lie superalgebra, and in particular the application of the theory of *W*-algebras to these areas. This research has involved a blend of algebraic, combinatorial and geometric methods. He also works in problems in the related to the representation theory of finite groups of Lie type and their structure.

## Dr Corneliu Hoffman

**Senior Lecturer**

Corneliu Hoffman interests lie in Group Theory, Representation Theory and Galois Theory. Recently he applied Bass-Serre theory and similar homological means to the question of classifying amalgams of linear groups. he is also interested in geometric group theory, more precisely in expander graphs via groups with property (T).

## Dr Richard Kaye

**Senior Lecturer**

Structural properties of models of Peano arithmetic, and in particular their initial segments.

Richard is one of the main workers in the area of models of first-order arithmetic. There are a number of themes to this research, but most structural information about models of arithmetic relates to the order structure of the model. Richard's work includes linking this order structure to the automorphism group of models of PA, and to looking at new families of initial segments, such as generic cuts. In many cases the structural properties are best understood through a language expanding that of PA by adding other predicates and functions - one representing the cut in question for example. This leads to new ways of looking at second order theories of arithmetic (utilising coding devices for example). The recent and on-going work with Tin Lok Wong, a PhD student of Richard's at Birmingham illustrates these ideas very well.

Strengthenings of the notion of recursive saturation and resplendency, in particular arithmetical saturation and transplendency.

Recursive saturation is a very natural and useful property that many nonstandard models of arithmetic have. In some cases (e.g. when the model has a nonstandard truth definition) recursive saturation is available "for free". Recursive saturation is closely related to the idea of resplendence in second order model theory. However, recursive saturation alone is often not enough for some results. An older result by Kotlarski, Kossak and Kaye shows that a countable recursively saturated model of PA has an automorphism moving every nondefinable point if and only if the model satisfies the stronger property of being arithmetically saturated. In recent work Richard and his PhD student Fredrik Engstrom looked at a powerful extension of this to form expansions of the model simultaneously omitting a type. The resulting notion - transplendency - is very powerful and not as yet fully understood and is still the subject of current research.

Properties of nonstandard algebraic structures, in particular nonstandard finite symmetric groups, abelian groups and linear groups.

Algebra and logic combine very well. Richard has instigated a study of finite algebraic objects inside nonstandard models. This leads to some very attractive algebraic objects, including nonstandard symmetric groups (studied for example by Richard and his research student John Allsup) and nonstandard finite linear groups. Even nonstandard cyclic groups have interesting structure which is being investigated currently by another PhD student, Reading. Results in these areas show that symmetric groups are closely connected with so-called sofic groups, and nonstandard groups often have natural quotients with analytic structure and often with interesting measures. The work results in interesting new examples of algebraic objects with new means of reasoning about them. Nonstandard methods of this type can be applied to other areas too. Another PhD student of Richard's is currently investigating Conway-style Combinatorial Games and the sorts of nonstandard games that arise from model theoretic considerations applied to these.

## Dr David Craven

**Senior Birmingham Fellow**

The main areas of Dr Craven's research are in group theory, both in the structure and representations of finite groups, and fusion systems. In the structure of finite groups, he mainly works on understanding the maximal subgroups of simple groups, particularly of exceptional groups of Lie type. For representations of finite groups, he focuses primarily on the local-global principle, attempting to glean information about the representation theory of a finite group from information about the representations of certain small subgroups, called local subgroups. This is related to the nascent area of generic representation theory, which attempts to allow the characteristic of the field over which representations are taken to become a variable rather than a fixed prime. In fusion systems his interests are in understanding the internal structure of fusion systems, and understand so-called exotic fusion systems, which do not arise from finite groups.