ENGINEERING AND PHYSICAL SCIENCES RESEARCH COUNCIL EPSRC Mathematical Sciences Community Overview Documents Research, discover, innovate CONTEXT As part of EPSRC’s Balancing Capability strategy, the Mathematical Sciences theme has been working with the research community, and other stakeholders, to establish an evidence base, which has provided insight into the research and training landscape of each of the mathematical sciences research areas. Community perspectives have been sought through a number of mechanisms, including a call for evidence across EPSRC, regional events and research area specific workshops. The approach taken for each research area has been tailored to complement EPSRC’s current evidence base and previous portfolio monitoring activities. For several research areas (the pure mathematics areas, applied mathematics areas and Mathematical Physics), an adopted approach was to build upon the landscape documents produced for the International Review of Mathematical Sciences 2010. This was undertaken by inviting the lead authors (or, where they were unavailable, their suggested alternatives) of these documents to co-ordinate community input and produce a community overview document. (To note: These documents were written in spring 2016 and the landscape may have developed since then) For other areas, notably Statistics and Applied Probability and Operational Research, we have built upon recent engagement activities, including discussions across council, and community overview documents have therefore not been produced. Activities to further develop the evidence base for these areas are planned for the near future and the outputs from these will be made available separately at a later date. The community overview documents provide a synopsis of the key features of the current landscape from a community perspective and highlight emergent trends since the 2010 landscape documents. Although the documents for some research areas provide examples of key activities, researchers and groups, it is important to note that they are not intended to provide exhaustive lists of all activities and people in the area. It should also be noted that these documents form one part of the overall evidence gathered for the Balancing Capability strategy and are being considered alongside information gathered from other engagement activities such as the EPSRC Mathematical Sciences workshops, the EPSRC Call for Evidence, and advice from the EPSRC Strategic Advisory Team. EPSRC would like to thank the authors for their efforts and hope that the community identifies these documents as a useful tool for providing a snapshot of the current landscape of each research area. Any comments or feedback should be sent to: [email protected] Algebra Lead author: Peter Symonds Supporting author: Iain Gordon Contributors: Martin Bridson, Derek Holt, Martin Liebeck, Kevin McGerty, Toby Stafford. 1. Subject breakdown: Research Area Number of Researchers % of Total Infinite Group Theory 38 18 Finite Group Theory 32 15 Representation Theory 83 39 Non-Commutative Algebra 23 11 Other 37 17 2. Discussion of research areas: Algebra pervades most of pure mathematics and it is impossible to draw any clear frontiers. As well as in algebraic geometry, algebraic topology and algebraic number theory, algebra is important in combinatorics, model theory and part of analysis. Some of the most effective research groups span several of these areas. The UK has been an international leader in algebra for a very long time. It maintains a strong presence across all the main sub-areas, although at the finer level there are gaps. What follows consists of a selection of recent highlights chosen by the contributors. It is not to be interpreted as a definitive overview of the subject. Awards International Congress of Mathematicians. Invited talks: Benson, Gordon (2010); Ardakov, Teleman (2014). Special session chair: Wadsley (2014). Selection committee for algebra: Smoktunowicz (2014). European Mathematical Society Prize. Smoktunowicz (2008). London Mathematical Society. Polya prize: Segal (2012). Berwick prize: Chuang and Kessar (2009). Senior Berwick prize: Teleman et al., (2014). Edinburgh Mathematical Society. Whittaker prize: Smoktunowicz (2009). Infinite Group Theory The shift in focus of UK research in infinite group theory reported in the 2010 review has continued and several new strands of research have gained in prominence and strength. Since the 1990s, the focus of work in the UK on infinite groups has shifted away from the study of varieties of groups, such as nilpotent or polycyclic groups, towards geometric group theory, mirroring a worldwide increase in the prominence and applications of geometric, topological and analytic methods in group theory. This has seen group theory play a leading role in advances in many adjacent subjects, particularly low-dimensional topology and geometry, in the wake of transformative work by Thurston and Gromov. There is also growing excitement about advances in profinite group theory: this has been a strength of the UK for some time, but its ever-increasing domains of application, from number theory to geometry, have brought a new intensity to the subject, as well as broad attention. 1 2 There are sizeable research groups in Oxford, Cambridge, Warwick, Bristol, Southampton, and further strength across London, Newcastle, Edinburgh, Glasgow and Saint Andrews. Of particular note since 2010 is an emerging school of Geometric Group Theory at Cambridge; Bristol also has a talented young group. Geometric group theory. Particularly dramatic is the way in which geometric group theory has solved the outstanding problems of 3-dimensional topology. The major figures in the final developments were Agol, Wise and Kahn-Markovic, with most of the action taking place in North America, but the UK's involvement was considerable: many aspects of the key technology of CAT(0) cube complexes were developed in the UK (Oxford and Southampton), and many extensions and applications of the Agol-Wise technology have been developed by Wilton, Bridson and others. Indeed the UK is world-leading in its work on fundamental groups of 3-manifolds In Oxford, there is a strong focus on Geometric Group Theory - Bridson, Drutu, Lackenby and Papasoglu all have close links with the topology group. Southampton remains a large and thriving centre for geometric group theory. Warwick, too, has a group with strong connections to low-dimensional topology. · Minasyan (Southampton) [2] has various notable results in geometric and combinatorial group theory. · Brendle (Glasgow), Margalit and Putman [12] resolved the Hain Conjecture. · Grabowski (Warwick) [25] studied `2-cohomology and the Atiyah conjecture. · Nikolov's (Oxford) [1] work on rank gradients and the growth of Betti numbers with various coauthors has attracted a lot of attention. · The study of cohomological aspects of group theory continued to be a national strength, with Kropholler and Leary (Southampton) and Nucinkis (RHUL) playing prominent roles. Profinite groups. Currently a lot of the excitement is in intimately related areas. For example, see the sections on asymptotic methods and p-adic groups below. Bridson (Oxford) and Wilton (Cambridge) [16] consider the existence of a finite quotient for a finitely presented group. Finite Group Theory Finite simple groups and conjectures in local representation theory. There have been breakthroughs in a number of well-known conjectures concerning the local representation theory of finite groups by Craven (Birmingham), Eaton (Manchester), Kessar and Linckelmann (City) { for example, the Donovan conjecture and Brauer's height zero conjecture, a major part of which has been proved by Kessar and Malle in [31]. One of the main strategies in attacking these conjectures has been to reduce them to statements about simple groups, and then, using substantial structural results on simple groups, together of course with a great deal of representation theory, to prove these statements. While this topic of course comes under the heading of Representation Theory, there is a substantial input from the structural theory of simple groups. Aspects of finite simple groups. There are several developments to discuss here. The first is fusion systems: these are categories based on p-groups, which capture the fusion properties of Sylow p-subgroups in finite group theory. Research in the area has developed rapidly because of their appearance in representation theory and homotopy theory, and also because of applications to simplifying parts of the classification of finite simple groups (CFSG). Henke's (Aberdeen) [30] theory of linking systems is particularly promising in this respect. 3 The second aspect is the major programme of writing a revised and unified proof of the CFSG: this is continuing, with substantial UK input from Capdebosq (Warwick), Magaard and Parker (Birmingham). Next, there is recent work in the theory of algebras related to the Monster sporadic simple group, pioneered by Ivanov (Imperial), Shpectorov (Birmingham) and their collaborators: this originated in the theory of vertex operator algebras, and aimed to provide an axiomatic setting for the Monster algebra, but it has been generalised into a new concept of axial algebras, which have been shown to have beautiful connections with Jordan algebras and 3-transposition groups. One should also mention the work of Wilson (Queen Mary) on sporadic groups, and also new constructions of families of exceptional groups of Lie type. Asymptotic group theory A