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 major development since 2010 is the interaction between additive combinatorics and group theory (both infinite and finite) provided by the theory of ’approximate groups’, with Green (Oxford) as a driving force in the UK. Achievements include the proof of Breuillard, Green and Tao [13] of Babai’s conjecture on diameters of finite groups of Lie type of bounded rank. This has been a tool in much subsequent work, including the construction of large families of new expander graphs by the same authors and Varju (Cambridge) [14, 9]. On a different front, Nikolov and Segal (Oxford) [41] have proved far-reaching asymptotic results on the generation of arbitrary finite groups, with applications to compact and profinite groups. There is also work of Burness (Bristol), Liebeck (Imperial) and collaborators [19] on bases of permutation groups, and on probabilistic aspects of finite groups, such as random generation and random walks.

Computational group theory One of the main publications to appear in this area during the period is the book [11] of Bray (Queen Mary), Holt (Warwick) and Roney-Dougal (St Andrews), classifying the maximal subgroups of classical groups of dimension up to 12; they include algorithms to write down specific representations of these. This has become a major resource. Other aspects of computation are dealt with later.

Representation Theory and Non-Commutative Algebra: These subjects are deeply intertwined in the UK, and they also have important interactions with algebraic and symplectic geometry, algebraic topology, combinatorics, integrable systems and number theory. Since the last IRM, several young people have moved to permanent positions in Britain (including Schedler (Imperial), Jordan, Sierra (Edinburgh), Bellamy (Glasgow), Ardakov, Kremnitzer, McGerty (Oxford). There have also been more senior arrivals, including Teleman (Oxford), and Vogtmann (Warwick), while some high profile individuals have moved elsewhere. Representation theory of finite groups. This is a well-established topic in the UK, with researchers around the country. Work on conjectures in local representation theory has already been mentioned. Other notable recent achievements are:

· Craven’s (Birmingham) work with Rouquier [23] on perverse equivalences and Brou´e’s conjecture, · The work of Benson, Henke (Aberdeen)and Grodal [7] on group cohomology and fusion systems. · Benson (Aberdeen), Iyengar and Krause [8], in a series of papers, classify localising and colocalising subcategories of certain triangulated categories, in particular for 4

modular representations of finite groups. They use techniques from representation theory, algebraic topology and commutative algebra. · A significant number of people work on the representation theory of the symmetric groups, led by Fayers (Queen Mary) and Lyle (East Anglia). · Bartel (Warwick) and Dokchitser (Bristol) have detailed results on the structure of representation rings; the original motivation was from number theory. p-adic groups and arithmetic geometry. The p-adic Langlands programme is a recent and rapidly developing aspect of number theory and representation theory. The UK has a number of researchers making fundamental contributions to this area: Ardakov (Oxford) and Wadsley (Cambridge) [5, 6] have done seminal work on the representation theory of p-adic analytic groups; this was the subject of an invited talk at the ICM 2014. Kremnitzer and Ben-Bassat (Oxford) are currently developing an approach to adic spaces as relative algebraic geometry, which should have manifold applications in this rapidly developing field. The arrival of Ciubotaru at Oxford has also reinforced the existing strength in the complex representation theory of p-adic groups represented by Stevens (UEA) and Bushnell (Kings).

Representations of algebras. Many significant results have been obtained on the structure of cluster algebras, cluster categories and cluster combinatorics. This is a young subject with many unexpected links to other areas. There are groups in Leeds and Durham and researchers in other places. Marsh (Leeds) is a leading figure and has produced a book on the subject [36]. There are also thriving groups working on quiver representations and quiver varieties, for example in work of Jordan [34] and Craw [24]. Others use model theoretic methods, a link with logic.

Lie theory and topological aspects. There is now a well established community in Lie-theoretic representation theory: Edinburgh, Glasgow, Manchester, Oxford being among the key institutions. This subject has rich interaction with combinatorics and algebraic geometry. Notable additions to the research community in this area are the arrival of Jordan in Edinburgh (quantum groups, topological field theory) and Teleman (Loop groups, field theories) at Oxford; the latter was an invited speaker at ICM 2014. Highlights of recent work in this area are: · Teleman’s [45] classification of 2-D semisimple field theories is a milestone in the study of topological field theories, · Gordon’s elegant proof [28] of Macdonald positivity using the theory of Cherednik algebras. · Hausel and collaborators [29] resolved the Kac conjecture using the moduli of quiver representations and point counting techniques over finite fields, developing an approach pioneered by Crawley-Boevey (Leeds) and van den Bergh. · Clarke and Premet’s (Manchester) [22] characteristic-free description of the unipotent and nilpotent varieties for connected reductive groups thereby generalizing case-by-case verifications of Lusztig. Premet has also pioneered the use of W-algebras to ellucidate the structure of representations of Lie algebras. · Bridgeland (Sheffield) [15] has given a beautiful construction of the full quantum group using Hall algebras.

Noncommutative geometry. This subject has strong interactions with algebraic geometry and many related areas, with highlights including: · Iyama and Wemyss (Edinburgh) [32, 33] have used noncommutative resolutions of singularities and their theory of maximal modifications to obtain significant results on 5

the structure of (commutative) resolutions of singularities and the minimal model programme. · Using the machinery of noncommutative algebraic geometry, Sierra (Edinburgh) and Walton [43] answered a 20 year old question on the structure of the Witt Lie algebra. · There are significant applications to moduli questions, for example, in noncommutative geometry in the work of Nevins and Sierra [40], and to quantum symplectic resolutions in work of McGerty (Oxford) and Nevins [38]. · Closely related is Pym’s (Oxford) classification of the generic types of noncommutative projective 3-spaces [42] via symplectic geometry. In recent work of Pym, Szendroi (Oxford) [20] and others there are conjectural connections between noncommutative geometry and Donaldson-Thomas invariants.

Category theoretic methods. We have already seen in the representation theory of finite groups that the study of various categories has become an important theme. In addition we mention: · An important theme in representation theory in recent years has been the construction of (higher) categorical structures. This language allows deep parallels to be drawn with topology and geometry (for example the group under Joyce at Oxford studying Donaldson-Thomas invariants study categorification of DT invariants using perverse sheaves or the work of Benson et al. already mentioned). Moreover, many interesting examples can be obtained using moduli of representations of quivers, giving connections to the foundational work of Crawley-Boevey. More purely algebraic techniques in categorification have been developed by researchers such as Miemietz (UEA) Turner (Aberdeen) and Chuang (City). · Broomhead, Pauksztello (Manchester) and Ploog [17, 18] study discrete derived categories in detail. · Braun, Chuang (City) and Lazarev (Lancaster) [10] construct a derived localisation theory for DG algebras.

Other: Commutative algebra. The UK is not as strong in classical commutative algebra as it used to be, despite the importance of the subject internationally. But there are many people in other areas who use the techniques, which have been very influential: for example in algebraic geometry and, more specifically, ring spectra in algebraic topology and the work of Benson et al. already mentioned. There are individuals such as Hering (Edinburgh) and Maclagan (Warwick) and there is a significant number of people with interests in invariant theory, for example in Kent. Symonds (Manchester) [44] found an explicit bound on the degrees of the generators of a ring of invariants in finite characteristic, a longstanding problem.

Semigroup theory. There is a well-established group of researchers in semigroup theory. The subject has many cross-connections to other areas, with current UK research linking to category theory, combinatorics, functional analysis, geometric group theory, representation theory, model theory, theoretical computer science and tropical geometry/max-plus algebra. Recent highlights include: · Gray (East Anglia) and Ruskuc’s (St Andrews) [26] resolution of the longstanding maximal subgroup problem for free idempotent-generated semigroups, which has reawakened interest and prompted new developments in an area of the subject previously believed to be too hard for progress. · The extension of techniques of geometric group theory to semigroup theory by Gray and Kambites (Manchester) [27]. 6

· Perhaps the most exciting development is the synchronising groups programme, in which a longstanding open problem in semigroup and automata theory (the Cern´yˇ conjecture) has motivated the development of an entire new area of finite group theory, and brought together leading researchers from all three fields, including Cameron (St Andrews) to attack the resulting problems [3, 4].

Computational Algebra. The GAP Computer Algebra System, which is the most widely used such system in the world for computing with groups, and algebraic structures in general, is administered by the CIRCA group in St Andrews. GAP packages have been written by various UK mathematicians recently, including Soicher (Queen Mary) in combinatorics and Mitchell (St Andrews) in semigroups. Holt (Warwick) has also made significant contributions to the facilities available in Magma. Schleimer (Warwick) and students have developed software for various computations with mapping class groups, and also for the ‘compressed word problem’ in hyperbolic groups. · Wilson and Bray (Queen Mary) are involved in the maintenance and updating of the widely used online ATLAS of group representations. · Leedham-Green (Queen Mary) has co-ordinated and spearheaded the major international project (which has been going now for more than 25 years) involving the theoretical and practical development of algorithms for computing with large finite matrix groups. The impact of this project has been at its greatest during the past three or four years, since user-friendly code has finally become available in GAP and Magma. · The book [11] dealing with the maximal subgroups of classical groups of dimension up to 12 has already been mentioned under finite groups.

3. Discussion of research community: There are approximately 214 algebraists employed in UK universities. Of these we count 44 as first permanent appointments since 2010. This is not enough to compensate for the large number of retirements. Only 9 of these new appointments were of women. There are 94 EPSRC-funded postgraduate students associated with algebra(64 FTE). From information from departments, we estimate that there are about 250 PhD students in total. Most come from abroad and bring their own funding. Given the international reputation of UK algebra and the numbers of researchers, this is well below capacity. There has been a significant number of new appointments at institutions with no tradition of work in algebra, or even pure mathematics. Given the current state of the job market, it has been possible to recruit very good researchers, but it will be a challenge to integrate these people into the research community. First permanent appointments (since 2010): Gramain, Henke, Izhakian (Aberdeen); Fairbairn (Birkbeck); Craven (Birmingham); McKay (Bristol); Wilton (Cambridge via UCL); Gildea (Chester); Felikson, Stasinski, Tumarkin, Vishik (Durham); Grant, Gray (East Anglia); Hering, Jordan, Sierra (Edinburgh); Carvalho, P´eresse, Young (Hertfordshire); Bellamy, Fourier, Voigt (Glasgow); Bentz (Hull); MacDonald (Lancaster); Tange (Leeds via Exeter); Johnson (Manchester); Fonick (Manchester Metropolitan); Elmer (Middlesex); Balagovic, Kolb (Newcastle); Vishik (Nottingham); Ciubotaru, Nikolov, McGerty (Oxford); Robertz (Plymouth); Fink (Queen Mary); Wildon (Royal Holloway); Bleak, Huczynska (St. Andrews); Kar, Petrosyan, Spakula (Southampton); Louder (UCL).

4. Inter/Intra-disciplinary activities and engagement activities: We have already described how a large part of algebra is intrinsically also part of another subject area and fundamentally intra-disciplinary. Many key themes are geometric in nature, 7 for example geometric group theory and geometric representation theory; non-commutative algebra is now firmly tied to algebraic and symplectic geometry. Connections with number theory are important, particularly through the Langlands programme and Iwasawa theory. Representation theory overlaps with logic in model theory. A relatively new development is the interaction of the theory of cluster algebras with many other areas: combinatorics, Poisson geometry, higher Teichmuller¨ theory, discrete integrable systems, categorification, Donaldson-Thomas invariants, scattering amplitudes, etc.. Tropical mathematics crosses the divide between pure and applied mathematics, as does coding theory. There is also a large variety of work somewhere between algebra and numerical analysis. The language of higher categories reveals profound and unexpected connections between algebra, topology and geometry; this has great potential. There are links with theoretical physics through string theory, statistical mechanics and topological quantum field theory, for example via diagram categories such as the Temperley-Lieb category and the partition category, which are also linked to quantum information. Perhaps as a consequence of this, the algebra community is only very loosely connected internally. Most practitioners regard themselves as working in some sub-discipline or in some other area. The networks funded by the LMS and the EPSRC play a crucial role in bringing people together and disseminating new ideas. LMS funded research groups: ARTIN (representation theory); BLOC (representation theory); NBGGT (geometric group theory); NBSAN (semigroups); Groups and geometry in S.E.; S. England profinite groups; Manchester, Bristol, London groups and their applications. EPSRC funded: RepNet (Anglo-Franco-German Representation Theory), CoDiMa (Computation in Discrete Mathematics).

5. Future direction/opportunities: We have seen that the UK is very strong in exploiting connections between algebra and geometry, topology, physics and combinatorics. Fewer algebraists work on links with number theory. We have already mentioned the importance of category theoretic language in a lot of new work. The theory of higher categories is likely to be crucial in the future for exploiting some of the connections mentioned above. Research in most sub-areas at an international level is cyclical, with periods of rapid advance and then some time for consolidation. It is very difficult to predict where the next exciting advance will be. What is important is that the UK should maintain a high level of expertise in all major sub-areas, so that it can participate at the highest level in the most exciting developments wherever and whenever they arise. Many of the most important sub-areas are perceived to be highly technical and this can be off-putting to PhD students and other researchers who are under pressure to produce results quickly; all the more so if the topic is intra- or inter-disciplinary. This could put the UK at a disadvantage internationally. The funding environment could do more to encourage people to take up these challenges.

6. Further comments: There is concern about the lack of structure in the career path between PhD and first permanent appointment. RA positions on grants are tailored to the needs of the PI, not the RA, and the availability of a suitable post when a student graduates is a matter of luck. The 8

Heilbronn Institute in Bristol plays a significant role here, employing several postdocs in algebra. It is probably not a coincidence that the number of women who survive past this stage is low. It is important to have a diverse collection of research groups, both in subject area and geographically. The challenge is then to enable them to interact effectively. Many research groups are vulnerable to key people leaving.

7. Main research groups in algebra: University employees. Aberdeen: Benson, Gramain, Henke, Izhakian, Martin, Sevastyanov, Turner. Birkbeck: Bowler, Fairbairn, Hart. Bath: Craw, King, Smith, Su, Traustassen. Birmingham: Craven, Evseev, Flavell, Goodwin, Hoffman, Magaard, Parker, Shpectorov. Bristol: Bell, Burness, McInroy, McKay, Rickard, Saunders, Semarao, Thomas. Cambridge: Bouayad, Brookes, Button, Camina, Grojnowski, Hagen, Lawther, Martin, Stewart, Varju, Wadsley, Wilton. Cardiff: Pugh. Chester: Gildea. City: Bowman, Chuang, Cox, De Visscher, Kessar, Linckelmann. Durham: Felikson, Stasinski, Tumarkin. East Anglia: Grant, Gray, Lyle, Miemetz. Essex: Higgins, Vernitski, Williams. Edinburgh: Gordon, Hering, Iyudu, Jordan, Kalck, Lanini, Leinster, Sierra, Smoktunowicz, Wemyss. Glasgow: Adams-Florou, Bellamy, Brendle, Brown, Feigin, Fourier, Kraehmer, Voigt, White. Heriot-Watt: Gilbert, Howie, Lawson. Hertfordshire: Carvalho, P´eresse, Young. Hull: Bentz. Imperial: Britnell, Evans, Ivanov, Liebeck, Schedler. Kent: Fleischmann, Hone, Launois, Paget, Shank, Woodcock. Kings: Rietsch. Lancaster: Grabowski, Levy, MacDonald, Mazza, Towers. Leeds: Crawley-Boevey, Marsh, Martin, Parker, Tange. Leicester: Baranov, Canakci, Schroll, Snashall. Lincoln: Khukro, Mattarei. Manchester: Bazlov, Borovik, Eaton, Gregory, Johnson, Kambites, Livesey, Pauksztello, Premet, Prest, Petukhov, Rowley, Stafford, Stohr, Symonds, Tressl. Manchester Metropolitan: Foniok. Middlesex: Elmer. Newcastle: Balagovic, Duncan, Jorgensen, King, Kolb, Rees, Vdovina, Waldron. Nottingham: Edjevet, Pumplun, Vishik. Oxford: Ardakov, Bridson, Ciubotaru, Cliff, Drutu, Dufresne, Green, Hollings, Kremnitzer, McGerty, Nikolov, Papazoglou, Pym, Segal, Szendroi, Teleman. Plymouth: Robertz. Queen Mary: Bray, Fayers, Fink, Majid, Muller,¨ Soicher, Wilson. Royal Holloway: Barnea, Blackburn, Nucinkis, Wildon. St. Andrews: Bleak, Cameron, Huczynska, Linton, Mitchell, Pfeiffer, Quick, Roney-Dougal, Ruskuc. Sheffield: Bavula, Bridgeland, Johnson, Katzmann. Southampton: Brodzki, Guest, Kar, Kropholler, Leary, Martino, Minasyan, Moutuou, Niblo, Petrosyan, Renshaw, Spakula, Wright. 9

South Wales: Gill. Swansea: Beggs, Brzezinski, Crossley, Garkusha. UCL: Hill, L´opez Pena,˜ Louder. Warwick: Capdebosq, Grabowski, Holt, Krammer, Maclagan, Rumynin, Vogtmann. York: Bate, Donkin, Geranios, Gould, Hawkins, Nazarov. Total:213.

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[32] O. Iyama, M. Wemyss, On the noncommutative Bondal-Orlov conjecture, J. Reine Angew. Math. 683 (2013), 119–128. [33] O. Iyama, M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), 521-586. [34] D. Jordan, Quantized multiplicative quiver varieties, Adv. Math. 250 (2014), 420–466. [35] M.W. Liebeck, D. Segal, A. Shalev, The density of representation degrees, J. Eur. Math. Soc. (JEMS) 14 (2012), 1519–1537. [36] R.J. Marsh, Lecture notes on cluster algebras, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Z¨urich, 2013. [37] P.P. Martin, The decomposition matrices of the Brauer algebra over the complex field, Trans. Amer. Math. Soc. 367 (2015), 1797–1825. [38] K. McGerty, Microlocal KZ-functors and rational Cherednik algebras, Duke Math. J. 161 (2012), 1657–1709. [39] K. McGerty, T. Nevins, Derived equivalence for quantum symplectic resolutions, Selecta Math. (N.S.) 20 (2014), 675-717. [40] T.A. Nevins, S.J. Sierra, Moduli spaces for point modules on na¨ıveblowups, Algebra Number Theory 7 (2013), 795–834. [41] N. Nikolov, D. Segal, Generators and commutators in finite groups; abstract quotients of compact groups, Invent. Math. 190 (2012), 513-602. [42] B. Pym, Quantum deformations of projective three-space, Adv. Math. 281 (2015), 1216-1241. [43] S.J. Sierra, C. Walton, The universal enveloping algebra of the Witt algebra is not noetherian, Adv. Math. 262 (2014), 239–260. [44] P. Symonds, On the Castelnuovo-Mumford regularity of rings of polynomial invariants, Ann. Math. 174 (2011), 499–517. [45] C. Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), 525588. MATHEMATICAL ANALYSIS

Lead A.Carbery (Edinburgh), J. Norris (Cambridge), P.Topping (Warwick). authors: ​

People we consulted: J. Ball (Oxford), K. Ball (Warwick), J. Bennett (Birmingham), J. Carrillo (Imperial), G.­Q. Chen ​ (Oxford), M. Dafermos (Cambridge), C. Elliott (Warwick), M. Hairer (Warwick), G. Holzegel (Imperial), Y. Kurylev (UCL), M. Mathieu (QUB), M. Pollicott (Warwick), D. Preiss (Warwick), M. Ruzhansky (Imperial), E. Shargorodsky (Kings), G. Stallard (Open U), S. van Strien (Imperial), A. Stuart (Warwick), E. Suli (Oxford), J. Toland (INI), S. White (Glasgow).

: Overview​ Mathematical analysis has undergone a period of transformation in the UK over the past 5­10 years. A number of key strategic areas have finally been developed to an internationally competitive level. UK activity has delivered a series of high­profile breakthroughs, recognised by a string of prizes, including Hairer’s 2014 Fields medal, 12 invited talks at the Seoul ICM 2014 and 5 at the Hyderabad ICM 2010. The subject has always been closely integrated with applications, but the number of topics that it influences (and which influence it) is expanding dramatically, both outside and inside mathematics, and we expect that growth to continue. This blurring of the boundaries of the subject is a positive trend, but it has meant that it was not always clear where the remit of this report should end. We have covered core topics such as PDE theory, harmonic analysis, classical analysis, stochastic analysis and dynamical systems etc., and even reached into computational analysis briefly, although this latter topic will be covered principally by the applied mathematics review. The size of the subject, measured by outputs, represents about half of all pure mathematics internationally.

Statistical overview: Data has been collected concerning the volume of outputs in analysis and its relation to the total output in pure mathematics. We have also received data on the number of researchers in the UK in analysis. This data comes with caveats since it is becoming harder and harder to make precise delineation between different areas, and because some of the data we have received is only partially complete. Our first data is derived from publicly accessible publication data from Math Reviews. We have assigned outputs in MSC classifications 26­49 inclusive to Analysis. The first table below gives numbers of publications reviewed in Analysis and “Pure Mathematics” (Algebra, G&T, Combinatorics, Number Theory, Logic, Probability, Relativity) with primary MSCs in the relevant range in the calendar years 2010­­2014. (See the annexe for full details.) We have also received partial data from EPSRC and some UK HoDs (via Iain Gordon) concerning numbers of researchers in certain subareas of pure mathematics. Although we do not agree with all categorisations made in those exercises, and many people were missing from the data altogether, a conclusion is that the numbers of UK researchers working in Analysis, Algebra and Geometry & Topology are roughly similar. The second table below gives numbers of publications reviewed against numbers of active UK researchers provided by EPSRC.

Math Reviews outputs 2010­2014 inclusive

UK outputs % share of UK World outputs % share of UK % outputs world outputs contribution to world total

Analysis 2947 27 118947 46 2.5

Pure Maths 10887 100 261337 100 4.2 UK Productivities 2010-2014 inclusive (excluding Probability MSC 60, Relativity MSC 83) UK Outputs UK researchers Productivity

Analysis 2947 225 13.1

Pure Maths 8992 877 10.3

The broad conclusion is that despite UK analysts producing more papers than average in the UK, analysis represents only 27% of UK mathematics, compared with 46% internationally, by volume of outputs. This suggests that analysis remains substantially under-represented in the UK.

Discussion of research areas: Discussions of individuals and research groups are necessarily only partial, and many people and groups will have been omitted. This document is meant as an indication of current activity in the UK and is not suitable as evidence in hiring/promotion cases, for example.

Geometric Analysis UK activity was tiny by international standards 10-15 years ago. Since then it has developed very successfully, with several departments making very high quality appointments. UK activity represents a small proportion of global output in the area, but the quality over recent years has been exceptionally high, and the subject has contributed substantially to the prestige of UK mathematics. Although there are no obvious boundaries between this area and several other areas covered below and in other Landscape Documents, we highlight minimal surfaces and geometric flows as areas of particular importance to the UK. In the recent 2014 ICM, Neves, Topping, Markovic and Malchiodi gave invited addresses. Neves won a Whitehead prize in 2013, the New Horizons in Mathematics prize 2016 and the Veblen prize 2016 for his development (with Marques) of min-max theory, leading, for example, to a proof of the Willmore conjecture. Donaldson’s breakthroughs on Kahler-Einstein metrics (joint with Chen and Sun) were based heavily on analysis, and led to the award of the $3 million Breakthrough Prize 2014. Other significant developments include Wickramasekera’s work on regularity theory for minimal surfaces, and the work on geometric flows of the Warwick group.​EPSRC,​ ERC and the Leverhulme Trust have given substantial and important funding to the area over this period, including a Programme grant (Topping, Neves, Dafermos), an ERC starting grant (Neves), Leverhulme Research Leadership award (Topping) plus numerous smaller awards. Several landmark workshops and conferences have taken place in the UK during this period, including at ICMS (2011) and Warwick (2015). The major groups in the UK are at Warwick and Imperial with a high quality presence at Cambridge, Oxford, Bath, QMUL, UCL, Kings and elsewhere.

Mathematical Relativity This area has also been developing very well in the UK recently. Dafermos is a leading figure, and a highlight is his proof (with Rodnianski and Shlapentokh-Rothman) of the linear stability of Kerr to scalar perturbations. Dafermos spoke at the 2014 ICM, and Gibbons spoke at the 2010 ICM. Luk won the Silver Prize of the New World Mathematics Award, 2014. Aside from Programme grant support mentioned in the previous section, ERC starting grants were held by Dafermos and Holzegel. The major groups in the UK are at Cambridge, Imperial and Oxford.

Calculus of Variations, materials science, homogenisation The UK has a small but high quality international presence in topics such as quasiconvexity (Kristensen, J.Ball, Rindler) and homogenization (Capdeboscq, Pavliotis, Smyshlyaev, A.Stuart). In analysis applied to materials science (including solid mechanics generally and liquid crystals) there is a larger and broader effort with considerable international impact. Specific areas of expertise include analysis applied to solid phase transformations (displacive and diffusive), nonlinear elasticity, atomic-to-continuum, crystallization and dislocations. The UK presence in the analysis of liquid crystals (Ball, Zarnescu, Majumdar, Nguyen, Robbins, Slastikov) is largely responsible for the international resurgence of interest by mathematicians in this area, demonstrated in the INI Liquid Crystals programme 2013. Ball holds an ERC Advanced Grant.

Nonlinear PDE of fluids, kinetic theory; other nonlinear PDE including free boundary problems, fully nonlinear PDE, pure nonlinear PDE not covered above.

As for areas of nonlinear PDE discussed above,​the​ UK contribution to these areas, including mathematical fluid mechanics, nonlinear diffusions, dispersive PDE and kinetic theory has​had​ a tremendous boost both in quality and in volume in the last 5 years due to excellent hiring in these fields in different institutions. For the first time in many years, the UK is recognized to have booming activity in these areas, reflected in a landmark conference on the subject at Oxford in 2012, INI programmes on Kinetic Theory 2010 and Free Boundary Problems 2014, and in activity particularly at ICMS, Oxford, Cambridge, Imperial, Warwick, and Edinburgh. The CDTs in PDEs in Oxford and in Analysis in Cambridge and Edinburgh are examples of large grants going hand in hand with the increasing activity of this area. Concerning evidence of national/international recognition, we mention the Whitehead Prizes of the LMS awarded to Varvaruca (2012), Mouhot (2014) and Niethammer (2011) and the ICM addresses of Markowich (quantum mechanics) and Seregin (fluid mechanics) in Hyderabad in 2010. The laudation of the Fields medal awarded to Villani in 2010 mentioned several works in collaboration with Mouhot and in particular their landmark contribution in Landau damping. Toland was awarded the Royal Society Sylvester Medal in 2012 and J. Ball received SIAM’s John von Neumann Prize in 2012. G.-Q. Chen was awarded SIAM’s SIAG/Analysis of Partial Differential Equations Prize in 2011 for joint work with Feldman. J.B. McLeod won the LMS Naylor prize 2011. Ruzhansky won a EPSRC leadership fellowship, and 2010 Daiwa Adrian Prize. Ben-Artzi won a EPSRC career acceleration fellowship. ERC grants are held by Mouhot, Rodrigo and Oh.

Inverse problems The rigorous, analysis side of inverse problems (IP) has also grown substantially during the last 10 years. The UK is now among the world leaders in such subareas as Geometric IP (Paternain, Oksanen), Statistical/stochastic IP (A.Stuart) and Spectral IP (focussed on the London linear analysis community). There is increasing activity also in applications to image analysis (Capdeboscq, Schoenlieb, and from other directions K.Zhang). There are directions that are not properly covered in the UK, for example methods based on harmonic and microlocal analysis, quasiconformal methods, numerical analysis associated with IP etc. However, the UK has become very visible internationally; for example INI had a semester on IP in 2011. A very positive move in UK IP life is the appearance of several centres that cover a broad range of IP research (applied, numerical and theoretical) and use diverse mathematical methods, including UCL’s Centre for IP, which has research groups in several departments (Mathematics, Statistics, Computer Science, Medical Physics). The other strong centres are Cambridge, Warwick and Manchester. EPSRC has been supporting theoretical IP, but to a far lesser extent than applied IP.

Harmonic Analysis Harmonic analysis is very lively internationally, continuing to attract researchers of extraordinary quality, and is continually adapting its techniques to challenges from outside its traditional remit and broadening its connectivity with other areas. This has led to important cross-fertilisation of ideas with aspects of areas such as combinatorial incidence geometry (e.g. the solution of the Erdos distance problem), algebraic geometry, topology, convex geometry, structure and organisation of big data and theoretical computer science. Earlier work of Bennett, Carbery and Tao on multilinear Kakeya has led to the Bourgain-Guth-Demeter “decoupling inequalities” programme, with its profound consequences for dispersive PDE and analytic number theory (cf. the complete resolution of Vinogradov’s mean value conjectures), and to the mainstream international development of multilinear ​geometric​harmonic analysis which over the last 5 years has transformed recognition of the underlying combinatorial nature of the subject, and in which the UK plays a central role. A major strength its intra-disciplinarity, with important links with activity in PDE (notably Edinburgh/Heriot-Watt, also Birmingham, Imperial, Bath, Oxford, Sussex, Warwick and elsewhere) making it an integrated part of the broader research landscape. Aspects of contemporary harmonic analysis also feature prominently in UK research in Geometric Measure Theory (notably Warwick), Additive Combinatorics and Number Theory (notably Bristol, Cambridge, Oxford, Edinburgh) and Combinatorics (Edinburgh, Birmingham). Research quality is typically high: Bennett holds an ERC grant and won a Whitehead Prize in 2011; a highlight of international collaborations is Wright’s work on variational Carleson theorems​with​ Oberlin, Seeger, Tao and Thiele. The major UK groups are Edinburgh & Birmingham.

Dynamical Systems and Ergodic Theory These areas have grown in importance internationally and in their connectivity with other areas of pure mathematics, as indicated by the award of Fields Medals to (non-UK mathematicians) Lindenstrauss, Avila and Mirzakhani. The subject has great strength across several areas, including complex dynamics, (Rempe-Gillen, van Strien; Rippon and Stallard, with work on Eremenko's conjecture and density of hyperbolicity using and developing sophisticated tools from classical complex analysis). The subject contributed 4 UK speakers in recent ICMs: Pollicott, Marklof, van Strien (2014) and Turaev (2010), though some sub-areas, such as pure smooth dynamical systems and dynamics of several complex variables are relatively under-populated. Funding includes ERC Advanced grants (Marklof, Melbourne, van Strien), and an EPSRC Advanced Research Fellowship (Rempe-Gillen). Marklof won a Whitehead prize (2010) and was made FRS. Rempe-Gillen was awarded a Whitehead prize (2010) and a Philip Leverhulme prize (2012). Ulcigrai won a Whitehead prize (2013), EMS prize (2012), ERC starting grant (2014), Philip Leverhulme prize (2014). The main centres are Warwick, Bristol, Imperial, Liverpool, Manchester and Open University.

Stochastic Analysis The UK is well-recognised in this area. For example, Grimmett, Hairer and Lyons spoke in the 2014 ICM. Recent foci of international activity were INI programmes in Stochastic PDE 2010 and Random Geometry 2015, and the Stochastic Processes and Applications Conference at Oxford 2014. Research is typically done in international collaborations. In the pathwise approach to stochastic analysis -- rough paths and regularity structures -- the UK has a globally leading role, driven by Lyons and more recently Hairer. Hairer’s theory of regularity structures opens up a whole new field of singular stochastic PDE, which now appears to offer the right models for some important physical phenomena previously inaccessible to mathematics. Hairer was awarded the Fields Medal in 2014 for this work, also LMS Frohlich Prize, EMS Prize, FRS. He currently holds an ERC grant. Another field generating excitement in stochastic analysis is two-dimensional probability, stemming from the breakthroughs of Schramm, Lawler, Werner and Smirnov. UK activity includes the Cambridge EPSRC Programme Grant on Random Geometry, where the recent appointment of Miller is a major boost. There are well-recognised and substantial groups in probability, contributing to the UK reputation in stochastic analysis, in Bath, Bristol, Cambridge, Imperial, Oxford and Warwick, with centres of more specialized excellence in Edinburgh, Durham, Loughborough, Manchester, Queen Mary, Swansea and York. The many strong links to applied probability include work in statistical mechanics, mathematical finance, inverse problems, random networks, population genetics, and high-dimensional computation.

Linear PDE & Spectral Theory The UK is among the world leaders in spectral theory and there is arguably no city in the world that matches the combined strength of the London universities, with spectral theory of Schrodinger operators (work of Parnovski and Sobolev) particularly prominent. It is also the world leader in rigorous computational spectral analysis, e.g. E.B.Davies, N.Higham, and Trefethen. Related fields in which the UK is strong include multiscale PDE related to non-classical homogenisation (Smyshlyaev, Cherednichenko) and random matrices. Loughborough hosts a well-known group in integrable systems. Davies was awarded the LMS Polya prize and Pushnitski won an LMS Whitehead Prize in 2011; Parnovski spoke at ECM 2012. Activity includes the INI programme on Periodic and Ergodic Spectral Problems, 2015.

Complex Analysis Much of this area now consists of complex dynamics, which has already been addressed in the section on Dynamical Systems. Halburd and Mazzocco are prominent in classical complex analysis. Crowdy uses conformal mapping techniques to analyse shapes and shape evolution within applied and computational complex analysis. Several complex variables is currently absent in the UK. Crowdy holds an EPSRC established career fellowship.

Real Analysis, Convexity, Fractal Geometry, Banach Spaces, Geometric Measure Theory, Geometric Functional Analysis These areas have a large number of hard, interesting interconnected problems that are also of substantial interest in other areas (e.g. harmonic analysis, theoretical computer science) and have reached a maturity that demands very strong individual researchers in order to make substantial advances. The UK has nevertheless been prominent in this regard with the world-leading contributions of K.Ball and Preiss, and others by Maleva and Mathe. Fractal geometry is healthy: St Andrews is still strong, diophantine approximation connections are internationally leading (e.g. Velani and Beresnevich in York); recent work of Fraser (and partly also Sidorov and Walkden) and Jordan originated in ergodic theory/dynamical systems (Pollicott et al.). UK-based harmonic analysis has led to important results in both multilinear geometric functional analysis and geometric measure theory.​After​ departures and retirements at UCL there is now less UK presence in classical real analysis, convex geometry, geometry of Banach spaces etc.; in the first case this reflects worldwide trends but the latter fields have been successful internationally with 4 European prizes. Csornyei spoke at the 2010 ICM. Preiss holds an ERC Advanced grant.

Operator Algebras; Operator Theory and Banach Algebras These areas are somewhat disjoint from the rest of the analysis spectrum both intellectually and geographically. The UK has significant involvement in the structure of C*-algebras, subfactors and conformal field theory, coarse geometry. Individual researchers in other areas have contributed significantly (e.g. Todorov, Voigt). UK has no real involvement in deformation-rigidity theory of finite von Neumann algebras (a la Popa/Vaes), and limited involvement in free probability. UK highlights in Operator Algebras include: major contribution to structure of simple nuclear C*-algebras culminating in the complete classification of algebras of finite nuclear dimension with UCT, an international endeavour going on for 25 years (Santiago, Tikuisis, White, Zacharias, as well as numerous researchers worldwide). Evans’ breakthrough with Gannon (Alberta) establishing how the Haagerup subfactor can be seen through conformal field theory has led to large scale international efforts involving V. Jones to connect these subjects. A major INI programme is scheduled in this direction for 2017. 14 EPSRC grants and 1 EU network grant have been held over the period by 12 distinct researchers, totalling c. £2.3M.​Operator​ Theory and Banach Algebras have been historically strong in the north (centred around NBFAS with a large number of researchers spread across a dozen universities), and remain particularly so in Belfast, Lancaster and Newcastle, with other leading individuals elsewhere (e.g. Leeds).

Computational Analysis While primarily covered by the applied mathematics review, this area has substantial overlap with analysis, and that overlap is strategically important for the development of fundamental work in analysis, and for its applications. Areas of particular strength in the UK, and representative researchers from those areas, include Kinetic Equations (Markowich, Carrillo and Degond), Computational PDE (Barrett, Elliott and Suli), Inverse Problems (A. Stuart), Spectral Theory (Marletta), Computational S(P)DEs (Hairer, Gyongy), Approximation Theory (Trefethen), Compressed Sensing and Computational Harmonic Analysis (Tanner). The work has attracted significant recognition, including: Trefethen (LMS Naylor Prize 2013), Morton (LMS De Morgan Prize 2010), Elliott (SIAM Fellow 2015), Ortner (Whitehead Prize 2015, ERC Starting Grant), Suli (LMS-NZMS 2015 Forder Lecturer). Speaking at ICMs were A. Stuart (2014) and Markowich (2010). Speaking at the 2012 ECM were N.Higham, Iserles and A. Stuart.

Discussion of research community The demographics of UK analysis is very dependent on the subarea, as we describe below. However there are some key themes. First, it is striking how reliant most of the subject is on hiring from abroad.​This​ has had some benefits in terms of making the UK highly internationally connected in several key areas. However, overall it is a structural weakness that both EPSRC and the community must work to address. The CDTs in analysis can be expected to partly address this in the medium term; DTA funding remains essential to maintain training across all areas. A second key theme is that several areas are struggling with retention. This may be partly linked to our first point, since researchers hired from abroad may be more likely to leave. Retention is a serious issue; important areas cannot afford to lose their leaders, and it is essential that the UK provides good conditions for these people. A third key theme is that several important areas have finally begun to grow in the UK, and have done so by hiring staff of high quality, as we describe below. We make further comments by subject area, making a coarser subdivision than in the previous section.

Nonlinear PDE (including geometric analysis, mathematical relativity, calculus of variations) This currently represents a substantial proportion of total international activity in analysis, and indeed in mathematics as a whole. Many of the sub-areas linked with nonlinear PDE have begun to strengthen and grow very well in the UK over the past 5-10 years. New hiring has been done across all areas at a remarkably high level given that the subject was so small before that. However, there is a considerable way to go before the subject reaches a size comparable to other major developed nations, and the UK is still very reliant on hiring from abroad. These issues should become less acute as new recruits train new PhD students, and CDTs should be expected to help in certain areas. The UK postdoc pipeline has expanded dramatically with, for example, over a dozen postdocs moving into academic positions around the world from the Warwick geometric analysis group alone over the 5 years to 2015. A few of the key leaders in some areas of nonlinear PDE are expected to retire soon, and it is important that they are replaced. However, on average, the UK has a very young demographic in the area. Some key departments have had difficulty retaining staff. Some key areas with a young demographic are particularly vulnerable to losing their leaders abroad.

Harmonic Analysis UK activity in Harmonic Analysis is distinctly international, with most primary research -- centred around Edinburgh and Birmingham with other contributions from e.g. Bath and Imperial -- conducted with collaborators overseas. In international terms, the UK workforce in harmonic analysis is relatively small, but has grown moderately with a number of high quality early-career appointments in Bath, Birmingham and Edinburgh/Heriot-Watt (Fischer, Martini, Morris, Reguera, Oh, Gimperlein, Pocovnicu). Edinburgh/Heriot-Watt is particularly strong in both harmonic analysis and its interactions with PDE and number theory. Oh diversifies the traditional interface with PDE into deterministic and probabilistic dispersive equations, and Martini, Fischer and Gimperlein grow expertise in noncommutative harmonic analysis balancing the departure of Dooley. The CDT in Edinburgh/Heriot-Watt will help accommodate some of the large number of bright students wishing to work in this area; student numbers have remained stable elsewhere. Nevertheless, reliance upon only two main geographical locations (Edinburgh & Birmingham) continues to be a moderate threat to mid-term stability.

Stochastic Analysis This has seen rapid expansion over 10-15 years, driven for a time by mathematical finance, but now reflecting a global trend towards probabilistic ideas across both pure and applied mathematics. The UK has been well recognised for a while in the area and has been able to recruit well, mainly at a junior level and mainly from abroad. The resulting community has a balance of ages and is international in outlook. Some important areas, such as two-dimensional probability, are under-represented in the UK, but it can be hoped that this will be corrected through the capacity of the UK to recruit internationally.

Real analysis, convexity, fractals, Banach spaces; complex analysis There is a sentiment that the UK has suffered from very substantial losses of key mid-career staff and that several subareas are at risk of extinction. There have also been several excellent young appointments. In high-dimensional geometry a small number of students continue to work with K. Ball, but the overwhelming majority of young researchers in the field come from the US, France and Israel. Banach spaces generally has a better demographic. Preiss has produced a number of students in GMT, and is training a group of postdocs.​Some​ of these areas (e.g. geometry of Banach spaces) seem particularly prone to losing high quality researchers to other areas and to other locations.

Operator algebras; Banach algebras and operator theory These subareas have a different feel from other areas in that the universities involved are substantially disjoint from the centres specialising in other areas of analysis. Banach algebras and operator theory has seen a decline in numbers of researchers while operator algebras has remained fairly stable overall, with new researchers brought in particularly from Germany and Canada to replace retiring staff. In operator algebras most researchers are towards the beginning of their careers; the strength of this group of researchers as indicated by their success rate with EPSRC first grants looks optimistic for the medium to long term. After Winter’s move, Nottingham no longer has a presence. Operator Theory and Banach Algebras have been historically strong in the north (centred around NBFAS with a large number of researchers spread across a dozen universities), and remain particularly so in Lancaster, Newcastle and Belfast with other leading individuals elsewhere (e.g. Leeds).

Linear PDE, spectral theory and applications This subarea also has distinct demographics. The field has been established and successful in the UK for some time, but is not regenerating itself internally. There is a marked geographical concentration of activity in London, dominated by the Russian school. Of senior people, Safarov is no longer with us and Davies has retired.

Inverse problems Inverse Problems has recently established itself in the UK (e.g. Centre for Inverse Problems in UCL) but in theoretical IP, the UK is viewed as being behind USA, France, Germany and Finland.

Dynamical systems and Ergodic theory In several universities dynamical systems has grown or even strengthened over the last 10 years (Bristol, Warwick, Imperial), whereas other groups have shrunk (e.g. Surrey) or have had changes of emphasis and personnel (Queen Mary, Liverpool and Manchester).

Computational Analysis The CDTs in Analysis have included Computational Analysis, which has strengthened links at that interface in those universities. From the Computational Analysis community, there is however a concern that the UK pipeline for researchers will not deliver research leaders in that area soon enough to replace retiring staff and, the CDTs notwithstanding, will not deliver in the numbers needed in the long term.

Interdisciplinary and intra-disciplinary connections We review firstly interdisciplinary connections, then intradisciplinary, emphasising recent achievements and promising developments.

Nonlinear PDE makes connections with areas in Engineering, Physics and more recently Biology and Social Sciences; in particular kinetic, hydrodynamic and hyperbolic/parabolic/elliptic PDE have had important successes in modelling chemotaxis, collective behaviour of animals, cells and bacteria, computational neuroscience, crowd dynamics, and networks and opinion formation, while the classical physical origin of these ideas in rarefied gas dynamics, semiconductors and plasmas also remain important fields of research.

Ideas from harmonic, linear, complex, geometric and stochastic analysis, and inverse problems, are critical to the efficiency of algorithms for large-scale computations, across many fields of application, for example in medical tomography, signal and image processing, compressed sensing, mathematical finance, and the analysis of algorithms in computer science. Data science, to which analysis has already made key contributions, will pose important new challenges for analysis over the coming decade. We give three particular examples of work already underway.​The​ work of A.Stuart on Bayesian inverse problems in infinite dimensions, in particular addressing problems in weather forecasting, builds on many techniques from stochastic analysis. Secondly, topological data analysis brings ideas from analysis and geometry to bear on the complex structures present in data, for example, the engagement of Brodzki with groups working in energy and clinical trials. Thirdly, the Centre for Inverse Problems at UCL is building links with fields from tomography to relativity.

Some​of​the currently most exciting areas in Analysis address questions from Theoretical Physics. The mathematics of General Relativity has seen an explosion of activity and successes. Key groups at Cambridge and Imperial have strong connections to the physics community. Secondly, continuum limits in Statistical Physics are progressively coming within the range of rigorous mathematics, first in the planar case (Schramm-Loewner evolutions) and now more generally through the emerging theory of singular stochastic PDE created by Hairer.

By way of an example to show that mathematical understanding has long-term benefits, we mention the work of J. Ball on martensites in Materials Science 25 years ago, which continues to influence current technological advances, for example in alloys that can change shape in response to temperature millions of times without significant damage.

We turn to the connections of Analysis within mathematics. As a first comment, the inter-connectivity of many sub-areas in Analysis is markedly greater than 20 years ago, perhaps as the trend to address more concrete problems calls for all available techniques acting together. This is also noticeable in and encouraged by the form of the Analysis CDTs.

Geometric Analysis has multiple connections with most of the other sections covered by this document, and it is common to have papers and workshops crossing the boundaries. It has recently solved a succession of famous problems in other areas, such as differential geometry and topology. The field also crosses over to applied mathematics in areas such as physics and biology (particularly problems in which there is a natural underlying energy such as materials science or the study of cell membranes) and image processing.

Harmonic analysis has important and diverse interactions across mathematics. While PDEs remains the largest single area of interaction, other well-established and active connections within mathematics are numerous, including additive combinatorics (Gowers, Green, Sanders, Wolf in the UK), combinatorial and incidence geometry, algebraic geometry, convex and geometric analysis, functional analysis, geometric measure theory, information theory, analytic number theory (cf. Woolley’s programme, Heath-Brown, Maynard), probability and statistics, representation theory, spectral theory and quantum mechanics.

Ergodic theory and dynamical systems have solved problems in analytic number theory (Davenport Conjecture and progress on Littlewood Conjecture), geometry (e.g. geodesic flows, group actions) and analysis (e.g. dimension of attractors of certain PDE’s, theory of quasi-conformal mappings).

Operator Algebras are the foundational objects for non-commutative differential geometry, and interact with topology via K-theory (e.g. the Baum-Connes conjecture approach to the Novikov conjecture), ergodic theory (e.g. Popa's work on finite von Neumann factors), groups (connections to geometric group theory, ongoing developments of approximation and rigidity properties in groups) and stochastic analysis via non-commutative probability. Through subfactors they connect with algebraic, conformal and topological quantum field theories, and with vertex operator algebras and link invariants including Khovanov homology.

Finally, notwithstanding the difficulties in communicating analysis to a general audience, there have been some effective public lectures, notably in schemes run at ICMS and INI, and by the LMS (e.g. Hairer’s LMS Popular Lecture).

Future directions and opportunities We highlight a selection of the many directions within analysis that we expect to be particularly fruitful in the future. The current trend of increased connections with other areas of mathematics should continue, with exchange of ideas and applications in both directions, and we will surely see continued applications of analysis to science and for society in general. The demand for young people in analysis appears likely to exceed supply for the foreseeable future.

Applications to Data Science. This is a major new area to which analysis expertise is relevant and presents a new challenge to our community to connect. A clear success already, relying deeply on Analysis, has been the development of Compressed Sensing. We believe there is much more to come, not only in Compressed Sensing but in the deployment of techniques from stochastic, harmonic and other parts of analysis such as the real variable methods being developed by Coifman et al.. One area of data science that will continue to draw on techniques from many different parts of analysis is that of image analysis. The new Alan Turing institute should be helpful in facilitating connections between analysis and data science in general.

There are many important challenges in Inverse Problems. These include developing rigorous underpinnings for statistical / stochastic methods in nonlinear inverse problems, understanding inverse problems for nonlinear PDE (such as for models in fluids, elasticity, relativity) and understanding discretization effects in inverse problems, such as applying a continuous model to discrete data. Photo-acoustic medical tomography, and other problems with a multi-type response, also present new challenges.

Promising connections between geometric analysis and other areas of mathematics are being made faster than they can be explored, and this should lead to rapid development of the subject over the medium term. Emerging areas include the fusion of probability theory and geometry in the context of the analysis of metric spaces. Following Perelman’s proof of the Poincare conjecture, there is increasing interest in PDEs governing geometric flows on manifolds as analytic techniques open new possibilities in geometry. This topic is being led partly by researchers in the UK, as is the hot topic of min-max theory.

In mathematical relativity, the big open problems are ahead of us: stability and instability for black holes without symmetry, the celebrated cosmic censorship conjectures of Penrose, and the analysis of asymptotically anti de Sitter spacetimes—the latter playing a key role in the theoretical physics community. The interdisciplinary overlap with theoretical physics is continuing to grow as the new mathematical results in these areas have sparked the interest of physicists, as well as connections with microlocal analysis.

We can expect a strengthening of cross-disciplinary links between mathematicians in kinetic, hydrodynamic and hyperbolic PDE and quantitative modelling in mathematical biology and social sciences.

Singular stochastic PDE. Just as the natural driving noise in stochastic ODE is too rough to use classical theory, so it is true in PDE. Although stochastic PDE is a well-established area, until Hairer’s recent breakthrough, many of the physically natural driving noises had appeared too rough for a mathematical theory, but no longer, presenting a huge opportunity to mathematics. The next decade will see intense activity as this is worked out in detail, in which the UK can take a leading role.

The new landscape in harmonic analysis described above, which has already had dramatic interactions with PDE and number theory, is likely to lead to exciting future developments even in the short term. The multilinear perspective is complemented by other emerging movements within harmonic analysis which also aim to understand oscillatory and cancellative objects, such as oscillatory and singular integrals, in much more transparent geometric terms. This is likely to lead to yet deeper connections with discrete and algebraic geometry and with geometric measure theory. Challenges in functional analysis include the development of analysis on metric spaces, quantitative geometry and the theory of embeddings into Banach spaces.

Ergodic theory is an area which keeps reinventing itself, with applications in mathematics and physics. Ideas from complex dynamics are likely to keep impacting complex analysis heavily, through such notions as quasiconformal and quasisymmetric rigidity.​Avila’s​ Fields winning work shows that a dynamical systems approach can be key to solve seemingly inaccessible problems in analysis and geometry (see e.g. Ghys’ laudation talk on the central role of renormalisation, a concept originating in Mathematical Physics).

There are strong connections between operator algebras and ergodic theory; these connections are less present in the UK, despite strength in both communities, and there is potential for the UK to develop a leading position in this area. A forthcoming INI programme focusing on operator algebras, subfactors and applications will help connect UK representation theory and algebraic geometry to progress in subfactors as has happened elsewhere in the world.

The interplay between spectral theory, random matrices, stochastic processes and number theory is becoming more important.

ANNEXE

Main Research Groups in Mathematical Analysis:

Geometric analysis: ​Principally Warwick and Imperial, with a high quality presence at Cambridge, Oxford, Bath, QMUL, UCL, Kings.

Mathematical Relativity: ​Cambridge, Imperial and Oxford.

Calculus of Variations, materials science, homogenisation: ​Oxford, Warwick, Bath, Bristol, Sussex.

Other ​nonlinear PDE:​Oxford, Warwick​ ,​Cambridge,​ Reading, Edinburgh.

Inverse problems:​UCL, Warwick. ; Harmonic Analysis: ​Edinburgh/Heriot-Watt, Birmingham also Imperial, Bath.

Dynamical systems: ​Warwick, Bristol, Imperial, Liverpool, Manchester, Open University. ; Linear PDE & Spectral Theory:​London Colleges also Cardiff, Edinburgh, Heriot-Watt, Loughborough.

Complex Analysis:​Open University, Imperial, individuals in several other institutions

Operator Algebras:​Glasgow, Aberdeen, Lancaster, Cardiff, Southampton. ; Operator Theory and Banach Algebras: ​Newcastle, Lancaster, Belfast also Leeds. ; Real Analysis etc.:​St Andrews, Warwick also Manchester, York. ; Stochastic Analysis:​Bath, Bristol, Cambridge, Imperial, Oxford, Warwick also Edinburgh, Durham, Loughborough, Manchester, Queen Mary, Swansea, York.

Computational analysis:​Imperial, Warwick, Cambridge, Oxford.

Details of statistical analysis

The immediately following statistics were derived from data culled from Mathematical Reviews on two separate dates in late 2015 by Rowena Stewart, the University of Edinburgh Science Librarian. In each case the raw data refers to the total numbers of papers reviewed in Mathematical Reviews over the time period (calendar years) in question with indicated primary MSC, on one hand for papers including a UK authorial address, and on the other for papers without prejudice to address.​Some​ caution is required in interpreting this data: for example there is no requirement that a UK author be associated to a department of mathematics, but could instead be associated to a department of computer science or philosophy, for example. (This effect may be particularly significant for Logic.)

Scenario 1:​Analysis is greedy and includes MSC classifications 22-52 inclusive plus 54, 58, 60.

Scenario 2​: Analysis is shy and includes only MSC classifications 26-49 inclusive. MSC 22 (Topological groups and Lie Groups) is allocated to Algebra; 52, 54 and 58 are allocated to Geometry and Topology, and 60 (Probability and Stochastic Processes) is counted separately.

Note: (i) Relativity (MSC 83) is counted separately in all cases as it is not possible to delineate between contributions in analysis, geometry and applied mathematics here. (ii) Algebraic geometry (MSC 14) and K theory (MSC 19) have been allocated to Geometry and Topology in all cases. (iii) Category Theory and Homological Algebra (MSC 18) has been allocated to Algebra in all cases. (iv) Numerical analysis has not been included, nor any aspects of analysis or any other area appearing under MSCs after 61, with the exception of Relativity (MSC 83).

TABLE 1: Scenario 1 outputs 2010-2014 inclusive

UK outpuss% share of UK World outputs% share of world UK % contribution to world outputs outputs total

Comb’torics 1140 10 23125 9 4.9

No. Thy 718 7 13978 5 5.1

Logic 1042 10 12729 5 8.2

G & T 1132 10 23505 9 4.8

Algebra 1524 14 28035 11 5.4

Analysis 4724 43 152738 58 3.1

Relativity 607 6 7227 3 8.4

Totals: 10887 100 261337 100 4.2 (avge)

TABLE 2: Scenario 2 outputs 2010-2014 inclusive

UK outputs% share of UK World outputs% share of world UK % contribution to world outputs outputs total

Comb’torics 1140 10 23125 9 4.9

No. Thy 718 7 13978 5 5.1

Logic 1042 10 12729 5 8.2

G & T 1556 14 37878 14 4.1

Algebra 1589 15 29677 11 5.4

Analysis 2947 27 118947 46 2.5

Probability 1288 12 17776 7 7.2

Relativity 607 6 7227 3 8.4

Totals: 10887 101 261337 100 4.2 (avge)

TABLE 3: Scenario 1 outputs calendar year 2014

UK outputs% share of UK World outputs% share of world UK % contribution to world outputs outputs total

Comb’torics 237 11 4482 9 5.3

No. Thy 143 7 2746 5 5.2

Logic 187 9 2463 5 7.6

G & T 236 11 4666 9 5.0

Algebra 259 12 5538 11 4.7

Analysis 949 44 29486 58 3.2

Relativity 154 7 1351 3 11.4

Totals: 2165 101 50732 100 4.3 (avge)

TABLE 4: Scenario 2 outputs calendar year 2014

UK outputs% share of UK World outputs% share of world UK % contribution to world outputs outputs total

Comb’torics 237 11 4482 9 5.3

No. Thy 143 7 2746 5 5.2

Logic 187 9 2463 5 7.6

G & T 320 15 7263 14 4.4

Algebra 271 12.5 5812 11 4.7

Analysis 575 27 23189 46 2.5

Probability 278 13 3426 7 8.1

Relativity 154 7 1351 3 11.4

Totals: 2165 101.5 50732 100 4.3 (avge)

For purposes of replication we have made the following allocations: ; ; ; Combinatorics: MSC 5 Number Theory: MSC 11 Logic: MSC 3,6 Relativity MSC 83 and​ ​: ; ; Scenario 1​: G&T: MSC 14,19,53,55,57 Algebra: MSC 8,12,13,15,16,17,18,20 Analysis: MSC 22-52 inclusive, 54,58,60 ; ; Scenario 2​: G&T: MSC 14,19, 51-58 inclusive Algebra MSC 8,12,13,15,16,17,18,20,22 Analysis MSC 26-49 inclusive; Probability MSC 60.

We have not provided any more detailed analysis of subgroups within Analysis as boundaries are very artificial.

EPSRC has provided estimated numbers of UK active researchers working in each of the areas Combinatorics (100), Number Theory (87) , Logic (27) , Geometry and Topology (210), Algebra (228) and Analysis (225). These are used in the following table of productivities defined as (Number of UK papers in the area over 2010-2014)/(number of UK academics in the area) under Scenario 2 as defined above. Pure Maths is taken to be the union of the areas mentioned above. The estimates below come with the caveat that the classification in the EPSRC data may correspond only roughly with the classification implicit in our use of MathReviews data.

TABLE 5: Scenario 2 UK productivities 2010-2014 inclusive (excluding relativity and probability)

UK outputs UK researchers Productivity

Analysis 2947 225 13.1

Pure Maths 8992 877 10.3

Conclusions: (i) One broad conclusion from the EPSRC data, (roughly consistent with the figures obtained by Iain Gordon), is that Algebra, Geometry & Topology and Analysis have broadly comparable numbers of UK researchers. A second broad conclusion from Table 5 is that Analysis is performing well on productivity.

(ii) Comparison of the cumulative 5 year data with the 2014 snapshot (Tables 1-4) reveals a good deal of stability, especially world-wide.

(iii) If for the sake of argument we consider Pure Mathematics as comprising MSC 2 -- 60 (with the endpoints included or excluded) we see that Analysis accounts for around 58% of Pure Mathematical activity as measured by papers reviewed in Math Reviews internationally and 43% within the UK under Scenario 1, and 46% internationally and 27% UK under Scenario 2. It therefore appears that Analysis in the UK is significantly underrepresented in comparison with international norms.

MATHEMATICS IN BIOLOGY & MEDICINE

Definition and Overall Comments:

Mathematical Biology and Medicine covers research into the development and application of state- of-the-art mathematical and/or statistical tools and techniques to investigate biological processes and systems, including those of relevance to the medical sciences. It includes research into the mathematical and/or statistical treatment of biological processes operating at any spatial or temporal scale, or over multiple scales, from the genetic and molecular levels to the whole population level. The UK has a long tradition of world leaders in this field (Fisher, D'Arcy Thompson, Turing, Lighthill, Murray, Anderson, May, to name but a few).The breadth and depth of UK research in this area (both in enriching other mathematical disciplines as well as impacting the biological and medical sciences) is such that the UK is widely acknowledged as the world leader in this area. As such, it is surprising that the subject area is not recognized by the EPSRC on its website: https://www.epsrc.ac.uk/research/ourportfolio/themes/mathematics/

The purpose of this document is to provide an updated view of the landscape in Mathematical Biology and Medicine reflecting several significant changes since the 2010 landscape document. More specifically, it answers the questions posed in the matrix.

Statistical Overview:

In a highly multi- and inter-disciplinary field like mathematical biology and medicine, to estimate numbers is an extremely difficult if not impossible task. Based on REF data, EPSRC estimate about 170 researchers who are core mathematical biologists. However, this is in mathematics and therefore does not take into account researchers based in biology, biochemistry, clinical medicine, computer science, ecology, engineering, pharmacology, physiology, physics, zoology, etc. who would have been submitted to different panels. Since the last document we have seen a very significant increase in the number of mathematical researchers who have now added mathematical biology to their research portfolio. We would therefore estimate that a more accurate figure would be in excess of 400 UK faculty researchers working in this area. Furthermore, EPSRC data suggest that demand for EPSRC Fellowships is strong across the career stages, indicating a healthy distribution of researchers from post-doctoral to established career stage.

Subject breakdown:

There is a strong tradition of researchers within Mathematics departments in the UK contributing to research on biological and medical questions. This has not only contributed to answering specific questions in the life and medical sciences, but also it has enhanced existing and developed novel mathematical methodologies in creative and innovative ways. Examples include continuum mechanics, numerical analysis, nonlinear systems, applied analysis, discrete mathematics, stochastic modelling and statistical methodologies. Over the last decade, the life and medical sciences have become progressively more data-driven and quantitative. This has led to a steadily growing demand for researchers in mathematical biology and medicine investigating a rapidly widening diversity of problems. Application areas include ageing, agriculture, antimicrobial resistance, bacterial research, biological fluid dynamics, clinical medicine (including cancer and cardiology), developmental biology, ecology, epidemiology, evolution, immunology, molecular mathematics, networks, neuroscience, plants and fungi, pharmacology, regenerative medicine, soft tissue mechanics, systems biology. Discussion of research areas:

By subject area, a frank assessment (both positive and negative) of past and current international standing of the field, supported by reference to highlights as appropriate

The mathematical biology and medicine community within the UK goes from strength to strength and is widely acknowledged to be leading the world in this field. A small sample of examples of recognition (since 2010):

 Fellows of the Royal Society (Etheridge, Goldstein, Maini, McVean and Tavare)  Fellows of the Royal Society of Edinburgh (Grebogi)  ERC grants (Gudelj and Roose)  The two most distinguished awards internationally in the science of mechanics (G K Batchelor Prize for Fluid Mechanics and Rodney Hill Prize for Solid Mechanics) are both going to members of UK mathematics departments for their work in biological applications (Goldstein and Ogden, respectively).  Humboldt Senior Prize (Politi)  Tim Pedley FRS has been elected the Raman Chair for 2016 (Indian Academy of Sciences), which is awarded every two years and covers the whole of science.  LMS Whitehead Prizes (Baker, Porter and Waters)  Royal Society Gabor Medal (McLean and Simons)  EPSRC Leadership Fellows (Beardmore)  Numerous Wolfson Merit Awards

The UK mathematical biology and medicine community is highly active and very well represented at international conferences and there is also a great deal of activity within the UK. Due to the fact that the subject now pervades applied mathematics, statistics, numerical analysis, pure mathematics, chemistry, the biological and medical sciences, as well as ecology and epidemiology, it is impossible to list the incredibly vast amount of activity in which UK researchers are involved. Therefore, we give only a small sample of examples which do not even begin to cover the tip of the iceberg:

 The joint ECMTB and SMB meeting (the biggest conference of the decade in the field) will be held in Nottingham this year, with 2 of the 8 plenary speakers from the UK (Baker (Oxford) and Gog (Cambridge)).  UK mathematicians (eg, Ashwin (Exeter), Coombes (Nottingham), Maini (Oxford) and Twarock (Leeds)) are involved in the organisation of semester programs at the NSF-funded Mathematical Biosciences Institute (MBI) in Ohio, which operates similarly to the Isaac Newton Institute.  The EPSRC-funded interdisciplinary network MMEMS (Mathematical Models and Experimental Microbial Systems) run by Gudelj and Beardmore (both Exeter) has so far organised 7 international conferences on a range of evolutionary topics designed to bring together mathematical and biological communities. Held across the UK the conference hosted invited speakers from 4 continents and 13 countries (details can be found at www.mmems.org)  The Newton Institute held a short programme on infectious disease dynamics (Aug – Sept 2013), a 6-month programme on Partial Differential Equations (July-Dec 2015) with the major focus being mathematical biology and is currently running a programme on Stochastic Dynamical Systems in Biology (Jan – Jun 2016).  Several mathematical biology programmes have been held at ICMS, including in 2015, Computational and Multiscale Mathematical Modelling of Cancer Growth and Spread (Chaplain (St Andrews), Lin, Trucu (both Dundee)) and on Mathematics for Health and Disease (Lythe and Molina-Paris (both Leeds)).  Lythe and Molina-Paris coordinate the FP7 Initial Training Network on Quantitative Immunology (2013-2017) and the IRSES FP7 INDO-EUROPEAN Research Network on Mathematics for Health and Disease (2013-2017).  Molina-París will chair the first ever session on Mathematical and Computational Immunology hosted in the Immunology Congress (the largest immunology congress that brings together the American Immunology Association, as well as the EU and the Australasian Associations) (Melbourne, August 2016).  Madzvamuse (Sussex) is a key partner in the Europe Horizon2020 MSCA-ITN on Research Training Network Integrated Component Cycling in Epithelial Cell Motility – an international PhD programme offering early-stage researchers the opportunity to improve their research and entrepreneurial skills and enhance their career prospects (11 universities, 4 research institutes and 4 industrial companies).  Many UK trained mathematical biologists are now major figures in centres worldwide, allowing the UK to have a significant impact on the subject across the globe.

Discussion of research community

A brief description of demographic trends, e.g. composition, origin and volume of research student pipeline, age distribution of academic staff, etc.

A substantial number of new full time faculty appointments of young people has been made since 2010 in many universities, including Aberdeen, Bath, Birmingham, Bristol, Cambridge, Dundee, Durham, Edinburgh, Exeter, Glasgow, Imperial, Liverpool, Nottingham, Oxford, Sheffield, St Andrews, Surrey, Swansea, York, Warwick, so there are many young people coming into the field. The lack of EPSRC funded CDTs in mathematical biology, coupled with the limited DTA funding, is a big blow for the subject area and will cut off its life supply, challenging the long-term sustainability of the excellence for which the UK is renowned. This also comes at a time when most mathematics departments now have at least one undergraduate lecture course in mathematical biology and these tend to be very popular, increasing the number of people wishing to pursue the subject at graduate level. It is also very noticeable that the subject area seems to attract more women than many other areas of mathematics.

There are still relatively few senior people in this field (as it is a comparatively young field) compared to many other subject areas and so the subject suffers from lack of representation on policy making boards. However, there are many UK researchers on editorial boards of major journals.

Inter/Intra-disciplinary activities and Engagement activities:

A description of connections with other areas of mathematics, science, engineering, etc

There has been a huge increase in the ways in which mathematics is now integrated into the scientific method outside university mathematics departments. Again, it is impossible to cover the vast amount of activity in this area so we illustrate with a small sample of the many examples of this:

 Mathematical models encoded in software are now being used widely. Examples include: the Medical Devices Division of the FDA in the US and two pharmaceutical companies are now testing cardiotoxicity of potential new drugs (Gavaghan, Oxford); development of a drug by a major pharmaceutical company to treat dementia and in an epilepsy industrial project in Switzerland (Schelter, Aberdeen); novel 2-D and 3-D whole cell tracking algorithms based on optimal control of PDEs are currently under commercialisation with IBiDi GBMH (Germany) as well as being adopted by experimental laboratories in the Netherlands and Germany (Madzvamuse).

 Tindall (Reading) leads the UK network on Quantitative Systems Pharmacology (QSP) funded by the EPSRC and MRC. The network is jointly organised by academia and industry. QSP is a new emerging area of science bringing together people integrating lower level genetic/molecular scale (systems biology type) models with higher scale ones at the pharmacokinetic pharmacodynamic (PKPD) level in order to aid drug development.

 Ogden's (Glasgow) constitutive law for large arteries is now incorporated in commercial software packages such as Abaqus.

 An EPSRC sandpit on Predictive Modelling for Healthcare Technologies through Maths was held in September 2012, and around £4.5M was allocated to projects developed at the sandpit. Follow-on funding from EPSRC led to the POEMS network (Predictive mOdelling for hEalthcare through MathS) , which aims to stimulate new mathematical modelling approaches for applications in medicine and healthcare. Its remit ranges from multiscale models of cells, tissues and organs, through to models of individual patients and their trajectory through healthcare.

 Sheffield has invested in the creation of an institute (Insigneo) entirely dedicated to in silico medicine, with close links to MultiSim 2015, the EPSRC Frontier Award centre which is developing a computational framework for multiscale modelling of the musculoskeletal system.

 Since the last document, funding for the 6 BBSRC Systems Biology Centres has ceased and several of them no longer exist. However, they have all seeded significant ongoing research activity, mostly funded by BBSRC, while EPSRC is establishing new centres (on a smaller scale) through the Healthcare Initiative. To date, 5 have been supported: o EPSRC Centre for Predictive Modelling in Healthcare led by Professor John Terry (University of Exeter) o EPSRC Centre for New Mathematical Sciences Capabilities for Healthcare Technologies led by Professor K Chen (University of Liverpool) o EPSRC Centre for Multiscale Soft Tissue Mechanics - with application to heart & cancer led by Professor Raymond Ogden (University of Glasgow) o EPSRC Centre for Mathematics of Precision Healthcare led by Professor Mauricio Barahona (Imperial College London) o EPSRC Centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging led by Professor John Aston (University of Cambridge)

 At least three of the RCUK’s Synthetic Biology Research Centres (Edinburgh, Nottingham and Warwick) are incorporating mathematical modelling.

 The BBSRC has funded a Multiscale Biology Network, led by Nottingham, within which mathematical biologists play a major role; the 10-person steering group includes five mathematicians (Owen and King (both Nottingham), Baker, Byrne, Molina-Paris).

 The University of Exeter is rapidly growing its capacity for research in mathematical biology and healthcare. The “Living Systems Institute” (led by Professor P. Ingham FRS and opening in 2016) will act as an interdisciplinary research centre that links mathematics, physics and computer science to a range of biological and medical teams. This will supplement the current activities of the EPSRC Centre for Predictive Modelling in Healthcare (PI Professor J. Terry, 2015-2018) and the Wellcome Centre for Biomedical Modelling and Analysis.

 The Durham Biophysical Institute involves various departments, including mathematics and is pursuing several interdisciplinary projects.  The Wellcome Trust has just funded 15 studentships at the University of Manchester in “Quantitative & Biophysical Biology”. This demonstrates external demand for mathematical expertise combined with structured cross-disciplinary training, a gap which EPSRC could be filling (perhaps in alliance with other Research Councils).

 In ecology, the Forestry Commission has implemented new policies on preventing the grey squirrel invasion in Northern Scotland based on model predictions by White and colleagues (Heriott-Watt).

 In Cambridge, Professor S Tavare FRS (probability and statistics, and now working in mathematical models of cancer), has been Director of the Cancer Research UK Cambridge Institute since 2013.

Future Direction/Opportunities:

Emerging subject areas, potential topics of cross-disciplinary overlap.

As more data become available, the major challenge for the subject will be to develop models that span the full range of length and time scales characterising biological systems and that can be validated experimentally through the predict-test-refine-predict cycle.

Modelling across scales requires “traditional” applied mathematics to merge with stochastic and computational approaches. There are opportunities for collaborations with statisticians on parameter estimation and identifiability, model selection, uncertainty quantification etc. and with computer scientists to develop rigorous approaches to move from one scale to another. Furthermore, comparing spatiotemporal models with the vast amounts of imaging (itself a highly mathematical area requiring further development) data now available requires statistical and spatially geometric descriptions of complex, time evolving structures, and analysis of large systems of ordinary differential equations, such as for whole-cell models, should benefit from advances in algebra and network theory. Moreover, there is a pressing need for open-source, well-engineered, fully tested, validated, and easily reusable software. However, it is important that funding core applied mathematics which, in many cases, underlies the modelling work that precedes the above and is essential in guiding further experiments, is not compromised.

It is becoming more and more apparent that some problems inspired by biology require approaches from the pure disciplines (such as analysis, algebra, probability, or operational research). Theory in differential geometry is now being applied innovatively through the use of geometric PDEs to unravel new mathematical models characterising the spatio-temporal dynamics in cellular biology, while in cancer it is becoming clear that the spatiotemporal distribution of the vasculature plays a very significant role in determining the success of many therapies, requiring us to use ideas from geometry to inform therapeutic protocols.

It is important to distinguish these challenges from Big Data. Questions in the latter may be more tractable (at the moment), but will only provide us with information on statistical correlations (important though this information is). In contrast, mathematical biology aims to deliver an understanding of mechanism, which is altogether more powerful, giving fundamental insights into biology and having greater potential for transforming biotechnology and medicine. Developing such an understanding is, however, a longer term investment. The immediate big challenges for the subject are of two types: first there are the increasing demands of biologists for relevance to their complex applications; second, there is the need for the subject to maintain contact with the broader mathematical community. Examples of new areas of mathematics include the work of Leinster (Edinburgh) in measures of diversity for biological populations, Twarock on the geometry of viruses, Buck (Imperial) on using ideas in topology in DNA, Girolami (Warwick) and Husmeier (Glasgow) in statistical parameter estimation for models (eg PDEs) in biology which brings potential connections to data science.

Many opportunities exist for embedding mathematics in the biological and medical sciences, including plant science and ecology (for example with respect to climate change). Furthermore, the new area of mathematical pharmacology and mathematical modelling of resistance evolution brings in strong links with industry, including Unilever, Pfizer, Syngenta, AstraZeneca, GlaxoSmithKline etc involved in drug development and we are increasingly seeing journal articles jointly authored with researchers from the pharmaceutical industry.

Further opportunities exist to exploit programs with NC3Rs and MRC which currently focus heavily on immediate practical applications, leaving a gap to fundamental research, which could be bridged by EPSRC. An often overlooked skill is that of modelling which is not so prevalent in traditional areas of applied mathematics where the models are known, but is a great challenge facing researchers in biology, ecology and medicine, where the science determines the type of model that is necessary and this then requires modellers who have a broad range of technical skills which are not necessary in other areas of applied mathematics. This is best taught at the graduate level, once again emphasising the need for graduate programmes in mathematical biology.

In the context of “Big Data”, one of the major challenges is to use mathematical descriptions of genomic and transcriptomic datasets to help our understanding of antimicrobial resistance evolution and the genetic basis of wider health problems. At some level, this is data mining that seeks human health correlates, while at other levels it is about understanding the mechanistic basis behind evolutionary trajectories in pathogens.

Further Comments:

Any further comments (both positive and negative) which are related to subjects not covered by the other section headings as appropriate.

Mathematical biology encompasses a vast breadth of mathematics. Depending on the problem being solved, researchers in mathematical biology and medicine need expertise in ODEs, PDEs, discrete systems, stochastic differential equations, probability, combinatorics, game theory, network theory, control theory, integro-differential equations, numerical analysis and scientific computing. However, it is no longer enough just to develop a model as the emphasis is now on the modeller to validate the model, and this requires a whole set of other mathematical tools. Therefore, research in this area now follows the pipeline from basic science through model building, analysis, validation, refinement and prediction. Funding mechanisms need to be sufficiently broad and flexible to encompass this inherent complexity.

There seems to be a misconception amongst the research councils that a single grant is sufficient for a research project to span this whole pipeline. This is simply not plausible and it would greatly benefit the mathematical biology community if the research councils and charities (eg Wellcome, CRUK etc) could work together to provide realistic levels of funding for doing this (which would also result in postdoctoral researchers acquiring the broad range of multidisciplinary research skills which industry requires, and in realising real impact of mathematics), as well as EPSRC continuing to hold its unique position in funding research focussing on particular aspects of the pipeline. In the first round of nationwide calls for CDTs, no CDT in mathematical biology was funded, although the subject does obtain some funding through systems biology, synthetic biology and industrial CDTs. A large part of the community recognised that the theme “New mathematics for biology” represented a major misunderstanding of what mathematical biology is. In the past 5-10 years the subject has gone beyond producing mathematical models that give behaviour “looking like” the biology, and today’s focus is scientifically rigorous validation by comparison with data. In short, it is now “mathematics leading to new insights into biology” – the unnecessary emphasis on new mathematics takes away the huge opportunities that exist for using established mathematics in novel, creative ways to make an impact on biology (this can be either through new application areas, or through a combination of different mathematical approaches). The present-day lack of a graduate student pipeline is a serious blow for mathematical biology and will have substantial long term consequences on the UK’s international standing in the field unless it is quickly addressed. Furthermore, students do not have sufficient funding to attend research programmes within the UK.

In short, the lack of any CDTs in mathematical biology is very debilitating for the subject (coupled with the fact that individual departments have very little, if indeed any graduate student funding). Mathematical biology courses are typically very popular at undergraduate level and there is huge demand from students to continue working in this area.

Furthermore, there is an insufficient number of thematic calls, resulting in a lack of postdoc funding that is essential to provide young researchers with a viable career track. The view that research must contain both new mathematics and new biology means that researchers in this interdisciplinary area need to be twice as good as their mono-disciplinary colleagues in order to compete. This is totally unrealistic and is a major issue that we faced in the 1990s and was satisfactorily resolved by joint EPSRC-BBSRC initiatives. These initiatives were extremely successful and it has always puzzled the community as to why they were discontinued. It would also be beneficial for the community to have more opportunities for mathematicians and experimental biologists to jointly apply on an equal footing for other large scale funding, e.g. Programme Grants, via joint EPSRC-BBSRC/NERC/MRC/Wellcome/CRUK/NC3Rs initiatives.

Moreover, the expectation that interdisciplinary research can do done on the same timescale as single discipline research (doctoral theses, postdoctoral fellowships), and explained in detail on the same forms for grant applications, is detrimental to the pursuit of world class, truly interdisciplinary, research. To do cutting edge mathematical biology research requires experimental validation, and the BBSRC and MRC appear to be more open to this through postdoctoral funding than does EPSRC.

The REF can have a very negative impact on interdisciplinary areas. For example, the mathematics panel will be primarily judging mathematical content and therefore this can discourage truly interdisciplinary research. This has been recognised by a number of bodies (it is not only an issue for mathematics) and is being fed into the present REF consultation. On the other hand, the area offers the opportunity for impact which can only be useful for the health of mathematics as a whole.

Key Research Groups in the area:

A summary of the key research groups active in this area.

There are several research groups, not necessarily in mathematics departments, which cover a huge range of subject areas in mathematical biology (agro-ecology, biological evolution, brain disease, cancer, cardio-vascular, development, ecology, epidemiology, immunology, network science, predictive medicine, pharmacology, systems biology, synthetic biology, tissue engineering, plants, diabetes, to name but a few areas). These include: Aberdeen, Bath, Birmingham, Bristol, Cambridge, Dundee, Durham, Edinburgh, Exeter, Glasgow, Heriot-Watt, Institute of Food Research (Norwich – BBSRC funded), Imperial College London, John Innes Centre (Norwich), Leeds, Leicester, Liverpool, Loughborough, Manchester, Nottingham, Oxford, Reading, Royal Holloway, Sheffield, Southampton, St Andrews, Stirling, Strathclyde, Surrey, Sussex, Swansea, University College London, Warwick, York.

Bibliography:

Any references made in completing this matrix We contacted 40 mathematical biologists from 30 UK universities and research institutes for written comments on all the above issues before putting this document together. Combinatorics

Lead authors: D. Kr´al’ (Warwick), D. Kuhn¨ (Birmingham), J. Wolf (Bristol) Contributors: S. Blackburn (RHUL), P. Cameron (St Andrews), A. Czumaj (Warwick), A. Dawar (Cambridge), B. Green (Oxford), J. Hirschfeld (Sussex), D. Osthus (Birmingham), O. Riordan (Oxford), N. Ruˇskuc (St Andrews), A. Scott (Oxford), E. Steingr´ımsson (Strathclyde), A. Thomason (Cambridge)

1. Overview Combinatorics is a rapidly evolving field of mathematics with connections to many areas, e.g. prob- ability, algebra, analysis, number theory, discrete optimisation, statistics, theoretical computer science and statistical physics. UK research in combinatorics has traditionally been very strong. In recent years several universities have further invested in combinatorics, resulting for example in significant expansion of the combinatorics groups at Birmingham, LSE, Oxford, Warwick and the establishment of a new group at Strathclyde. Many of these newly appointed staff have been attracted from overseas. Accord- ing to data provided by EPSRC (based on REF 2014 submissions and institutional websites) this gives the following picture in 2015: UK researchers in Combinatorics UK researchers in Pure Mathematics 100 877 The excellent standing of UK combinatorics research is evidenced by the following signs of interna- tional recognition. Invited ICM talks in 2010 and 2014 • David Conlon (Oxford) “Combinatorial theorems relative to a random set”, 2014 • Daniela K¨uhnand Deryk Osthus (both Birmingham) “Hamilton cycles in graphs and hyper- graphs: an extremal perspective”, 2014 • Oliver Riordan (Oxford) “Percolation on sequences of graphs”, 2010 In addition, there were two ICM talks at the interface of combinatorics and related areas in 2014: the plenary talk by Ben Green (Oxford) on “Approximate algebraic structure” as well as the invited talk by Tom Sanders (Oxford) on “Roth’s theorem: an application of approximate groups”. ERC and other major grants UK combinatorialists have been awarded 9 out of the 50 ERC grants going to UK mathematicians between 2007–2015, i.e. 18% of all such grants. For comparison, according to the data supplied by EPSRC, only about 11% of all UK pure mathematics researchers are combinatorialists. (We have not been able to obtain data about the number of UK researchers for all of mathematics.) • David Conlon (Oxford), ERC starting grant on “Randomness and pseudorandomness in discrete mathematics”, 2016–21 • Agelos Georgakopoulos (Warwick), ERC starting grant on “Random graph geometry and con- vergence”, 2015–20 • Peter Keevash (Oxford), ERC starting grant on “Extremal combinatorics”, 2010–15 • Peter Keevash (Oxford), ERC consolidator grant on “Combinatorial construction”, 2015–20 • Daniel Kr´al’(Warwick), ERC starting grant on “Classes of combinatorial objects: from structure to algorithms”, 2010–15 • Daniel Kr´al’(Warwick), ERC consolidator grant on “Large discrete structures”, 2015–20 • Daniela K¨uhn,(Birmingham), ERC starting grant on “Quasirandomness in graphs and hyper- graphs”, 2010–15 • Deryk Osthus (Birmingham), ERC starting grant on “Asymptotic graph properties”, 2012–17 • Oleg Pikhurko (Warwick), ERC starting grant on “Extremal combinatorics”, 2012–17 An additional ERC grant at the interface of combinatorics and algebra/number theory has been awarded to Ben Green (Oxford). Moreover, two ERC grants at the interface of combinatorics and computer science have been awarded to Leslie Goldberg (Oxford) and Thomas Sauerwald (Cambridge). A further ERC grant at the interface of statistics and combinatorics has been awarded to Sofia Olhede (UCL). 1 2

Other large grants include a Royal Society Research Professorship for Timothy Gowers (Cambridge) and EPSRC Fellowships for Christina Goldschmidt (Oxford), Daniela K¨uhn (Birmingham) and Alexandre Stauffer (Bath). Prizes Recent highlights include a Fulkerson Prize for Mark Jerrum (Queen Mary), EMS Prizes for Ben Green and Tom Sanders (both Oxford), EATCS Award for Martin Dyer (Leeds), European Prizes in Combina- torics for David Conlon, Peter Keevash (both Oxford), Dan Kr´al’(Warwick), Wojciech Samotij (then Cambridge) and Tom Sanders (Oxford), D´enesK¨onigPrizes for Wojciech Samotij (then Cambridge) and Lutz Warnke (Cambridge), an LMS Senior Whitehead Prize for B´elaBollob´as(Cambridge) and LMS Whitehead Prizes for Peter Keevash (Oxford), Daniela K¨uhn,Deryk Osthus (both Birmingham), Oliver Riordan and Tom Sanders (both Oxford).

2. Discussion of research areas For clarity, we have divided the discussion into the areas of extremal combinatorics, probabilistic com- binatorics, structural & enumerative combinatorics, arithmetic combinatorics, algebraic combinatorics, combinatorial results in computer science and statistical physics as well as designs, coding theory & cryptography. Obviously, there are significant overlaps between these (only some of which are pointed out explicitly). Indeed, some of the major trends have been at the interfaces between these areas, leading to increasingly blurred boundaries – in particular, a number of the developments could have been listed under more than one heading.

2.1. Extremal combinatorics. Much of extremal combinatorics studies the interplay of local and global parameters of discrete structures. The area has seen major growth (both in the UK and internationally) over the last several years. It is an area of exceptional strength in the UK, as e.g. evidenced by two invited ICM talks in 2014 and 7 ERC grants in the UK. As discussed below, many recent breakthroughs were sparked by an increased interplay of combi- natorics with other areas, such as fourier analysis, optimisation (e.g. via the flag algebra method), linear algebra and above all probabilistic methods. Many of these exciting developments were driven by researchers based in the UK. In many respects, there is now a remarkably successful theory in the dense graph setting. For sparse graphs as well as hypergraphs the picture is much less coherent, but there have been major advances. In particular, a recent and extremely fruitful trend has been to consider the sparse random graph setting. Conlon (Oxford) and Gowers (Cambridge), and independently Schacht, developed general sparsity transference principles in the spirit of the celebrated Green-Tao theorem, which yield probabilistic analogues of classical extremal results (such as Ramsey’s theorem, Szemer´edi’stheorem and Tur´an’s theorem) in the much more challenging sparse random setting. This has now turned into a flourishing area, with significant further contributions e.g. made by Conlon (Oxford), Fox and Zhao (Oxford), who provided a counting lemma for sparse structures, as well as by Allen and B¨ottcher(both LSE); see also Section 2.4. A further breakthrough in this direction was provided by the hypergraph container methods intro- duced by Saxton and Thomason (Cambridge), and independently by Balogh, Morris and Samotij (then Cambridge). Containers offer a unified approach to many combinatorial questions concerned (usually implicitly) with independent sets. The method yields simple proofs of many hitherto difficult results, in- cluding the number of H-free hypergraphs, the above sparsity theorems of Conlon-Gowers and Schacht, and the Kohayakawa-R¨odl-Luczak-conjecture. There are ongoing applications to many further areas, in- cluding extremal set theory, Ramsey theory as well as additive number theory (indeed, related container results had been obtained earlier by Green (Oxford) in this context). Quasirandomness and (generalisations of) Szemer´edi’sregularity lemma have driven many of the developments in extremal combinatorics and beyond. For example, Allen, B¨ottcher(both LSE), Griffiths (Oxford), Kohayakawa and Morris used the regularity lemma to resolve a long-standing problem of Erd˝os and Simonovits on chromatic thresholds of graphs. In the sparse setting, a novel version of the regularity lemma was proved by Scott (Oxford). The digraph version was instrumental in the proof of Sumner’s universal tournament conjecture by K¨uhn,Mycroft and Osthus (all Birmingham). The hypergraph 3 regularity setting led to major progress on extremal problems involving e.g. hypergraph matchings and cycles, with contributions by Allen, B¨ottcher,Skokan (all LSE), Keevash (Oxford), K¨uhn,Lo, Mycroft, Osthus and Treglown (all Birmingham). Important advances in our understanding of bounds involved in applying the regularity and associated removal lemmas were made by Conlon (Oxford) and Fox, leading on from earlier work of Gowers (Cambridge). K¨uhnand Osthus (both Birmingham) developed (iterative) absorbing methods combined with quasir- andomness in the context of decomposition and factorisation problems. Their resolution of Kelly’s 1968 conjecture on Hamilton decompositions of tournaments extended to the setting of “robust expansion” (a relaxation of quasirandomness). This in turn led to the proof of the Hamilton decomposition con- jecture and the 1-factorisation conjecture by Csaba, K¨uhn,Lo, Osthus and Treglown (all Birmingham). The robust expansion approach was also key e.g. to the proof of the Bollob´as-H¨aggkvistconjecture on the Hamiltonicity of regular graphs. The above methods and tools have led to further packing and decomposition results (e.g. concerning the tree packing conjecture). A spectacular highlight (discussed in Section 2.6) is Keevash’s (Oxford) result on hypergraph decom- positions and more generally of the existence of designs – again quasi-randomness is a key concept. Probabilistic ideas have also transformed other areas such as Ramsey theory. For example, the study of a natural random greedy process has led to new insights into the classical problem of determining the Ramsey number R(3, t) by Bohman and Keevash (Oxford) as well as independently Fiz Pontiveros, Griffiths (Oxford) and Morris. Advances on the notoriously difficult hypergraph version were made by Conlon (Oxford), Fox and Sudakov. Progress on longstanding graph Ramsey problems includes work by Skokan (LSE) and co-authors, in particular the proof of a conjecture of Erd˝oson the Ramsey numbers of cubes and cliques. More generally, new directions in the area of infinite Ramsey theory were opened up by Leader (Cambridge), including results concerning partition regularity of infinite systems. An important new direction in extremal combinatorics has been the theory of combinatorial limits. Combinatorial limits allow the representation of large discrete objects by analytic objects (functions, measures), which can be examined using tools from mathematical analysis (with quasi-randomness and the regularity lemma playing a key role). UK combinatorialists contributed to extending the range of applications of such methods in a very significant way. In the dense setting, Elek (Lancaster), Kr´al’, Math´eand Pikhurko (all Warwick) with coauthors developed new analytic representations and provided new insights into existing models; this resulted e.g. in resolving a question of Graham on quasirandom permutations by Kr´al’and Pikhurko (both Warwick) or several problems of Lov´aszand Szegedy on the structure of dense graph limits by Kr´al’(Warwick) and his collaborators. In the sparse setting, one of the standard convergence metrics follows from the work of Bollob´as(Cambridge) and Riordan (Oxford). Elek (Lancaster) developed analytic representations of sparse graphs (graphings) and linked them to various problems in algebra. Combinatorial limits are closely linked to the flag algebra method of Razborov. This method turns problems from extremal combinatorics into optimisation problems, which can be approached with the assistance of computers. The flag algebra method resulted in the solution of a number of problems that had not seen any progress in many decades, for example the edge-triangle density problem. Several UK researchers, in particular, Baber (UCL), Kr´al’,Pikhurko (both Warwick) and Talbot (UCL) contributed to this stream of results in a significant way. As examples, we mention the proof of the 2/3 conjecture on domination in three-coloured graphs by Baber and Talbot (both UCL), and the solution of a problem 3 of Erd˝osand S´oson the uniform Tur´andensity of hypergraphs containing no copy of K4 − by Glebov (then Warwick), Kr´al’(Warwick) and Volec (then Warwick). Vaughan (QMUL) developed a publicly available tool for basic applications of flag algebra, called flagmatic. Another area of strength in the UK is the extremal theory of set systems. A recent highlight here is the use of Fourier analytic tools by Ellis (QMUL), Filmus and Friedgut to resolve the Simonovits-S´os conjecture on the size of triangle-intersecting families.

2.2. Probabilistic combinatorics. The area of probabilistic combinatorics is a very broad one because of its wide applicability to discrete mathematics and theoretical computer science, as touched on else- where in this document. Significant parts of the UK activity in this field are linked to other parts of combinatorics (or mathematics more generally), see e.g. Section 2.1. In this section we focus on the study of probabilistic structures in their own right, which forms the theoretical backbone of the subject. 4

This is an area in which the UK has traditionally been very strong: indeed the two standard refer- ences are the books by Bollob´as(Cambridge) on Random Graphs and by Penrose (Bath) on Random Geometric Graphs. The classical study of random graphs remains strong, with groups in Cambridge, Oxford, LSE and Birmingham. One highlight is Montgomery’s (Cambridge) settling of the 20-year old tree-threshold conjecture of Kahn on the existence of arbitrary spanning trees in random graphs. His methods allow many embedding questions of this kind in random graphs to be resolved, and there are close connections with the techniques developed by K¨uhn,Osthus and the Birmingham group to resolve some old and important embedding questions in non-random graphs. Bollob´asand Riordan (Oxford) have greatly extended classical results on the phase transition in both random graphs and hypergraphs, in a series of papers ending in an extremely precise result that they then used to prove a fundamental enumeration result for hypergraphs. Striking work of Riordan (Oxford) and Warnke (Cambridge) concerns phase transitions in Achlioptas processes. Confounding the perceived wisdom that percolation in such systems behaves “explosively”, i.e. with a sudden phase transition, they showed that the rescaled size of the giant component has a limit that is in fact continuous (a mathematical result published nonetheless by Science). At the interface between classical random graphs and random geometric graphs, the comprehensive study of the important class of random planar graphs was initiated by McDiarmid and Welsh (both Ox- ford) as well as others two decades ago. This field has developed greatly, with very detailed knowledge of planar graphs now available, and with substantial extensions to other surfaces (including the recent proof of the McDiarmid-Steger-Welsh conjecture by Chapuy and Perarnau (Birmingham) and the proof of the Benjamini and Schramm conjecture on harmonic functions on planar graphs by Georgakopoulos (Warwick)). There are a number of further UK researchers working on random geometric graphs, in- cluding Georgakopoulos (Warwick), Fountoulakis (Birmingham), Penrose (Bath), Perkins (Birmingham) and Walters (QMUL). Bootstrap percolation is a percolation model studied since the 1970s by leading researchers in statis- tical physics, probability and combinatorics. Recent work, which includes a wide extension of this model called monotone cellular automata in a random environment, has shed much light on the classical model by analyzing behaviour in a much wider context. UK researchers at the forefront of these developments include Bollob´as(Cambridge), Przykucki (Oxford) and Smith (Cambridge). Geometric random graphs as well as percolation are connected to the analysis of random models for complex networks and the study of random processes on such networks. This has become a major area at the interface of combinatorics, probability and statistics, which is very active in the UK, through researchers such as Cooper (KCL), M¨orters(Bath), Fountoulakis (Birmingham), Sauerwald (Cambridge) and Stauffer (Bath). Remarkable advances in the theory of expander graphs, specifically Cayley graphs of groups of the form SL(d, Z/qZ) have been made recently by Varju (Cambridge), sometimes with co-authors. He has a number of outstanding results on random walks in groups. We mention just one recent application, a breakthrough on an 80-year old problem: using his work with Breuillard on the entropy of random walks, P∞ i he has shown that for many algebraic numbers the distribution of a Bernoulli convolution i=0 iλ is absolutely continuous. Methods of probabilistic combinatorics have lately cast new light on integer factorisation: Lee (Ox- ford) found a suitable randomisation of key aspects of the number theory ingredients to prove that an NFS-type algorithm produces congruences of squares in expected time L(1/3, 2.88), the first rig- orous result of this type, and Balister, Bollob´as(Cambridge) and Morris finally resolved a question of Pomerance by establishing the sharp threshold for a uniform random sequence of integers to con- tain a subsequence whose product is square. Further applications include the highly influential use of probabilistic combinatorics in computational counting, actively developed by Dyer (Leeds), L. Goldberg (Oxford) and Jerrum (QMUL). There are countless other examples.

2.3. Structural and enumerative combinatorics. Structural and enumerative questions have always been at the core of graph theory (and combinatorics more generally). There have been exciting devel- opments in the recent years, with major contributions by UK researchers. 5

Some of the most studied problems in graph theory involve graph colouring and structural decom- position problems. A cornerstone of the field is the Strong Perfect Graph Theorem of Chudnovsky, Robertson, Seymour and Thomas, which gives a forbidden subgraph characterisation of perfect graphs (and was recognised by the award of the 2009 Fulkerson Prize). Recent work in the area has focused on developing techniques that handle a wider range of problems. A notable example is the extensive study by Chudnovsky, Scott (Oxford) and Seymour of graphs with large chromatic number, which includes proofs of two well-known conjectures of Gy´arf´asfrom the 1980s. Vuˇskovic (Leeds) with various col- laborators has developed new algorithmic tools for structural decompositions of graphs with forbidden subgraphs, which can be used in colouring algorithms. Probabilistic techniques were used by Havet, van den Heuvel (LSE), McDiarmid (Oxford) and Reed to prove an asymptotic version of Wegner’s conjecture on colouring planar graphs. These techniques were extended by Amini, Esperet and van den Heuvel (LSE) to a general framework for proving asymptotic results on colourings of graphs embedded in surfaces (which they applied e.g. to Borodin’s Cyclic Coloring Conjecture). General structural results on colourings of planar graphs with restricted triangles were given by Dvoˇr´ak,Kr´al’(Warwick) and Thomas, and led to a proof of a conjecture of Havel from 1969 on colouring planar graphs whose triangles are far apart (this result was mentioned in the citation of Dvoˇr´ak’s2015 European Prize in Combinatorics). Combinatorics has always been closely linked to combinatorial optimisation and a substantial amount of research in combinatorics is linked to optimisation. The areas of particular strength in the UK include flow problems, matching theory and matroid theory. A highly recognised contribution to this field is that of V´egh(LSE) to algorithms for combinatorial optimisation problems such as connectivity, flows and matchings. A recent highlight concerning matchings in graphs is a proof of a conjecture of Lov´aszand Plummer from the 1960’s on the number of matchings in cubic graphs. The proof found by Esperet, Kardoˇs,King, Kr´al’(Warwick) and Norine is based on a combination of probabilistic techniques and techniques from discrete optimisation, and was mentioned in the citation of the Kr´al’s2011 European Prize in Combinatorics. Enumerative problems involving matchings (and related structures) are also highly relevant from the view of statistical physics (see Section 2.8) and computer science. UK researchers continue to advance the frontiers of matroid theory, with Welsh (Oxford) considered one of the world leaders in this area. In addition to results on classical matroid questions related to, amongst others, enumerative problems involving the Tutte polynomial by Noble (Brunel), Welsh (Oxford) and their collaborators, the work of van den Heuvel (LSE) and Thomass´eon covering bases of matroids is particularly noteworthy, having generalised classical results of Edmonds, Nash-Williams and Tutte on covering and packing bases in matroids and spanning trees in graphs.

2.4. Arithmetic combinatorics. The area of additive (or arithmetic) combinatorics is particularly dif- ficult to define, as it touches on many other mathematical subdisciplines, including but not limited to analytic number theory, group theory, harmonic analysis, algebra, ergodic theory, geometry and theo- retical computer science (the latter three connections are somewhat less well developed in the UK). Despite being a relatively young and small area, additive combinatorics has a very strong track record internationally (Fields medals to Gowers (Cambridge) in 1998, Tao in 2010, Abel Prize to Szemer´ediin 2012), and the UK is world-leading in this field, with a concentration of expertise at Oxford (Conlon, Green, Sanders), Cambridge (Gowers, Varj´u)and a sizeable group at Bristol (Browning, Rudnev, Wolf, Wooley), where the interface with analytic number theory is particularly well developed. Between them the three groups run a regular 1-day meeting in additive combinatorics. One of the central pillars of the area has been the celebrated theorem of Green (Oxford) and Tao from 2004, which states that the primes contain arbitrarily long arithmetic progressions. It continues to drive developments at the interface with analytic number theory. Recent highlights with UK participation include the simplification of the original transference principle used in the proof of the Green-Tao theorem by Conlon (Oxford), Fox and Zhao (Oxford), which relies on an extension of the regularity method to sparse pseudorandom hypergraphs. Another recent result which has attracted a significant amount of international attention concerns long gaps between primes, with significant contributions of Green and Maynard (both Oxford). Substantial advances have been made in the past decade on the long-standing problem of improving bounds in Roth’s theorem on arithmetic progressions of length 3. Sanders (Oxford) improved, strikingly 6 and repeatedly, on well-known work of Bourgain, by refining the existing Fourier-analytic techniques with combinatorial input. The best known bound at the time of writing is due to Bloom (Bristol). Since Gowers’s harmonic analysis proof of Szemer´edi’s theorem on long arithmetic progressions in dense subsets of the integers in the late 1990s, there have been numerous consequential developments in higher-order Fourier analysis, notably inverse theorems for the higher-order uniformity norms by Green (Oxford), Tao and Ziegler. Higher-order regularity lemmas for functions on cyclic groups, for example, have been developed by Green (Oxford) and Tao, and Gowers (Cambridge) and Wolf (Bristol), with various applications. At an international level there has been a flurry of activity centred around the interface with algebra and geometry in the past five years, in particular with a view to exploiting sophisticated variants of the (not so new) polynomial method. The (almost) solution of the Erd˝os-distanceproblem by Guth and Katz has sparked a number of other breakthroughs. The UK is active in this area. For example, the best sum-product estimates in finite fields are due Rudnev (Bristol) and collaborators, making progress towards a finite-field analogue of a long-standing conjecture of Erd˝osand Szemer´edivia an incidence theorem on points and planes in three dimensions. At the interface with group theory, Gowers (Cambridge) obtained a characterisation of quasirandom groups and proved that any group with a non-trivial representation of constant dimension contains a large product-free subset. Much internationally acclaimed UK-based headway has also been made in the area of growth in groups in the past decade, notably through results of Helfgott (then Bristol) on growth in SL2(p), which have applications to expander graphs, and his bounds on the diameter of any transitive permutation group. More recently, a complete and very general characterisation of so-called approximate groups has been given by Breuillard, Green (Oxford) and Tao. In the more traditional abelian context, Sanders (Oxford) made a significant breakthrough towards the Polynomial Freiman-Ruzsa conjecture, by showing that any subset of a finite abelian group with bounded doubling is contained in a coset progression of dimension which is quasipolynomial in the doubling constant.

2.5. Algebraic combinatorics. Research on permutation patterns is prominent in algebraic combi- natorics in the UK. This area has grown significantly in the last couple of decades – the modern development can be traced back to considerations of sorting problems in theoretical computer science, and this interplay continues to be a feature in current work in the UK. Order-theoretic aspects of permutation classes (analogous to the structural study of graph classes) are being studied by Bevan and Brignall (both Open University) and by Linton and Ruˇskuc(both St Andrews). Brignall (Open University) and Ruˇskuc(St Andrews) have played a key role in a larger international effort to develop structural foundations of the theory of permutation pattern classes. Recently, key building blocks have emerged in the form of so-called grid classes. On one hand, this has enabled the entry of geometrical and language-theoretic methods into the theory, with particular applications to decidability questions and some generic enumeration results. On the other hand, the conceptual parallel with the structural graph theory, made transparent by considering permutations as relational structures with two linear orders, has enabled fruitful interactions between these two fields, as exemplified by the work of Brignall (Open University) and Lozin (Warwick). A crucial concept here is that of well quasi-orderedness. Bevan (Open University) has combined structural ideas with probabilistic methodology and spectral graph theory to obtain powerful new results about the asymptotic behaviour of some difficult permutation classes. Another recent direction is the study of topological properties, in particular the M¨obiusfunction, of the poset of permutations. Significant progress has been made in recent years in eliciting the topology of its intervals, by Smith and Steingr´ımsson(both Strathclyde) and their collaborators. There are several other areas of high-quality research at the interface of algebraic combinatorics with other areas of mathematics. Georgakopoulos (Warwick) provided a complete classification of planar cubic Cayley graphs, identifying some classes with unexpected properties which led to counterexamples to conjectures of Bonnington, Mohar and Watkins. Sir´aˇn(Openˇ University) is part of an international group working on embedding graphs in surfaces and the degree-diameter problem. Cameron (St An- drews) has mobilised a large international team dealing with the synchronisation problems for groups and semigroups. This arises from the long-standing open problem known as the Cern´yconjecture,ˇ and involves an intricate analysis of (semi)group actions on graphs by endomorphisms. At the other – 7 infinite – end of the spectrum, the study of group and semigroup actions on infinite, well-behaved (e.g. homogeneous) combinatorial structures, such as the Rado random graph or the linear order of rationals, is another area of traditional strength in the UK, at the interface between algebra, combinatorics and model theory, which we discuss in more detail in Section 4. 2.6. Designs, coding theory and cryptography. A spectacular highlight was the proof of the existence of combinatorial designs by Keevash (Oxford), which resolves a well-known question going back to the 19th century. In fact, his result extends beyond the initial setting – remarkably, it even extends to decompositions of quasirandom hypergraphs. The ingenious proof blends algebraic as well as design- theoretic ideas with an absorbing approach involving probabilistic techniques. The result has had an enormous impact which goes far beyond the immediate area. Algebraic and geometric approaches to the construction of codes and designs have been successfully pursued by several groups in the UK including those at QMUL, St Andrews and Sussex. These include the construction of codes from different combinatorial structures such as graphs, finite Desarguesian spaces, non-Desarguesian planes. An EPSRC-funded project at QMUL has produced a database of designs for use both in combinatorics and statistics (see below for the connection with experimental design). Coding theory has many applications in cryptography, which has been a significant strength at several UK universities, in particular, Bristol and RHUL. Recent advances in coding theory in the UK that rely on combinatorial techniques include the discovery of problems in implementations of Transport Layer Security (TLS), new key predistribution schemes for wireless sensor networks, and cryptanalysis of protocols proposed for Radio-frequency Identification (RFID). 2.7. Combinatorial results in computer science. Combinatorics has a long history of underpinning developments in computer science. In particular, combinatorial methods are of central importance for the analysis of algorithms and their complexity. The central conferences in this area (FOCS, ICALP, SODA, STOC, etc.) regularly have papers with strong combinatorial content. Combinatorial methods were essential to designing complex algorithms for approximate counting and sampling. The work of Jerrum (QMUL), Sinclair and Vigoda on rapidly mixing Markov chains on graph matchings led to the design of a fully polynomial approximation scheme for the permanent and was recognized by the award of the 2006 Fulkerson Prize. Dyer (Leeds), L. Goldberg (Oxford) and Paterson (Warwick) received the best ICALP paper award in 2006 for their work in this area, and L. Goldberg (Oxford) and Jerrum (QMUL) have been the recipients of best ICALP paper award twice (in 2010 and 2012) for their work on the approximation of the partition function of the ferromagnetic Potts model and their work on computing the sign of the Tutte polynomial. The regularity method of Szemer´edimentioned in Section 2.1 led to super-fast algorithms for testing properties of large dense inputs. Czumaj (Warwick) and collaborators have made fundamental contri- butions to such algorithms for sparse inputs by designing an algorithm for testing expansion properties of sparse graphs and providing a general characterisation of testable properties of such graphs. Combi- natorial notions and results stemming from the Graph Minor Project of Robertson and Seymour, e.g., tree-width and the Grid Minor Theorem, laid the foundations for fixed parameter algorithms, which form a focus of the research of Gutin (RHUL) and Paulusma (Durham). UK researchers are also among the world leaders in algorithmic game theory. Koutsoupias (Oxford) is one of the founders of this area, and was recognised as such in the citation of the 2012 G¨odelPrize, which he was awarded for “laying the foundations of growth in algorithmic game theory”. Daskalakis, P. Goldberg (Oxford) and Papadimitriou settled the complexity of computing Nash equilibria in normal form games, one of the most fundamental problems at the intersection of game theory and computer science. 2.8. Combinatorial results in statistical physics. There are also deep interactions between combi- natorics and statistical physics. Recent examples of results using combinatorial techniques include the work of Dukes (Strathclyde) on the abelien sandpile model, Koteck´y’s(Warwick) work on lattice models and related phase transition phenomena, new approaches by Perkins (Birmingham) related to occupancy fractions in the hard-core model, Sokal (UCL) and Jackson’s (QMUL) work on flow polynomials, Sokal (UCL) and Scott (Oxford)’s work on signs of Taylor coefficients, and Ueltschi’s (Warwick) work on 8 random permutations. Another example of work in this area is that of Dukes and Steingr´ımsson(both Strathclyde) on web worlds and web diagrams motivated by quantum chromodynamics. More generally, there are also important connections between statistical physics, computer science and combinatorics, as seen in the work of L. Goldberg (Oxford) and Jerrum (QMUL) on counting H-colourings and approximating the partition function of spin systems. Rapidly mixing Markov chains play a central role here, with pioneering contributions on approximating the permanent (which counts perfect matchings) by Dyer (Leeds) and Jerrum (QMUL), as mentioned in Section 2.7.

3. Discussion of research community It seems impossible to list all UK combinatorialists without missing anybody, and the list below contains only the most active and visible groups in the UK; we mention researchers with both permanent and temporary appointments but omit PhD students. It should be noted that there are many (very strong) individual researchers at other institutions (e.g. Dyer (Leeds), Elek (Lancaster) and Vuˇskovi´c (Leeds) to name a few), who are not listed below. • Bath. The University of Bath has a very strong probability group supported by its research cen- tre, the Probability Laboratory at Bath (Prob-L@B). The focus of the research centre contains many problems at the interface with combinatorics, in particular covering random graphs and networks, branching structures and percolation. Researchers working on such problems include Mailler, M¨orters,Penrose and Stauffer. • Birmingham. The University of Birmingham has a large and very active group of researchers, which has considerably expanded recently. The main focus is in extremal combinatorics and probabilistic combinatorics with an overlap to the analysis of random algorithms as well as problems at the interface of combinatorics and statistical physics. The combinatorialists working in Birmingham include Fountoulakis, Han, Joos, Kim, K¨uhn,Lo, McDowell, Mycroft, Osthus, Perarnau, Perkins and Treglown. • Bristol. The University of Bristol has a group of combinatorialists with research interests primarily concerning arithmetic and probabilistic combinatorics. A number of combinatorics activities in Bristol are linked to the Heilbronn Institute for Mathematical Research, which is run as a partnership between the University and the UK Government Communications Headquarters. Combinatorialists at Bristol include Barber, Bloom, Dettmann, Gillespie, McInroy, Rudnev, Semeraro, Skerman and Wolf; many researchers working in other areas, in particular computer science (Clifford, Montanaro, Sach and Sun) and number theory (Browning and Wooley), have close research ties with the combinatorics group. • Cambridge. The University of Cambridge is home to many strong combinatorialists. Their research interests range from algebraic, extremal and probabilistic combinatorics through graph theory to applications in computer science and number theory. The combinatorialists working in Cambridge include Bollob´as,Gowers, Leader, Letzter, Montgomery, Narayanan, Smith, Thoma- son, Varju and Warnke; Cambridge also has many other mathematicians and computer scientists (e.g. Dawar, Sauerwald, Winksel) whose research interests include combinatorial problems. • Durham. Durham University and its research group “Algorithms and Complexity in Durham” (ACiD) has a large number of combinatorialists and computer scientists with mutually over- lapping research interests – Bordewich, Dabrowski, Dantchev, Erickson, Friedetzky, Johnson, Krokhin, Mertzios, Nichterlein, Paulusma and Stewart. Particularly well-represented areas of re- search in combinatorics and on the interface of combinatorics with other areas include constraint satisfaction problems, (graph) algorithms, graph theory and networks. • LSE. The London School of Economics and Political Science has a very visible combinatorics group formed by Allen, Anthony, Batu, Biggs (emeritus), B¨ottcher,Brightwell, van den Heuvel, Lewis-Pye, Skokan and Swanepoel. Many members of the group share common interests in extremal and probabilistic combinatorics; the group is also active in other areas such as combi- natorial geometry, structural graph theory and applications in computer science. Many members of both the operational research group and the game theory group at LSE (e.g., Sorkin, von Stengel, V´eghand Zambelli) have combinatorial research interests and keep intensive research links to combinatorialists based at LSE. 9

• Open University. The Open University has an active combinatorics group formed by Barbina, Bevan, Brignall, Chicot, Granell (emeritus), Griggs, Grimm, Holroyd, Quinn, Sir´aˇn,Webbˇ and Wilson (emeritus). The main focus of the group include combinatorial designs, graphs on surfaces, structural graph theory and structure of permutations. • Oxford. The University of Oxford is home to a large number of combinatorialists with very broad research interests; these range from extremal combinatorics, matroid theory, probabilistic combinatorics and structural graph theory to problems on the interface with computer science and number theory. Combinatorialists at Oxford include Conlon, Goldschmidt, Green, Keevash, Lee, McDiarmid, Prendiville, Przykucki, Riordan, Sanders, Scott, Welsh (emeritus) and Zhao. Oxford is also home to many computer scientists (e.g. L. and P. Goldberg, Gottlob, Jeavons, Kiefer, Koutsoupias, Lapinskas, Ouaknine, Santhanam and Zivn´y)whoˇ work on combinatorial problems related to their own research themes. • QMUL. Queen Mary University of London has a combinatorics group with broad ranging inter- ests established by Cameron (now primarily St Andrews), including Ellis, Fink, Jackson, Jerrum, Johnson, Stark and Walters. Their research interests include algebraic aspects of combinatorics, combinatorial geometry, probabilistic combinatorics, and applications of combinatorics in com- puter science. The group also benefits from the presence ofLuczak at QMUL who works on the interface of probability and combinatorics. • RHUL. Royal Holloway, University of London, has an active combinatorics group which includes Blackburn, Gerke, Moffatt and Wildon. Their research is focused on algebraic and probabilistic themes in combinatorics. In addition, many computer scientists at RHUL (Cohen, Gagarin, Gutin, Tzameret and Wahlstr¨om)have combinatorial research interests; the research contacts between combinatorialists and computer scientists at RHUL are particularly intensive. • St Andrews. The University of St Andrews has a group of combinatorialists focusing on al- gebraic problems in combinatorics and design theory. The members of the group are Bleak, Cameron, Huczynska, Mitchell, Quick, Roney-Dougal and Ruˇskuc. The group has strong re- search links to statisticians (e.g. Bailey) and computer scientists (e.g. Linton) in St Andrews who also have combinatorial research interests. • Strathclyde. There is an emerging combinatorics group at the University of Strathclyde, currently formed by Dukes, Kitaev, Selig, Smith and Steingr´ımsson. The primary research focus of the group includes the structure of permutations and algebraic aspects of combinatorics. • UCL. University College London is home to a small but lively and diverse group of combinato- rialists, which include Bar´any, Pena, Sokal and Talbot. Their research interests cover extremal combinatorics, geometric combinatorics and problems on the interface of combinatorics and mathematical physics, in particular. • Warwick. The University of Warwick and its Centre for Discrete Mathematics and its Ap- plications (DIMAP) have recently attracted a large number of combinatorialists to Warwick. Combinatorialists at Warwick include Georgakopoulos, Haslegrave, Hong, Hu, Hubai, Kr´al’, Lozin, Pikhurko, Staden and Tyros, and there are a number of others (e.g. Ball, Maclagan, Math´e)whose research has a combinatorial overlap. The most active research areas at Warwick are algebraic combinatorics, extremal combinatorics and structural graph theory. The DIMAP centre also stimulates research at the interface of combinatorics and computer science, which is exceptionally well developed in Warwick; many computer scientists there (e.g. Cormode, Czu- maj, Ene, Englert, Jurdzinski, Lazic, Murawski, Paterson and Tiskin) have strong combinatorial research interests.

The British combinatorics community is very active and visible both internationally and on the na- tional level. British combinatorialists regularly deliver invited talks at international conferences on math- ematics in general and major conferences focused on combinatorics (such as the European Conference on Combinatorics, Graph Theory and Applications, SIAM Conference on Discrete Mathematics), and are members of their scientific committees. At the national level, major events that have been organised regularly include the biannual British Combinatorial Conference (organised by the British Combinatorial Committee), the annual two-day Colloquia in Combinatorics in London (organised by LSE and QMUL), the annual One-Day Meeting in Combinatorics in Oxford, and the annual Open University Winter 10

Combinatorics Meeting in Milton Keynes. In addition, several dozens of other workshops and research schools have been held on combinatorics in general or on particular topics in combinatorics (including workshops at ICMS, the LMS-EPSRC Durham symposia, and the Scottish Combinatorics Meetings). Many of these events received funding from CMI, EPSRC and/or the LMS.

4. Inter/Intra-disciplinary connections Combinatorics has close connections to many other areas of mathematics (including algebra, anal- ysis, discrete optimisation, geometry, number theory, probability and statistics) and other sciences (in particular, computer science and statistical physics). Many of these have been touched upon in the previous sections. In what follows, we highlight some additional aspects.

Connections to probability and statistics. The connection between combinatorics and probability has seen major growth over the last few decades, with the pioneering work of Bollob´as (Cambridge) being hugely influential. Nowadays, almost all of the main combinatorics research groups are contributing to some aspects of this interface. There are two main (but interconnected) directions: firstly, the use of (increasingly sophisticated) probabilistic tools to resolve questions of a combinatorial nature, and secondly, the study of random discrete structures and networks. For example, the analysis of random models for complex networks and the study of random processes on such networks has become a major area at the interface of combinatorics, probability and statistics. Numerous events and workshops along this interface have been held. For example, the 2016 Newton Institute program on Theoretical Foundations for Statistical Network Analysis is devoted to such questions. Another connection with statistics, which is well-established and still active in the UK (especially through groups in St Andrews and Southampton) involves experimental design. The subject has many applications, including designing clinical trials in medicine.

Connections to algebra and group theory. Combinatorics has links with several areas of algebra, which are actively studied in the UK. The most important tools for the study of finite permutation groups have been combinatorial: in particular, coherent configurations were invented for this purpose (and in- dependently for the graph isomorphism problem). The classification of finite simple groups has changed the landscape here but there are also important new developments in semigroup theory. Geometric group theory has developed out of the subject known as combinatorial group theory, and combinatorial methods are still important in this area (one in which the UK has considerable strength, see the algebra landscape document). There are many exciting developments at the interface of algebra, combinatorics and computer science. Expander graphs play an important role in many aspects of computer science, in particular in the derandomisation of algorithms. Many explicit constructions involve finite groups, and in the other direction, results about expanders are used to prove generation and word-length properties for finite groups. The explicit (combinatorial) construction of expanders based on the zig-zag product is behind a recent proof of the equivalence of logspace and symmetric logspace computations. This is a significant step towards a possible proof of the equivalence of logspace and non-deterministic logspace computation, the space analogue of the P vs. NP problem. One should also mention the work on homogeneous structures at the interface of algebra, combina- torics and model theory, which is carried out by various groups in the UK, including those in Imperial, Leeds, St Andrews, UEA and York. The initial theory of homogeneous structures was developed as part of model theory but the subject combines methods from various parts of mathematics besides model theory, including permutation group theory, descriptive set theory and combinatorics (in partic- ular combinatorial enumeration and Ramsey theory), and has connections to topological dynamics and constraint satisfaction. A classification of homogenous digraphs by Cherlin includes examples of great complexity and sheds light on the richness of homogeneous structures in general. It would not be possible to design efficient tools for computations in algebra without a good under- standing of the combinatorial structure and symmetries of algebraic objects. The two main computer algebra systems, GAP and Magma, have many practical as well as theoretical uses. The former is administered by the CIRCA group in St Andrews, which is also responsible for the constraint solver 11

MINION. The phenomenon known as combinatorial explosion means that these tools need to be at the cutting edge to achieve significant results. Connections to geometry. The UK combinatorics community is also visible in the area of combinatorial geometry. There is a beautiful recent solution by Green (Oxford) and Tao of the conjecture by Dirac and Motzkin. It combines the classical tools from combinatorial geometry with the classification of the cubic curves and also with advanced tools from additive combinatorics. A traditionally strong line of research in the UK is that related to convex bodies and polytopes, which is led by B´ar´any(UCL) and his collaborators. Connections to computer science and logic. Combinatorial methods are of central importance in the- oretical areas of computer science that deal with the analysis of algorithms and their complexity. The majority of research in this field has traditionally been carried out outside the UK, in particular, in the US. However, in recent years a number of significant research groups, in particular, in Birmingham, Bristol, Cambridge, Durham, Edinburgh, KCL, Leicester, Liverpool, Oxford, RHUL and Warwick, have established themselves and begun to take a lead in this space. These provide a significant focus for interactions between combinatorics research and its applications, as well as a source of new challenges for fundamental research in combinatorics. In addition to the results concerning algorithm design and algorithmic game theory mentioned in Section 2.7, the area of constraint satisfaction problems deserves to be highlighted as being of fun- damental both theoretical and practical importance. Recent advances in this area combine algebraic and combinatorial methods. Several groups in the UK (Durham and Oxford) are leaders in this area of computer science and have made significant contributions towards the classification of tractable con- straint satisfaction problems and the Feder–Vardi Conjecture, which offers a classification of tractable problems through algebraic means. It comes as no surprise that combinatorial methods have been essential for many other recent devel- opments in computer science. An example of this is model checking, which is an area at the interface of combinatorics, computer science and logic where the UK is very strong. We should highlight here the work of Dawar (Cambridge) and his collaborators on algorithms for sparse graphs, and the work Pilipczuk (then Warwick) and his collaborators on fixed parameter algorithms. In the other direction, methods from logic have also been important in the analysis of large graphs and led to the notion of first-order graph convergence. The strength of several computer science groups in the UK is supported by a close interaction with combinatorialists at the same institution; Oxford and Warwick are notable examples that support this claim. Connections to operational research. Combinatorics and operational research (OR) were closely con- nected, providing inspiration for research, methods and approaches. Many fundamental optimisation problems rely on combinatorial methods to understand their fundamental properties and design efficient solution methods. Integer programming lies at the heart of many modern OR solution approaches, and many techniques for solving integer programmes draw directly on geometric ideas in polyhedral combinatorics. The behaviour of other typical instances of many archetypal OR problems leads directly to problems in probabilistic combinatorics. There is also substantial interplay between combinatorics, operational research and theoretical physics, in particular, in the areas of the analysis and design of optimisation algorithms. In more recent times, the fields have inspired and found inspiration in the algorithmic game theory and mechanism design. Another real-world problem of great significance is that of DNA sequencing. Here the frequency of ambiguity in reconstruction from “shotgun” data is analysed and reduced to a question about directed graphs, and in particular a new graph polynomial structurally similar to the Tutte polynomial. More recent research challenges such as data analytics and the theory of complex networks are likely to lead to further growth of this interface. Strong OR groups can be found in mathematics departments at LSE, Southampton and Warwick. Because of the inherently interdisciplinary nature of the work, researchers working at this interface are often located in other departments such as management departments or business schools (Lancaster, Warwick), and computer science departments (Durham, KCL, Leicester, Liverpool, Oxford, QMUL, RHUL, Warwick). 12

5. Further directions and opportunities As can be seen from the above discussion, UK research in most areas of combinatorics is world-leading. Some of this prominence and growth has been relatively recent. The EU support for combinatorics in the UK is proportionally higher than for other areas, which is evidenced e.g. by the number of ERC grants, and combinatorics is still one of the smaller areas of mathematics in the UK. To keep the UK competitive on an international level, it is vital that the current areas of strength continue to be supported by EPSRC. In this context, we note that EPSRC (in response to the comments from International Reviews) supported the foundation of the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick, which attracted several top combinatorialists to the UK and became one of the leaders in the area of combinatorics and computer science. However, despite recent growth, the interface between algorithms, combinatorial optimisation and combinatorics remains underrepresented in the UK. We would like to conclude by highlighting some possible opportunities to further expand and strengthen the UK’s research profile in combinatorics. A number of exciting future directions have already been touched upon in previous sections, so we will primarily focus on those that have been discussed in less detail so far. Firstly, combinatorics is in an excellent position to contribute to the “big data” challenge (perhaps, along with statistics, more than any other mathematical discipline), one of the top scientific priorities acknowledged by the government and EPSRC. This is an opportunity to further tighten the links between (randomised) algorithms, combinatorics and computational complexity. In particular, one of the core topics of the newly established Alan Turing Institute is the (statistical) analysis of data via algorithms. This area forms the main interface between algorithms (and computer science in general), combinatorics and statistics. As progress is likely to involve randomised techniques, there is opportunity to capitalise on UK strength in probabilistic combinatorics. Secondly, some parts of algebraic combinatorics appear to be under-represented in the UK, possibly to the extent that it may be thought of as a weakness in the UK portfolio. Examples of recent advances that attracted significant attention internationally and that have little intersection with current research in the UK include the work of Adiprasito, Huh and Katz on the coefficients of characteristic polynomials of graphs and matroids, and the work of Marcus, Spielman and Srinivasa on interlacing polynomials and its use to resolve the important Kadison-Singer problem. Another area that seems to be under-represented in the UK is analytic combinatorics. This area also led to important results over the last few years (e.g. on random maps and random graphs on surfaces), as evidenced by an ICM plenary talk by LeGall and a European Prize in Combinatorics to Schaeffer, amongst others. We believe that further strengthening the areas of UK combinatorics that are currently at the forefront internationally, combined with well-channelled and high-quality growth in underrepresented areas, would allow UK combinatorics to consolidate its position as a world leader on the international stage over the coming years. CONTINUUM MECHANICS

Lead Authors: Alexander Movchan, Nigel Peake

Statistical Overview

The best way to estimate this is from REF2014 data. According to the statistical summary provided by HEFCE, the number of UK researchers working in the area of continuum mechanics is estimated at 235. This exceeds the numbers linked to Mathematical Biology (171), Nonlinear Systems (225) and Numerical Analysis (175). Out of 235 researchers in Continuum Mechanics, for 222 cases this area is considered to be the primary area of their research. Continuum Mechanics is therefore seen to account for some 29% of the total applied mathematics contingent submitted to REF2014.

Fluid Mechanics:

One (no doubt imperfect) way of estimating trends in activity is to analyse submission and publication data for UK authors in the Journal of Fluid Mechanics, one of the leading journals in the field. In 2010, at the time of the last International Review, 157 papers with a UK academic as corresponding author were submitted, with a success rate of 62%. In 2014 the corresponding figures were 212 and 68% respectively. This increase in both volume and quality is in fact the result of a steady upward trend over the last decade. Internationally, the UK is behind the US in terms of volume (286 US paper submitted in 2014), but ahead in terms of success rate (63% acceptance rate for US papers). The UK Fluid Mechanics research community considers the world class quality, excellence and high impact as the important criteria for publications. In 2010 we identified 167 academics based in UK mathematics departments who were working exclusively or predominantly in fluid mechanics. Although the number of JFM submissions paints only a small part of the picture, it seems unlikely that the subject has contracted in the intervening period. Internationally the blend of Applied Mathematics and Engineering and the interaction between them is unique. Fluid mechanics is a very vibrant research area with lots of potential for impact.

Solid Mechanics:

If we count generally and include models of solid mechanics, phase transition, solid/fluid interaction, non-linear analysis stimulated by materials science, waves, fracture, then the number of people well exceeds 120-150. However, the Solid Mechanics community in the UK nowadays includes researchers in Mathematical Sciences as well as Engineering, on many occasions interacting with each other very actively. Hence the above number of 120-150 is a lower bound, which indicates the strength of Solid Mechanics in the UK.

Subject breakdown

The number of active researchers in the area of continuum mechanics is estimated at 235, with 222 stating that continuum mechanics is their primary research area, with 12 and 1 considering continuum mechanics being their secondary and tertiary areas, respectively. According to the statistical data provided by EPSRC, Continuum Mechanics appears to be the larger area compared with Mathematical Biology, Nonlinear Systems and Numerical Analysis.

According to the Deloitte-EPSRC report of 2013, the contribution of mathematical sciences research to the UK economy was estimated at £208 billion in terms of GVA contribution, which makes around 16% of the total UK GVA. Fluid Mechanics was explicitly mentioned in the Deloitte report, Solid Mechanics was included through the areas such as differential equations and mathematical physics. With the new assessment indicator, ``impact’’, included in REF, connections with industry have become very important, and interaction between solid mechanics and fluid mechanics communities have been enhanced through the industrial interface. The overall area of Continuum Mechanics, which incorporates both Fluids and Solids appears to be one of the strongest research areas in Applied Mathematics in the UK.

Discussion of Research Areas

Fluid Mechanics:

The subject is traditionally very strong in the UK, and we believe it has maintained its international standing. Perhaps the most notable positive change in recent years is the prominence of the subject in the EPSRC Centre for Doctoral Training portfolio: two CDTs, at Imperial (Fluid Dynamics across Scales) and Leeds (CDT in Fluid Dynamics), specialise in fluid mechanics, and both with sizeable contributions from the corresponding mathematics departments. Other CDTs, while not being exclusively devoted to fluids, nevertheless cover fluid research to varying degrees. These include Oxford (Industrially Focused Mathematical Modelling) , Imperial (Mathematics of Planet Earth) and Cambridge (Cambridge Centre for Analysis).

Other large grants have been awarded in the area in recent years, including EPSRC Programme Grants for the Laminar Flow Control Centre at Imperial and for Mathematical Underpinning of Stratified Turbulence (joint between Bristol and Cambridge), and very recently an EPSRC Grand Challenge grant for Managing Air for Green Inner Cities (Cambridge and other partners). A key feature of all three of these large grants is that in each case the lead PI is based in a mathematics department, indicating success of mathematics academics as leaders of large multi-disciplinary teams.

Activity with a large fluid-mechanics content at the Isaac Newton Institute in the last 5 years has included programmes on Free Boundary Problems and Related Topics, The Mathematics of Liquid Crystals, Multiscale Numerics for the Atmosphere and Ocean and Water Waves, while a 6 month programme is planned in 2017 on Mathematics of Sea Ice Phenomena.

Solid Mechanics (including Solids/Fluids interaction):

The subject has developed very well in the recent years in the UK and has a high competitive international standing. The Fellows of the Royal Society working in the area are Willis, Ball, Ogden (awarded 2016 IUTAM Rodney Hill Prize), Ockendon and Atkinson. The success of Oxford is especially impressive, with two major Centres for Doctoral Training (Analysis and Industrial Maths) created and producing outstanding work. A special mention should be given to the UK effort in liquid crystals, a subject which mathematically is in a period of rapid growth. The research groups involve Oxford, Cambridge, Bristol, Bath, Strathclyde, Warwick, Southampton. There was also the 2013 programme on the Mathematics of Liquid Crystals at the Isaac Newton Institute https://www.newton.ac.uk/event/mlc , which reinforced the success of UK Applied Mathematics in the important area.

Major advance in the area of modelling metamaterials and waves in multi-scale systems in Manchester, Imperial and Liverpool is also acknowledged – these are excellent groups with proven record of success and EPSRC funding is at the level of major five-year grants. Liverpool and Glasgow solid mechanics groups have contributed strongly to the major EPSRC sponsored projects in Maths with Healthcare. Liverpool and Glasgow have been awarded two out of five centres funded by EPSRC. In particular, Liverpool Maths with Healthcare £2.4M EPSRC centre incorporates, among several strategic directions, modelling of a transient fluid-solid interaction in the design and analysis of the blood flow through stented blood vessels. Glasgow has the EPSRC £2M Mathematics in Healthcare Centre for 'SofTMech', a Centre for Multiscale Soft Tissue Mechanics - with application to heart & cancer, which has started in April 2016 and is funded for 4 years.

A serious achievement in the area of solid/fluid interaction was also a major (£1.5M) EU Commission grant on modelling of hydraulic Fracture (IAPP EU project “HYDROFRAC”) awarded to Aberystwyth University.

The international standing of the UK continuum mechanics community is confirmed by recent awards of top prizes to outstanding individuals, including both the Batchelor (fluid mechanics) and the Rodney Hill (solid mechanics) IUTAM prizes in 2016, and two of the first four winners of the Weissenberg Award from the European Rheology Society. The existence of very broad-based and well-attended meetings, such as the British Applied Mathematics Conference and the European Study Group for Industry, are seen as key strengths, especially in allowing the community to address challenges in new application areas.

Discussion of Research Community

Fluid Mechanics:

The new CDTs in fluid mechanics are certainly training a significant number of new PhD students in the area (between them of the order of 30 per year specialising in fluids). What is not clear, of course, is whether the CDTs have actually increased the number of PhD students entering the field, or whether the CDT funding has simply replaced other funding routes into the field (such as EPSRC Doctoral Training Accounts). However, it is clear that the areas supported by the CDTs and large programme grants are seeing an influx of talented young people. For instance, modelling of high Reynolds number flow, historically a strength of UK applied mathematics but somewhat reduced at the time of the last International Review in 2010, has seen renewed activity thanks to the Laminar Flow Centre at Imperial College London. Solid Mechanics:

World Class groups working on Mathematical Modelling of Solids include: Oxford (with two large CDTs in industrial maths and nonlinear analysis),Cambridge (DAMTP jointly with Engineering), Imperial (with a very successful program grant on mathematical foundations of metamaterials, jointly with Liverpool), Manchester (with substantial EPSRC and industrial funding in the area of waves in complex industrial and biological materials including metamaterials and a forthcoming ICMS meeting), Liverpool (with two major EPSRC grants, that includes the program grant and interdisciplinary grant maths/healthcare (solid/fluid interaction) and a significant amount of EU funded projects in waves and solid mechanics), Keele (nonlinear elasticity, waves and thin structures), Bath (non-linear analysis, numerical, industrial maths), UCL (multi-scale homogenisation), Aberystwyth (impressive track record of successful EU funded projects on modelling of Solids, Solid/Fluid interaction, biomechanics), Glasgow (nonlinear solids and biological applications), Nottingham (industrial maths), Strathclyde (liquid crystals and phase transition), Loughborough (linear and non-linear waves, and interface with engineering), Brunel (waves, theoretical models of fatigue, damage and fracture mechanics).

In particular, Impressive development, which has brought talented young people into the area of mathematical modelling of solids, has taken place in Oxford, with non-linear analysis of PDEs and industrial maths centres being in the forefront. The success of Imperial and Liverpool in modelling of elastic metamaterials (EPSRC program grant) has also established very strong links between Maths and Physics in Imperial College (Pendry group) and has brought amazingly talented young people. In particular, the Solids group in Liverpool has expanded with several excellent recent faculty appointments and impressive PhD students and postdocs. Manchester is very impressive indeed maintaining leadership in key areas of mathematical modelling of solids, waves and fluid/solid interaction.

Across the Applied Mathematics in the UK, the expertise in Solid Mechanics is diverse, and several important research topics may be covered in a single research group. A good example of a small but excellent Solid Mechanics group is in Aberystwyth University, whose focus is in dynamics in discrete structure, biomechanics, hydraulic fracture, supported by substantial EU Commission grants.

In the assessment of the level of activity in the UK in solid mechanics it is important to take into account interdisciplinary collaboration between UK mathematicians and engineers and physicists. In particular, the Deloitte-EPSRC report explicitly mentions Aeronautics and manufacturing, which use “algorithms to model real world events and optimise processes”. Solid Mechanics research provides a direct and invaluable contribution to this area.

Inter/Intra-disciplinary activities and Engagement activities

Fluid Mechanics:

The subject has always enjoyed excellent links with Industry, and this trend has very much continued, and perhaps accelerated, in recent years. Applications of fluid mechanics relevant to oil & gas exploration, aerospace, manufacturing, and construction industries continue to form an important part of research activity. The importance of research impact in the REF is no doubt partially responsible for driving this trend forward.

The Mathematics Study Groups with Industry, a feature of UK mathematics for almost 50 years, seem to attract ever greater interest. Continuum mechanics forms a smaller part of these events than it did at their inception, but this is perhaps due to the rapid expansion in the range of applications of other parts of mathematics rather than any inherent reduction in scale or importance of fluid & solid mechanics. Other opportunities for engagement of mathematicians of all kinds with industry have also developed in recent years, including the founding of the Turing Gateway at the INI in Cambridge as a national resource for bringing together academics and industry, and expansion of industrial engagement activities at ICMS. An increasing number of University mathematics departments have also employed their own Knowledge Transfer Facilitators or Industry Liaison Officers to increase impact (examples include Bath, Manchester, Cambridge).

Solid Mechanics:

The area of modelling of Solids in the UK has generated an impressive range of inter and intra disciplinary activities. This includes initiatives supported by EPSRC (Oxford, Imperial, Liverpool, Manchester) as well as EU funded initiatives (Aberystwyth, Liverpool, Loughbourough), which strongly link academic and industry, that include British Aerospace, Ministry of Defence, NHS as well as a range of EU based companies outside the UK.

The evidence of success is also provided by the recent (2015) Springer publication by Philip Aston et al. on “UK Success Stories in Industrial Mathematics”, which includes a number of successful impact cases in continuum mechanics, including solids, elastic waves and fluids, with the interface to industrial research and real life engineering and industrial applications.

Future Direction/Opportunities

Fluid Mechanics:

Nano-fluidics, `lab on a chip’, is an area of very significant growth potential, in which the UK still seems to lag behind international competitors, especially the US. The advent of new materials, as discussed below, will no doubt drive considerable research efforts. However, at the boundary between fluids and solids, the level of activity in research on `soft matter’ in UK mathematics departments can be enhanced further. At present, excellent work has been carried out on liquid crystals (Strathclyde), granular materials (Manchester and Bristol) and nonlinear viscoelasticity (Glasgow, Manchester). Biological Fluid Mechanics, always a strong subject in the UK, will no doubt see yet further development, in part driven by increasingly good experimental technique (for instance, Ray Goldstein’s work on topological changes during cell development and microbial locomotion have only been made possible by rapid improvements in visualisation techniques in recent years). There is no doubt that a number of areas which have already been identified as key challenges, including applications in medicine and climate change, will continue to gain in importance. Environmental modelling is a particularly broad area, with opportunities for further applications of both fluid and solid mechanics in prediction and mitigation of natural disasters, and in high-tech agriculture. Solid Mechanics:

An `explosion’ of academic and industrial interest was observed in the area of modelling of multi- scale metamaterials. Major discovery made in Manchester (Graphene success) and the pioneering work by the group of Pendry (Imperial) has stimulated this development greatly. The recent EPSRC program grant award to Imperial and Liverpool is a very exciting development, which has stimulated the research in this area. Oxford non-linear analysis of PDEs has generated several world leading directions in modelling of phase transition, damage of solids and plasticity. Very exciting new developments on the interface of biology, medicine and solid/fluid interaction is indeed very promising as well. There are opportunities in the modelling of smart structures, new designer materials and much else where the UK solid mechanics community will have the potential to make significant impact.

Two (out of five) EPSRC funded centres in Maths with Healthcare awarded to Liverpool and Glasgow, included a substantial bulk of challenging research work on solid mechanics, fluid-solid interaction, and soft tissue mechanics, which has a very high potential impact. The other three centres include Cambridge, Imperial and Exeter.

In general, significant opportunities are seen for further interaction of continuum modelling with both experiments and numerical analysis and computation, including greater involvement in uncertainty quantification and development of high-speed calculations for optimisation and control of systems in real time.

In the recent years mathematical modelling of solid/fluid interaction in the framework of a hydraulic fracturing has received a lot of attention and funding, especially from the European Commission. This is linked to a relatively new and environment sensitive industrial activity related to exploration of shale gas in the UK.

Further Comments

• CDTs in the areas incorporating modelling of Solids and multi-scale metamaterials would flourish in the UK and EPSRC support in this area would be much appreciated.

• A more secure funding base for Industry Study Groups would be helpful, as would a more long-term approach to funding key infrastructure in the form of the INI and the ICMS.

• It is suggested strongly that collaborative Programme grants/CDTs involving maths with materials/physics/biology in the areas of materials, should be further encouraged by EPSRC (many associated with fluids/solids).

• It is noted that Continuum Mechanics as a research area offers a great deal – in an engineering context but especially on the fundamental mathematical level.

• One issue which needs to be analysed is the extent to which fluids/solids work is being funded by mathematics from within EPSRC (many of the large grants mentioned earlier involve contributions from other disciplines and funding from other EPSRC themes). Of course, it is a good thing that the area is able to attract funding in an inter-disciplinary way, but it should also be remembered that fundamental study in continuum mechanics continues to be a significant part of UK mathematical sciences research. This is especially important for more fundamental or blue skies work, which might not necessarily have yet reached a point where it can have significant inter- disciplinary impact.

• Another issue for concern is the danger of research in continuum mechanics (particularly within mathematics) becoming more focused within a decreasing number of centres of excellence. This would not be a good thing for UK community as a whole, especially given the continued geographical diversity of world-leading research within the UK. The interdisciplinary nature of continuum mechanics means that funding routes which encourage collaboration beyond the discipline also need to be maintained.

• Professor David Abrahams, the newly appointed Director of Isaac Newton Institute, is a highly respected and internationally recognised expert in Solid Mechanics. With his innovative and creative approach, and his enthusiasm, the research area of Solid Mechanics in the UK will have additional opportunities to develop from strength to strength and to produce further major impact.

• Finally, the UK has a fine tradition of experimental work on continuum mechanics being conducted within mathematics departments, with the close-coupling yielding significant benefits for both theoreticians and experimentalists. This important resource needs to be maintained in the future.

Key Research Groups in the Area

Fluids: The list of excellent research groups working in this area in the UK includes Imperial, City, UCL, St Andrews, Edinburgh, Bristol, Bath, Exeter, Oxford, Cambridge, UEA, Nottingham, Manchester, Loughborough, Leeds, Birmingham, Reading, Glasgow, Nottingham, Strathclyde, Keele.

Solids and waves: The list of excellent research groups working in this area in the UK includes Oxford, Cambridge, Imperial, Manchester, Liverpool, Keele, Bath, UCL, Brunel, Aberystwyth, Glasgow, Nottingham, Strathclyde, Loughborough.

Bibliography

Some interesting impact cases are summarised in the recent Springer book by Philip Aston et al: UK Success Stories in Industrial Mathematics, 2015.

Deloitte-EPSRC report of 2013 was also used as a source of the factual information.

The following colleagues were approached for feedback and suggestions on the above Landscape Document: I.D. Abrahams, J.M. Ball, R. Craster, S.J.Chapman, , Y. Fu, N. Hill, O.Jensen, J. Kaplunov, J. King, G. Mishuris, S. Mikhailov, W. Parnell, G. Rogerson. Geometry and Topology

Lead Authors: J.P.C.Greenlees, U.Tillmann Consultants: T.Bridgeland, M.Haskins, N.Hitchin, M.Lackenby, G.Paternain, I.Smith and R.Thomas 1. Statistical Overview: We warn that the following figures from 2008 and 2015 are not directly comparable, as is apparent from the different descriptors. Total headcount attributed to Geometry and Topology in RAE2008: 178 Total headcount attributed to Geometry and Topology in 2015: 224 The proportion of Early Career Researchers’ Outputs submitted to RAE2008 was 23%. The proportion of G&T staff with a PhD after 2008 in 2015 was 21%. 2. Subject breakdown: The table indicates the number of active researchers in designated sub- areas. Data extracted from RAE 2008 submission and current staff lists; again the absolute numbers are not directly comparable. (Use of REF2014 data would be inappropriate because institutional submission strategies meant many active researchers were excluded for strategic reasons).

Research Area RAE2008 (No.) % of G&T Researchers in 2015 (No.) % of G&T Algebraic Geometry (AG) 44 25 65 29 Algebraic Topology (AT) 48 27 49 22 Geometric Topology (GT) 36 20 41 18 Differential Geometry (DG) 50 28 69 31 The growth in Algebraic and Differential Geometry capacity, and the shrinking in Algebraic Topology is consistent with the current international context.

3. Discussion of research areas: The period since the 2010 Landscape has been one of intense activity and progress in the area of Geometry and Topology worldwide, with UK highlights represented by the following signs of international recognition.

Invited ICM2014 talks: • Mihalis Dafermos (Cambridge) “The mathematical analysis of black holes in general relativity” • Mark Gross (Cambridge) “Local mirror symmetry in the tropics” • Andr´eNeves (Imperial) “New applications of Min-max theory” • Constantin Teleman (Oxford) “Gauge theory and mirror symmetry” • Peter Topping (Warwick) “Ricci flows with unbounded curvature” Major prizes: • New Horizons Prize 2016, Veblen Prize: Andr´eNeves (Imperial) (for work around his proof of the Willmore Conjecture)

1 • Breakthrough Prize 2014: Simon Donaldson (Imperial) (for his work on K¨ahler-Einsteinmetrics) • 2016 American Mathematical Society Moore Prize: C.Birkar (Cambridge) and P.Cascini (Imperial) The UK has several world-leading research groups and has been fully involved in international develop- ments as we describe below. Although a number of high-profile researchers have left the country (Totaro, Weiss, Markovic), the hiring of M.Gross, M.Dafermos, C.Teleman, and a number of very strong early stage appointments paint a general pattern of vigour and development. In what follows, particular centres and particular researchers are named as principal points of reference for the Review Panel. These lists are not exhaustive and it is inevitable that important names and places have been omitted. Generally, we have described areas of activity rather than particular results. The CDT ‘London school in Geometry and Number Theory’ has benefitted areas of London strength. Algebraic geometry Centres: Bath, Cambridge, Edinburgh, Imperial, Liverpool, Oxford, Sheffield, Warwick. Headlines: Outstanding UK contributions to analytic methods, mirror symmetry and enumerative geometry; a wealth of other important work, particularly in moduli theory. The main areas of algebraic geometry internationally can be described as: minimal model theory; moduli spaces; enumerative geometry and mirror symmetry; analytic methods; algebraic cycles; and combinatorial and computational algebraic geometry. The applications of algebraic geometry to other subjects such as number theory, commutative algebra, and representation theory should be discussed in other Landscape documents. Minimal model theory. This subject is still buzzing from the spectacular proof of finite generation for the canonical ring by C.Birkar (Cambridge), P.Cascini (Imperial), Hacon, and McKernan in 2006. Further important progress on the minimal model programme has been made in the decade since then. Attention is now turning to understanding how much of the programme can be made to work in characteristic p, with important progress made by P.Cascini. Overall the UK is strong in this area: M.Reid (Warwick) and A.Corti (Imperial) played major roles in the developments leading to finite generation. Other sig- nificant UK experts include G.Brown (Warwick), A.Pukhlikov(Liverpool), I.Cheltsov (Edinburgh) and N.Shepherd-Barron (KCL). Moduli spaces. The group around D.Joyce and B.Szendroi in Oxford made important progress on moduli spaces of coherent sheaves on Calabi-Yau threefolds using the new theory of derived algebraic ge- ometry developed by B.Toen, G.Vezzosi and J.Lurie. These homotopical techniques will undoubtedly be increasingly important in moduli theory as well as in many other areas of algebraic geometry. J.P.Pridham (Edinburgh) is a leading expert in homotopical algebraic geometry, but the UK would do well to build strength in this area. A.Bayer (Edinburgh) and E.Macri introduced important new derived category meth- ods for studying the birational geometry of moduli spaces of coherent sheaves on surfaces; this has led to a flurry of activity worldwide. M.Wemyss (Edinburgh) uses moduli spaces of representations of non- commutative algebras to elucidate homological aspects of the minimal model programme. Other activity connecting to representation theory includes T.Schedler (Imperial). B.Noohi (QMUL) is an expert on stacks. More classical areas of moduli theory are also well-represented in the UK: G.Sankaran (Bath) stud- ies moduli of K3 surfaces and hyperk¨ahlervarieties, N.Shepherd-Barron (KCL) is an expert on the moduli space of abelian varieties, A.Craw (Bath) and A.King (Bath) study moduli of quiver representations, and activity on vector bundles over curves continues through Liverpool and the VBAC group. F.Kirwan (Ox- ford) has worked on extending GIT to non reductive groups with applications in various areas, such as jet spaces and the Kobayashi hyperbolicity conjecture. Mirror symmetry and enumerative geometry. The UK contribution in this area is world-leading. M.Gross (Cambridge) and his collaborators have applied ideas from mirror symmetry to solve important conjectures on surface singularities and cluster varieties. R.P.Thomas (Imperial) with R.Pandharipande proved the Katz-Klemm-Vafa conjecture expressing the Gromov-Witten theory of K3 surfaces in terms of modular forms. R.P.Thomas, M.Kool (Imperial postdoc) and V.Shende solved the Gottsche conjec- ture about numbers of nodal curves in general linear systems on surfaces. Previous work of D.Joyce on wall-crossing has become much-studied internationally. T.Bridgeland has made important advances and is

2 building a significant group in Sheffield. D.Joyce, B.Szendroi and others have pursued an ambitious project to categorify Donaldson-Thomas invariants. T.Coates (Imperial) and A.Corti (Imperial) are working on an exciting new approach to the classification of Fano manifolds and orbifolds via mirror symmetry and Gromov-Witten calculations. Analytic methods. The highlight here is the historic proof by S.Donaldson (Imperial) (with X.Chen and S.Sun) of the existence of K¨ahler-Einsteinmetrics on K-stable Fano manifolds. This was a hugely important open problem, and Donaldson’s proof earned him the $3 million breakthrough prize. UK experts in this area also include J.Ross (Cambridge) and R.Dervan (Cambridge PhD student). Other analytic methods in algebraic geometry, typified by Siu’s nonvanishing theorems and Demailly’s work on hyperbolic complex manifolds, are not represented in the UK. Algebraic cycles. The theory of algebraic cycles, typified by the Hodge conjecture, is an important area of algebraic geometry. The theory embraces Hodge theory, motivic homotopy theory, and applications to quadratic forms and algebraic groups. Generally this area is under-represented in the UK, especially since the loss of B.Totaro to the US, but V.Guletskii (Liverpool), C.Vial (Cambridge) and A.Vishik (Nottingham) are all doing important work. Combinatorial algebraic geometry. Algebraic geometry has a strong combinatorial side which has developed internationally in recent decades, including subjects like algebraic combinatorics and Schubert calculus, Gr¨obnerbases, and tropical geometry. D.Maclagan (Warwick) and M.Hering (Edinburgh) are leading UK researchers in this area. Tropical and non-Archimedean geometries are beginning to play a major role in algebraic geometry and mirror symmetry. The UK has some expertise here in J.Nicaise (Imperial), M.Gross (Cambridge), D.Maclagan (Warwick), and recently saw exciting new foundations developed for both subjects by A.MacPherson (Imperial PhD student), J.Giansiracusa (Cardiff), K.Kremnizer (Oxford) and O.Ben-Bassatt (Oxford). Computational algebraic geometry. Many mathematicians and users of mathematics rely on computer algebra programs. The algorithms for computational algebraic geometry have mostly been developed outside the UK. There is much potential here for productive interaction between algebraic geometry and other subjects. Part of the Coates-Corti project referred to above joint with A.Kasprzyk (Nottingham) involves massive parallel computation and has led to big new contributions to the Matlab and Sage libraries. Algebraic topology Centres: Aberdeen, Oxford, Sheffield. Headlines: Strong participation in algebraic topology of manifolds. Unexploited opportunities there and elsewhere. The main areas of algebraic topology internationally can be described as: abstract homotopy theory; stable homotopy; algebraic topology of manifolds; applications to algebraic K-theory. Thanks to three successive waves of foundational revolution (structured ring spectra, model categories and ∞-categories), a number of dreams are becoming accessible. Internationally there is an influx of new talent, not yet well reflected in the UK. Abstract homotopy theory. The machinery of Quillen model categories and ∞-categories is being used to excellent effect in many areas of mathematics (commutative algebra, representation theory, K- theory, combinatorics, algebraic geometry and category theory), and homotopy type theory links to logic and computation. There is some UK activity (notably I.Moerdijk, N.Strickland (Sheffield) and A.Tonks (Leicester)), but much participation is secondary, and it should be enhanced. Related activity in higher category theory is represented by N.Gurski (Sheffield), S.Paoli (Leicester) and T.Leinster (Edinburgh). The simplicial approach to higher categories following Joyal and Lurie are not well represented in the UK. Algebraic topology of manifolds. The study of manifolds has seen major developments growing out of the Atiyah-Segal approach to topological quantum field theory (TQFT). These are on the one hand the proof by Madsen and Weiss of the Mumford conjecture and on the other hand the work of Lurie and Hopkins on the cobordism hypothesis. This has led to the study of diffeomorphisms of manifolds and associated homology stability questions, as well as the quest for explicit classifications of TQFTs in low dimensions. Major results in the first direction have been achieved by Galatius and O.Randal-Williams (Cambridge) who prove an analogue of the Mumford conjecture for higher dimensional manifolds (presented at ICM 2014

3 by Galatius). Using very different techniques, strong results on the classification of TQFTs have been proved by C.Teleman (presented at ICM 2014), A.Juhasz, as well as C.Douglas and collaborators (all Oxford). Significant related work by J.Giansiracusa (Swansea), R.Hepworth (Aberdeen), U.Tillmann (Oxford) and her students studies questions on operads, configuration spaces and homology stabilitiy. While the departure of M.Weiss has been a loss in this area, C.Teleman’s return to the UK represents a major gain, and O.Randal-Williams’s appointment in Cambridge is an opportunity. Other more classical approaches to the study of manifolds such as surgery theory and K-theory are represented in the UK by A.Ranicki (Edinburgh) and D.Crowley (Aberdeen). Stable homotopy theory. The use of structured spectra has been a great enabler, permitting analogies to graduate to constructions; examples in the study of moduli spaces and TQFTs are discussed above whilst those to geometric topology are not presently as active. Applications to algebraic geometry and mirror symmetry and descent are not strongly represented in the UK; Schlichting (Warwick) represents applica- tions to K-theory. Traditional stable homotopy theory and homotopy type theory have been represented by S.Whitehouse and N.Strickland (Sheffield). Commutative algebra of ring spectra continues to provide structural insights, for example J.Greenlees (Sheffield) and collaborators explain observed dualities in algebraic K-theory and elliptic cohomology with level structures. The pioneering work of A.Baker (Glasgow) and Richter on Brauer groups is feeding in to the striking results of Antieau and Gepner. Techniques originating in stable homotopy theory for the global study of derived categories and homotopy categories have continued to be a theme of (D.Benson (Aberdeen), most recently in the representation theory of group schemes (with Iyengar, Krause and Pevtsova). The classification of rational equivariant cohomology theories has developed beyond tori and free spaces to include the apparatus of representation theory J.Greenlees (Sheffield), D.Barnes (Belfast) and Kedziorek (Sheffield PhD student). Unstable homotopy theory. The local homotopy theory of classifying spaces (homotopical theory of groups and finite loop spaces) has made significant progress, including the classification of p-compact groups. R.Levi (as part of the leading Broto-Levi-Oliver team) and his group in Aberdeen have played a central role. This is intimately related to the representation theory of finite groups. More traditional homotopy theory now has a foothold in Southampton (J.Grbic, S.Theriault). Geometric topology (including geometric group theory; see also the Algebra Landscape) Centres: Oxford, Warwick, Cambridge, Southampton. Headlines: An area of increasing strength in the UK. The main areas of geometric topology can be described as low-dimensional geometry/topology and ge- ometric group theory. Traditionally, high-dimensional manifold theory is also part of this field, but has seen little activity in recent years, particularly within the UK. Internationally, geometric topology is highly- regarded and very fast-moving, contributing 6 Fields medalists over the past 30 years. In the UK, the subject underwent a period of decline about 15 years ago, but is now a world-leader. The most notable advance in the field in the past 5 years was the proof by Agol and Wise of the Virtually Fibred Conjecture. This resolved a longstanding question of Thurston about hyperbolic 3-manifolds, and it required dramatic developments in both low-dimensional topology and geometric group theory. As a result of this work, Agol won the 2015 Breakthrough Prize. Poorly represented in the UK. Low-dimensional geometry/topology. A cornerstone of the proof of the Virtually Fibred Conjecture was groundbreaking work of V.Markovic (Cambridge, but recently moved to Caltech) and J.Kahn, who showed that every hyperbolic 3-manifold group contains a surface group. M.Lackenby (Oxford) has remained very active at the interface between topology and geometry in dimension 3, and addressing fundamental questions about knots and links. At Warwick, S.Schleimer and B.Bowditch have continued to analyse mapping groups of surfaces, which form an important topic straddling both low-dimensional topology and geometric group theory. A notable area that has seen significant expansion internationally and in the UK is that of Heegaard Floer homology. This is concerned with new powerful invariants of low-dimensional manifolds. A.Juhasz (Oxford), J.Rasmussen(Cambridge), B.Owens (Glasgow) L.Watson (Glasgow) all work in this field. A.Lobb (Durham) works in the adjacent area of gauge-theoretic invariants of 3-manifolds.

4 The study of minimal surfaces has been a major branch of geometry for the past 100 years, and given their utility in manifold theory, this topic can also be viewed as part of geometric topology. In this field, the standout work has been by F.Cod´aMarques and A.Neves (Imperial), who solved the Willmore conjecture. Geometric group theory. This subject has grown tremendously in the past 20 years. It can now be considered a major part of geometric topology, closely related to hyperbolic geometry and low-dimensional topology. The UK is world-class in geometric group theory. Oxford has the strong team of M.Bridson, C.Drutu, and P.Papasoglou, as well as M.Lackenby in related aspects of 3-manifold theory. Warwick also has a particularly strong research group, with B.Bowditch, S.Scheimer, and new appointments K.Vogtmann and V.Gadre. Another recent appointment is H.Wilton (Cambridge), who has been doing some important work on the interaction between geometric and profinite group theory and 3-manifold theory. Southampton is also well represented in goemetric group theory with I.Leary, G.Niblo, A.Minasyan, J.Anderson and P.Kropholler. Connections to representation theory (cluster algebras of triangulated surfaces using hyperbolic geometry and geometric group theory) are reflected in the recent appointment of A.Felikson (Durham). Differential geometry Centres: Bath, Cambridge, Durham, Loughborough, Imperial, KCL, UCL Oxford, Warwick. Headlines: Apparent from major prizes of Dafermos, Donaldson, Neves, Topping. The main areas of differential geometry internationally can be described as: classical differential geometry; geometric analysis; symplectic and contact geometry. There are close links with dynamical systems as well as twistor theory and mathematical physics which we leave for other Landscapes. Two of the main achievements in this period were the work of S.Donaldson, Chen and Song on stability and K¨ahler-Einsteinmetrics and that of A.Neves (Imperial) and Marques on the proof of the Willmore conjecture. Donaldson’s main work was done at Imperial, where he retains a 40% contract (the remaining 60% is at Stony Brook). Donaldson’s shift is a significant loss to the UK but he leaves in his wake a thriving community of differential geometers in London. Mathematicians like A.Neves and M.Haskins at Imperial are US-trained in analytical techniques and have significant achievements. Neves is now internationally renowned for his work but Haskins, along with J.Nordstrom (Bath) and algebraic geometers, have made important advances in the study of G2 manifolds, a subject which Donaldson in his Breakthrough Prize address pinpointed as a future growth point. KCL has also developed in this period a strong research group in differential geometry (J.Berndt, S.Salamon, G.Tinaglia, D.Panov) as has UCL (M.Singer, J.Lotay, F.Schulze, C.Wendl). The CDT in Geometry and Number Theory reinforces this strength. There has been in general a noticeable increase in the area of geometric analysis (a subject which has in the past been a source of concern for reviews of UK mathematics) with mathematicians such as E.Hunsicker (Loughborough), R.Moser (Bath), R.Mueller (QMW), O.Post (Durham), M.Rupflin (Oxford) being appointed. P.Topping (Warwick) has long been a home-grown geometric analyst but there is now more expertise in the UK. Symplectic geometry and topology is a growth area worldwide especially in its categorical aspects. This is reflected in the UK, with I.Smith (Cambridge), C.Wendl (UCL) and A.Ritter (Oxford) representing the theme, with J.Evans (UCL), and Y.Lekili (KCL) as new appointments. D.Joyce in Oxford has produced foundational work in this context, linking it with the string theorists mirror symmetry and counting issues. J.Kedra (Aberdeen) reflects more topological approaches. In Sheffield and Manchester there is an active group in Poisson geometry and higher order structures. Much differential geometry is also done in the UK under the aegis of Theoretical Physics (in Oxford, Cam- bridge and Edinburgh in particular). There is also a strong link with integrable systems in Loughborough, Leeds and Bath for example. The UK has a significant number of researchers working at the interface of geometry and integrable systems. The main research groups in this area include Bath (led by F.Burstall, D.Calderbank), Lough- borough (J.Ferapantov, M.Mazzocco, S.Veselov) and Oxford (A.Dancer) along with smaller groups elsewhere.

5 Other UK strengths include the link between integrable systems, loop groups and harmonic maps to highly symmetric targets (notably F.Burstall, N.Hitchin). J.Wood (Leeds) has also made major contributions to harmonic maps and morphisms. More recent strengths are in special Frobenius manifolds (I.Strachan (Glasgow), S.Veselov and collaborators) and the development of generalised geometry by Hitchin and his recent students. Geometric analysis. Geometric analysis has emerged as one of the most powerful tools in differential geometry, and has been boosted since Perelman’s proof of the Poincar´econjecture. P.Topping (Warwick) is a leading figure in the field. There is an emerging group in geometric inverse problems (this is a kind of geometric analysis): L.Oksanen (UCL), S.Kurylev (UCL), G.Paternain (Cambridge), S.Holman (Manchester). Other activity under geometric analysis includes J.Lotay (UCL), and connecting geometric analysis to relativity G.Holzegel, C.Warnick (Imperial), L.Nguyen, Q.Wang (Oxford) and M.Dafermos (Cam- bridge).

4. Discussion of research community in geometry and topology Although a number of high-profile researchers have left the country, the general pattern has been of vigour and development. Amongst the active researchers, the numbers obtaining a PhD in 5-year blocks 1969–2009 are 15, 11, 18, 24, 26, 37, 44, 53, which supports the subjective impression that there was a low point around 1980 and that there has been a resurgence of interest more recently. The corresponding figures for female researchers only is 0, 1, 1, 1, 3, 5, 6, 5. Altogether, about 10% of the active researchers in G&T are female. Amongst those who received a PhD from 1990 onwards (i.e., including 2005-09), about 12 % are female, but this does not allow for the number of people who leave mathematics soon after a PhD. In any case this compares unfavourably with the proportion across the Mathematical Sciences (around 20%). Year of award PhDs in G&T Female PhDs in G&T 1965-69 12 1 1970-74 15 0 1975-79 11 1 1980-84 18 1 1985-89 24 1 1990-94 26 3 1995-99 37 5 2000-04 44 6 2005-09 53 5 For those outside the main centres, a major factor in maintaining the research community are the LMS- supported Scheme 3 seminars. These include the COW (Cambridge-Oxford-Warwick) algebraic geometry seminar, the Bristol-Oxford-Southampton geometric group theory seminar, and the Transpennine Topol- ogy Triangle (Leicester-Manchester-Sheffield), Applied Algebraic Topology (Aberdeen, Durham, QMUL, Southampton), Manifolds (Aberdeen-Cambridge-Oxford).

5. Inter/Intra-disciplinary activities and Engagement activities: There are numerous interactions between geometry, topology and other areas of mathematics. Con- nections between algebraic geometry and representation theory are strong (R.Bocklandt, T.Bridgeland, A.Craw, I.Grojnowski, A.King), as are those between algebraic topology and representation theory (K.Kremnitzer (Oxford), Benson (Aberdeen)), and are discussed in the Algebra Landscape document. The ties between algebraic geometry and number theory are fundamental, for instance at Cambridge, Im- perial and Oxford (see the Number Theory Landscape document); Geometric group theory is on the border between geometry and algebra. There are close relations between geometric topology and dynamical systems, most visible at Warwick. J.Hunton (Durham) has applied algebraic topology to the study of aperiodic tilings, D.Buck (Imperial)

6 uses three-manifold and knot theory to illuminate structural and mechanical features of DNA-protein inter- actions, while M.Farber (QMUL) has an active interest in robotics and stochastic topology. Topological Data Analysis has attracted much recent interest including from the Alan Turing Institute. UK topolo- gists actively involved include J.Brodzki (Southampton), V.Kurlin (Durham), R.Levi (Aberdeen), with Oxford emerging as a centre of activities. 6. Future Direction/Opportunities: The value of groups and centres in building strength is indis- putable, but for sustainable research capacity it is essential that there is broad geographical distribution. For the flexibility of capacity and the overall health of UK mathematics it is fundamental to have geographical spread without sacrificing intellectual and geographical connectivity. The following are picked out from the text in Section 3. Areas of significant activity and strength. Minimal model programme. Mirror Symmetry. Manifolds and field theories. Special holonomy. Geometric analysis. Opportunites to repair UK weakness. Homotopical algebraic geometry. ∞-categories. Hyperbolic mani- folds. Computational geometry and topology. Intra-disciplinary links. Representation theory, group theory, commutative algebra, algebraic K-theory, number theory, integrable systems, PDEs, relativity, theoretical physics, Cross-disciplinary opportunities. Topological data analysis. Theoretical computer science. Computer vision.

7 Logic

Lead Author: Dugald Macpherson, in discussion with Anand Pillay. Consulted: Michael Benedikt, Anuj Dawar, Mirna Dzamonja, Nicola Gambino, Volker Halbach, Leonid Libkin, Jeff Paris, Michael Rathjen, Philip Welch, Alex Wilkie.

1. Statistical Overview: Logic in the UK is diverse, with high level research both in pure mathematical logic, and in its interactions with several other parts of mathematics and with computer science (CS) and philosophy. There are far more logicians in CS than in mathematics departments, and a significant number also in philosophy departments. Most of these are not included in the statistics below as they are not likely to submit grant proposals to the EPSRC Mathematics Portfolio. Connections to computer science and to philosophy are discussed mainly in Section 5. Within UK mathematics departments, model theory has considerable strength and the largest number of logicians. There are growing and excellent groups in set theory and in proof theory/categorical logic. Computability theory within mathematics departments is currently small, due to recent loss of staff.

2. Subject breakdown: The table indicates the number of active researchers in designated sub-areas. It is based on consultations with representatives of the fields below. It includes retired but active logicians, and researchers with permanent UK positions and a strong logic connection, but excludes postdocs, and those on a series of fixed term positions (some of whom are mentioned in later text). On the logic/computer science border, it includes only those likely to submit EPSRC applications to the Mathematics Portfolio, so grossly underestimates the overall UK activity in logic in CS. Research Area Number of % of Total in Researchers 2015

Model theory 20 46 Proof theory, constructive 12 28 mathematics, categorical logic, reasoning Set theory 5 12 Computability theory 1 2 Logic in computer science 5 12

3. Discussion of research areas: The traditional areas of mathematical logic are model theory, set theory, proof theory, computability theory (aka recursion theory). Research in the first two is clearly part of mathematics, but proof theory and computability theory span mathematics, computer science, and philosophy. Model theory has a sophisticated internal theory, but the bulk of activity tends to be on interactions with other parts of mathematics, often through a close analysis of geometric and combinatorial properties of `definable sets’. Set theory and especially proof theory retain close connections with the foundations of mathematics in the sense of Hilbert and Gödel. Proof theory typically concerns the strength of axiom systems, for example calibrating (in `reverse mathematics’) the arithmetic axioms needed to prove major mathematical results, or extracting computational content from constructive proofs; there have been increasing interactions with categorical logic, which is concerned with applications of category theory within mathematical logic, and has links to other disciplines, e.g. to geometry via topos theory and to topology via homotopy type theory. We do not here discuss the role that category theory has in other parts of mathematics such as geometry, topology, and algebra. Set theory has links to proof theory (e.g. through constructive set theory) and to philosophy of science (through foundations of mathematics); much current set-theoretic activity is concerned with axiom systems extending ZF, independence results, and applications for example in analysis and topology. Computability theory has traditionally focussed on the mathematical theory of `effectiveness' or `computability in principle', in particular on the Turing degrees and related structures. More recently, the techniques have been applied to measure and `algorithmic’ randomness, and to reverse mathematics.

Proof theory, below viewed together with constructivism, categorical logic, and (uncertain) reasoning, is practised in both mathematics and CS departments, and sometimes also in philosophy departments. Computability theory has aspects (Turing degrees, randomness, connections to proof theory) which are clearly mathematics, and other aspects (theoretical underpinnings of computing) belonging more to computer science.

Model theory. The UK is a world leader in the more applied parts of model theory, oriented towards algebra, algebraic geometry, and number theory.

A striking development, reported in the 2010 Logic Landscape document, is the application of ideas from o-minimality (in model theory) to diophantine-geometric problems, using fundamental work of Pila and Wilkie (Oxford). It has led to the first unconditional proof, due to Pila, of cases of the André-Oort conjecture, and to big progress on the Zilber-Pink conjectures, and related functional transcendence questions. For example, Pila and Tsimmerman [25] proved an `Ax-Lindemann’ theorem for the moduli spaces of abelian varieties, verifying the André-Oort conjecture for these up to dimension 6. They also established [26] the Ax-Schanuel property for the j-function, and, with Habegger, Pila [18] gave a conditional proof of the Zilber-Pink conjecture for products of modular curves. Pila and Wilkie (with others) were awarded the Association of Symbolic Logic’s 2013 Karp Prize, and Pila (FRS 2015, ICM plenary speaker 2014) was also awarded Clay Research Award and LMS Senior Whitehead Prize in 2011. In addition to Pila and Wilkie, Jones [6,20,21], a recent lecturer appointee in Manchester, has important work in this area, partly proving cases of a conjecture of Wilkie on the number of rational points on a definable set in a real exponential field, and partly more number-theoretic but motivated by model theory.

The UK has been central for some 15 years in a fruitful programme, led by Zilber (Oxford, LMS Polya Prize 2015), to understand the logic of the complex exponential field. Combining ideas from transcendence theory (Schanuel’s conjecture), diophantine geometry, and model theory (abstract elementary classes), Zilber gave an amazing construction of a `pseudo- exponential field’ with good model-theoretic properties (quasi-minimality – every definable set is countable or co-countable) and conjectured that this structure is isomorphic to the complex exponential field. This work led to the Zilber-Pink diophantine-geometric conjectures mentioned above. Delicate issues around pseudo-exponentiation have now been clarified by Kirby (UEA), Mantova (starting a 3-year Leeds position in 2016), Zilber himself, and co-authors. Wilkie [31] continues to work on the influential conjecture that the complex exponential field is quasi-minimal, and Macintyre with others [12] showed that Shapiro’s conjecture of complex analysis follows from Schanuel’s conjecture. Schanuel-type conjectures arise in other work of Zilber [32] connecting model theory to Shimura varieties, and making links to the Grothendieck-André period conjecture. Zilber participates with Fesenko (PI), Hitchin, Kim, and Kremnizer in an intradisciplinary EPSRC Programme Grant.

Since the Ax-Kochen/Ershov work in the 1960s, the model theory of valued fields has been a powerful tool with diverse applications, and source of new model-theoretic concepts. UK experts include Derakhshan, Koenigsmann, and Macintyre (Oxford), Halupczok and Macpherson (Leeds), Anscombe (UCLan), and Fehm (moving to Manchester). In particular, Derakhshan and Macintyre have a recent body of work [13] on the model theory of the finite adeles, and Derakhshan and coauthors have applied motivic model theory to group theory. Halupczok [8,9], in important collaborations with Cluckers, Gordon, Yin, and others, has found applications of the model theory of valued fields, sometimes via motivic integration, for example to questions in p-adic representation theory. Macintyre, Derakhshan, Anscombe, Fehm, and Koenigsmann have all contributed on important definability questions for valued fields (e.g. in [7]), with implications around decidability and Hilbert’s Tenth problem. On the latter, Koenigsmann [22] showed that the integers are definable in the rationals by a universal formula; an existential definition would yield Hilbert’s Tenth problem for the rational field. The rich area of the model theory of fields with extra structure (valuations, differential operators, analytic structure, automorphisms) is also a focus of Tomasic (QMUL), a specialist in model theoretic-approaches to difference algebraic geometry.

There continue to be close interactions in the UK between model theory and algebra. Prest (Manchester), recently with two PDRAs (Gregory and Pauksztello), is a world leader in the model theory of modules and connections to category theory. He has recent results on the Ziegler spectrum (completely described for string algebras [23,27]), and (with Gregory [17]) on links between decidability and representation type. Tressl (Manchester) works on o- minimality, real algebraic geometry, spectral spaces and the spectra of rings, with a massive book in progress with Schwartz and Dickmann. Borovik (Manchester) has played a leading role in a long-running programme on simple groups of finite Morley rank (conjecturally algebraic groups, a conjecture now proved in `even type’). Insights from this and from model theory have driven his work on black box groups in computational group theory.

There has been UK strength since the 1980s on connections between model theory and permutation groups. This has been boosted recently through links to topological dynamics and Ramsey theory, and also to constraint satisfaction in theoretical computer science. Highlights here include a proof by Evans (Imperial) and Tsankov that a large class of automorphism groups has Kazhdan’s property (T), and a shift of focus towards transformation semigroups -- a 2015 Durham Symposium brought together model theorists and others (e.g. Cameron, Gould, Gray, Kambites, Mitchell) with different perspectives.

Activity developing the internal theory of model theory has been reduced since the departure of Pillay. It continues mainly through the continued exploration by Evans of `Hrushovski constructions’ as a rich source examples, through work of Anscombe and Kestner (UCLan) and Macpherson on pseudofinite model theory (exploring first order properties of families of finite structures – e.g. finite simple groups -- through general model theory applied to ultraproducts), and through Kirby’s work, motivated by Zilber’s exponential field, beyond first order model theory. Topological dynamics, mentioned above, also has connections to `pure model theory’ through investigation of definable groups acting on their type spaces – a subject developed by Pillay before he left Leeds and continued by Penazzi (UCLan). Kaye (Birmingham) continues to work on Peano Arithmetic, linking model theory and proof theory.

Model theory (stemming from a 2012 paper of Hrushovski) has fed into a major body of recent work of Breuillard, Green (Oxford), and Tao in additive combinatorics and approximate subgroups – see e.g. [4].

Proof theory, constructive mathematics, categorical logic, reasoning. UK strengths here span mathematics, philosophical foundations of mathematics, and computer science.

There is an internationally-prominent group in proof theory and constructivism at Leeds. This is led by Rathjen, who has done major work on the ordinal analysis of strong theories, with recent emphasis on the strength of type theories and their connections to intuitionistic set theories [28, 29]. The group has close links with Weiermann (Gent) and Harvey Friedman (Ohio), investigating themes at the interface of proof theory and combinatorics, and with Aczel (Manchester) and Lubarsky (Boca Raton) on themes relating to constructivism. On the CS side, there is also a major proof theory group in the Computer Science Department at Swansea University. Strengths include bounded arithmetic and logical complexity (Beckman), verification and the extraction of programs from proofs (Berger and Seisenberger), and type theory (Setzer). Seisenberger is involved in industrial collaborations with Siemens Rail Automation, Chippenham, UK, and Berger, Seisenberger, and Setzer are members of the Swansea Railway Verification Group. Oliva (QMUL) currently holds a Royal Society URF. He is working on classical, intuitionistic, and linear logic with an emphasis on proof translations and interpretati3ons, with applications in computable analysis.

Fujimoto (Bristol) has a joint appointment in mathematics and philosophy. One of his specialities is the proof theory of truth theories and set theories.

UK research in categorical logic maintains areas of traditional strength in exploring applications of category theory in logic and in theoretical computer science. The leading group in categorical logic in the UK is at the DPMMS in Cambridge, led by Hyland and Johnstone, two major international figures in the subject. Johnstone is currently completing the third volume of his definitive account of topos theory. A new group, focusing on connections with type theory and the theory of programming languages, has been established at the University of Strathclyde by Ghani, who moved there in 2008. The Quantum Group at the University of Oxford, recently joined by Staton (Royal Society URF), also uses methods of categorical logic. Categorical logic is also represented at Leeds, thanks to the appointment of Gambino in 2013, and in Manchester, where Schalk works on aspects related to linear logic. Recent work includes a proof by Gambino and Joyal [16] that operads and their bimodules can be organized in a cartesian closed bicategory.

The novel areas of research known as homotopy type theory and Voevodsky’s univalent foundations programme are well-represented, with significant activity by several groups (Cambridge, Strathclyde, Leeds, Nottingham), with links to computer-assisted proof- checking and the theory of programming languages (Microsoft Cambridge). Research in this area builds on the strong traditions in the UK in categorical logic (see above) and type theory (the LEGO proof assistant was designed and implemented at LFCS, University of Edinburgh; dependently-typed programming languages have been studied at Nottingham and Strathclyde). It benefits from the presence in the UK of experts in higher-dimensional category theory and homotopical algebra (e.g. Leinster in Edinburgh, Gurski in Sheffield and Paoli in Leicester). It should be noted that there is strong international competition on the subject, especially from INRIA (France), where the Coq proof assistant is developed, and the US, where the US Air Force for Scientific Research is funding a large grant on this topic, coordinated by Steve Awodey at CMU.

Homotopy type theory has strong links to research in theoretical computer science and ICT, especially in the groups at Strathclyde, Nottingham, Oxford and Cambridge.

In Manchester, Jeff Paris continues to direct a small but original research group in `uncertain reasoning’, around `foundations of probability theory in the setting of mathematical logic’ and `inductive logic’. Paris and Vencovska published a major monograph [24] in 2015. Several analytic philosophers have connections to this work, in particular a group at the University of Kent around J. Williamson, who work on inductive logic and Bayesian reasoning.

Set Theory. The UK set theory team is small, but the research of several individuals is at a high level, producing a stream of strong PhD students, some building academic careers.

Large cardinal axioms and forcing constitute a major international theme, with the UK prominent. It is emerging that singular cardinals are critical in this context from the combinatorial viewpoint, and clarify the boundaries of independence. This topical subject has been the subject of several recent conferences, and Dzamonja (UEA) [11,15] has recently published three key papers in the area. Among further work, a paper of hers solving a long- standing set-theoretic question on partial orders has led to a collaboration with computer scientists on the convergence of well-founded state systems; and towards analysis, she and Borodulin-Nadzieja classified Boolean algebras that carry a uniformly regular measure.

.[especially forcing axioms [1 ,1א Aspero (UEA) is expert on iteration techniques of forcing at In his work with Mota [2] he developed the technique of symmetric iterations which allows

This gives an .2א one to get the value of the continuum in the final model to be larger than

Brooke-Taylor (Bristol) works .(2א)important step into developing strong forcing axioms at H on large cardinals and connections to category theory and set theory, with long-term goals in algebraic topology. Through category theory, Vopenka’s Principle (VP, a large cardinal axiom) has had topological applications, and in the potentially important [5], Brooke-Taylor shows VP is indestructible under many forcing constructions.

Determinacy, and the fine structure of inner models of set theory, are central to the work of Welch (Bristol) [19,30]. He is using Global Reflection Principles to justify the use of Woodin cardinals in axiomatic set theory. Among other work on analytic aspects of set theory, he and his ex-student Le Sueur are starting to lift recent results of Montalban and Shore on the reverse mathematics of determinacy of two person perfect information games in subsystems of analysis, to results in weak subsystems of ZFC + `there is a measurable cardinal’.

A further UK theme is the connection between set theory, foundations, and the philosophy of mathematics. Meadows (Aberdeen) works on the set theory of theories of truth, at the interface of set theory and the proof theory of truth theories; at the opposite end of large cardinal theory, it contributes to theories of very weak subsystems of analysis and possible axiomatisations enlarged by a truth predicate. This area thus interacts with that of Fujimoto (Bristol, see Proof Theory), and Welch was also in 2011-13 an AHRC-funded Co-I in Oxford on such a research project. Dzamonja’s research on large cardinals also feeds into philosophy (work with Panza in Paris 1 on `asymptotic truth of axioms of ZFC’), as does the work of Aczel and Rathjen on constructive set theory for intuitionistic logic.

Computability Theory. With the recent death of Cooper (Leeds), Lewis-Pye (LSE), one of the world’s leading computability theorists, is now the only researcher with a permanent position in the UK working in core computability. Since he is also principally working in algorithmic game theory and AI at present, this leaves computability in the UK considerably weakened from its position five years ago. There remain basic, difficult and longstanding problems concerning structural properties of various “degrees of computability”: Turing degrees, recursively enumerable degrees etc. Harris (teaching fellow in Leeds) works in the enumeration degrees, while recent work by Lewis-Pye in this area [3,14] has concentrated on the study of algorithmic randomness and interactions with the Turing degree structure. Welch, a set theorist in Bristol, has done influential work on infinite time Turing machines, and recently on other forms of ‘higher order’ computation. There is a collaboration of Beggs (Swansea) with Tucker in CS at Swansea, on physical models of computation.

Logic in Computer science. UK logicians do influential research in several aspects of computer science, some covered above, but more under the remit of EPSRC’s ICT portfolio. A variety of activities are included there, including quantum information, automata theory and verification (with active groups in Edinburgh, Oxford, UCL, QMUL, Imperial, Warwick), logic and databases (Birkbeck, Edinburgh, Liverpool, Oxford), as well as description logics and logical aspects of the semantic web (Birkbeck, Liverpool, Manchester, Oxford).

4. Discussion of research community In model theory, Macintyre (QMUL) and Wilkie (Manchester) have retired since 2010, though both remain very active. Zilber (Oxford) is retiring from his Logic Chair in 2016 but is highly active, and Hrushovski, who has outstanding achievements over the last 30 years in model theory and applications, has been appointed to this Professorship. Pillay moved from Leeds to the USA in 2013. Evans moved in 2015 from UEA to Imperial. Model theorists newly with permanent UK posts include Fehm and Jones (Manchester), Halupczok (Leeds), and Kestner, Penazzi and Anscombe (a new group in UCLan). Derakhshan has done important research on fixed-term positions in Oxford, and Mantova has a 3-year position in Leeds.

In the area around proof theory, Schuster moved from Leeds to Italy, and was replaced in Leeds by Gambino. The group in Manchester in uncertain reasoning is somewhat reduced, with Wilmers now retired. The age profile raises concerns for the continuation of the excellent Cambridge group which has trained many UK logicians. Appointments of relatively young researchers (Gambino in Leeds, Staton in Oxford) may lead to other groups being created elsewhere. Leinster moved from Glasgow to Edinburgh. A significant departure from the UK is that of Alex Simpson (now in Slovenia).

Historically, young category theorists have tended in the UK to move to jobs in computer science departments. This is changing, partly through new appointments in mathematics departments and in other disciplines, and partly through EPSRC support.

Set theory in the UK is currently concentrated in Bristol (Welch) and UEA (Aspero, Dzamonja). Others linked to set theory include Meadows (Philosophy, Aberdeen), Forster (Cambridge), and some researchers in other subjects such as combinatorics and proof theory, who also publish and supervise PhD students in set theory (Leader in Cambridge, Rathjen and Truss in Leeds). There are other active researchers without permanent posts such as Morgan (Edinburgh), and Knight (Oxford) who works in set-theoretic topology. Brooke-Taylor (moving to Leeds in 2016) has held an EPSRC Early Career Fellowship in Bristol since 2013. The permanent positions of Aspero and Meadows were gained since 2010. There were two recent losses with the death of Kolman and departure from the UK of Bovykin (previously on a temporary post).

Computability theory suffered a big loss in 2015 with the death of Cooper (Leeds). Lewis- Pye moved in 2013 from Leeds to LSE in a shift towards algorithmic game theory and AI. Harris has a temporary post in Leeds, and Pauly (Cambridge 2011-15) moved to Brussels.

Overall, the age distribution in model theory, proof theory, and set theory is healthy, though model theory is hit by the recent and imminent retirement of very prominent senior researchers, and the departure of Pillay. There are PhD students at all the main logic centres, and postdocs, some with extended positions, at several. The group in Leeds is particularly large, with around 5 postdocs and 25 PhD students in the logic group in mathematics, and others in CS. Overall, there are around 15 postdoc logicians working in mathematics departments in the UK, and around 50 PhD students. Research-based employment prospects for logic PhDs are particularly strong, due to the close CS links.

The main Logic academic organisation in the UK is the British Logic Colloquium, a UK- registered charity with around 200 members. It runs an annual conference, and its level of activity has increased in recent years, with financial support offered to around 9-10 UK logic- related conferences and workshops in each of 2012, 2013, 2014, and 2015. EPSRC is the main funding source for UK research grants in logic. Other significant sources include the EU (Marie Curie and ERC), Leverhulme Trust, Templeton Foundation, LMS (for meetings), Royal Society (Fellowships), and the US Air Force (a grant).

5. Inter/Intra-disciplinary activities and Engagement activities Inter-disciplinary research in Mathematics/Computer Science. The strongest interdisciplinary links of mathematical logic are towards computer science. These arise in proof theory and category theory, which yield applications to automated reasoning and proof verification, programme extraction from proofs, data types, and semantics of programming languages. There is no clear dividing line between these areas and `logic in computer science’, which covers topics such as modal logics and description logics, computational complexity, database theory, aspects of AI, and aspects of quantum information theory.

Major developing currents include applications of logic to areas such as (i) systems biology (ii) quantum information and foundations of quantum mechanics (iii) computational game theory and economics (iv) database theory (v) description logics. Leading UK centres in (i) are Edinburgh (Hillston, Danos), and Microsoft Research Centre in Cambridge (Cardelli). In (ii), the quantum group in Oxford University’s Department of Computer Science, led by Abramsky and Coecke, has some 30 members and has been instrumental in building a new international community using new kinds of logical and category-theoretic methods. In (iii) Liverpool and Oxford (Wooldridge, Goldberg) are prominent. (iv) is led by groups in Edinburgh (Libkin, EPSRC Advanced Fellowship) and Oxford (e.g. Gottlob and Benedikt, with ERC and EPSRC Advanced Fellowships respectively).The semantic web has given increased recent prominence to (v), in which Horrocks (Oxford) is a leader. There are logic groups in the following Computer Science Departments (as well as individuals elsewhere): St. Andrews, Bath, Birmingham, Cambridge, Durham, Edinburgh, Imperial, Kings College London, Leeds, Leicester, Liverpool, Manchester, Nottingham, Oxford, QMUL, Swansea, UCL, and Warwick.

Interdisciplinary research in mathematics and philosophy. A number of logicians in philosophy departments carry out research at the interface of mathematical and philosophical logic. Departments with philosophers engaged in logical research include Aberdeen, Birmingham, Bristol, Cambridge, Leeds, Oxford, St Andrews, and Warwick. Philosophers take a strong interest in developments in mathematical logic, in particular, proof theory and set theory. Logic and Philosophy of Mathematics seminars (Birmingham, Leeds, Oxford) support links between logicians in Mathematics, Computer Science, and Philosophy Departments. Important areas of research with significant contributions by philosophers are the formal theories of paradoxes and truth, and neologicism. Both proof theory and set theory feed into foundations of mathematics, and logicians in this area include Rathjen, Meadows, Welch, and Dzamonja, and some working in homotopy type theory.

Intradisciplinary research, logic within mathematics. Internationally, the most wide- ranging intradisciplinary interactions of logic with other parts of mathematics arise through model theory. Major recent themes include: transcendence theory and Diophantine geometry; real algebraic and analytic geometry; algebra (geometric group theory, representation theory, algebraic groups, permutation groups); motivic integration, and rigid analytic geometry; combinatorics (additive combinatorics, Szemerédi regularity, Ramsey theory, extremal combinatorics). The UK has had a strong presence in all of these. Both model theory and set theory have interactions with topological dynamics, with research of Evans notable, and Dzamonja (UEA) is prominent in set-theoretic applications to analysis, and also towards computer science (convergence of algorithms). Brooke-Taylor’s EPSRC Fellowship explores applications of set theory via category theory to algebraic topology. The scope of logic through category theory has greatly increased through expanding connections between Martin-Löf type theory and homotopy theory. This appears most vividly in Voevodsky’s Univalent Foundations programme, supported by recent Princeton and IHP Paris programmes, and some big international grants. This has both interdisciplinary and intradisciplinary aspects, with connections to philosophy (foundations of mathematics), computer science (type theory, proof verification), and for example to topology.

Engagement. UK logicians remain highly visible nationally and internationally. Prominent roles have been held recently by Macintyre (LMS President), Hyland (LMS General Secretary), Wilkie (President of the Association of Symbolic Logic). Cooper was Chair of the committee which oversaw the highly successful international Turing Centenary in 2012 (see also [10]). He also initiated and became President of Computability in Europe (CiE), a major organisation with some 1200 members, a journal, and an annual conference. CiE has led to a widening of the perspective of computability theory, and there are researchers in the UK in many disciplines with a significant link to the subject. Rathjen is on the Scientific Committee of the Oberwolfach Research Institute. Dzamonja (followed by Benedikt in 2014) was Chair of the ASL Logic in Europe Committee 2008-2014, and was the UK delegate for the Individual Members in the European Mathematical Society 2012-14. She was President of the European Set Theory Society 2012-2014, a UK-registered charity. Dawar is head of the European Association for Computer Science Logic (CSL).

There have been four recent logic-centered programmes at the Isaac Newton Institute: `Semantics and Syntax: a legacy to Alan Turing’ (2012) and `Mathematical, foundational and computational aspects of the higher infinite (2015), as well as `Logic and Algorithms’ ( 2006), and `Model theory and applications to algebra and analysis’ (2005). Among many logic meetings, the UK hosted the Logic Colloquium (the main international logic meeting) in Manchester in 2012 and will host it in Leeds in 2016. The 2011 European Set Theory Conference was at the Edinburgh ICMS, and Edinburgh also hosted in 2010 the Federated Logic Conference, that brings together every four years 8 main conferences on CS logic.

The UK is prominent in logic-related networks and training events. Examples include the logic Marie Curie ITN MALOA (2009-13, coordinated by Leeds), and the large European Science Foundation network in set theory INFTY (2009-14). Logicians are active in Taught Course Centres, for example with three logic courses in 2015-16 for the MAGIC consortium.

6. Future Direction/Opportunities: The following areas seem especially ripe for further development in the next few years.

(a)Model theory and transcendental number theory. Here there has been a rapid stream of striking developments, partly emanating from the Pila-Wilkie work described earlier.

(b) Model theory and combinatorics. This will be one of the main themes of a Paris IHP model theory programme in 2018. UK-based model theorists have not been prominent here, apart from Pillay (when in the UK) and Hrushovski. There are model-theoretic aspects to the work of Green with Breuillard and Tao on approximate subgroups, mentioned earlier.

(c) Independence in set theory, forcing axioms, and forcing at large cardinals – these are thriving areas internationally, with connections to other fields and many strong researchers.

(d) Categorical logic and homotopy type theory. This wide-ranging field (see Section 3 and 5) has expanded rapidly in the last 6-8 years, driven by potential applications in mathematics, philosophy, and computer science. 7. Main Research Groups in Logic: There are substantial logic research groups in the following mathematics departments. Leeds (model theory, proof theory/categorical logic, set theory), Manchester (model theory, uncertain reasoning), Oxford (model theory, set-theoretic topology), UEA (model theory, set theory). Logic in DPMMS in Cambridge (categorical logic, set theory) has a high age profile. UCLan has a model theory group of three, and Bristol has a small but influential group with a strong logic tradition. There are other smaller groups and individuals, working in or close to logic, in Birmingham, Imperial, Leicester, QMUL, and Sheffield.

Computer Science and Philosophy departments with strength in logic are listed in Section 5.

8. Bibliography:

1. D. Aspero, P.B. Larson, J.T. Moore, `Forcing axioms and the continuum hypothesis’, Acta Math. 210 (2013), 1—29.

2. D. Aspero, M.A. Mota, `A generalization of Martin’s Axiom’, Isr. J. Math. 210 (2015), 193—231.

3. G. Barmpalias, A. Day, A. Lewis-Pye, `The Typical Turing degree’, Proceedings of the London Mathematical Society 109 (1), 1-39, 2014.

4. E. Breuillard, B. Green, T. Tao, `The structure of approximate groups’, Publ. Math. Inst. Hautes Etudes Sci. 116 (2012), 115-221.

5. A.D. Brooke-Taylor, `Indestructibility of Vopenka’s Principle’, Arch. Math. Logic 50 (2011), 515—529.

6. G. Boxall, J.O. Jones, `Algebraic values of certain analytic functions', Int. Math. Res. Not. 2015, 1141-1158.

7. R. Cluckers, J. Derakhshan, A.J. Macintyre, E. Leenknegt, `Uniformly defining valuation rings in Henselian valued fields with finite or pseudofinite residue field’, Annals of Pure and Applied Logic 164 (2013), 1236-1246.

8. R. Cluckers, J. Gordon, I Halupczok, `Integrability of oscillatory functions on local fields’, Duke J. Math. 63 (2014), 1549—1600.

9. R. Cluckers, J. Gordon, I Halupczok, `Local integrability results in harmonic analysis on reductive groups of large positive characteristic’, Ann. Sci. Ecole Norm. Sup. 47, no. 6, (2014), 1163--1195.

10. S.B. Cooper, The Machine as Data: a Computational View of Emergence and Definability, Synthese, published online July 2015.

11. J. Cummings, M. Dzamonja and C. Morgan, ‘Small universal families of graphs on .ω+1’, J. Symb. Logic, to appearא 12. P. D'Aquino, A. Macintyre, G. Terzo, `From Schanuel's conjecture to Shapiro's conjecture’, Comment. Math. Helv. 89 (2014), 507-616.

13. J. Derakhshan, A. Macintyre, `Some supplements to Fefferman Vaught, in relation to model theory of adeles’, Ann. Pure Appl. Logic 165 (2014), 1639-1679.

14. R. Downey, Kach, S. Lempp, A. Lewis-Pye, A. Montalban, D. Turetsky, `The Complexity of Computable Categoricity’, Advances in Mathematics 268, 423-466, 2015.

15. M. Dzamonja, J. Vaananen, `Chain models, trees of singular cardinality and dynamic EF games’, J. Math. Logic 11 (2011), 61—85.

16. N. Gambino, A. Joyal, On operads, bimodules and analytic functors, Memoirs of the American Mathematical Society, to appear.

17. L. Gregory, M. Prest, `Representation embeddings, interpretation functors and controlled wild algebras’, arXiv:1411.3221

18. P. Habegger, J. Pila, `o-minimality and certain atypical intersections’, Ann. Sci. Ecole Normale Sup., to appear.

19. P. Holy, P. Welch, L. Wu, `Local club condensation and L-likeness’, J. Symb. Logic 80 (2015), no.4, 1361—1378.

20. G.O. Jones, M.E.M. Thomas, A.J. Wilkie, `Integer-valued definable functions’, Bulletin of the London Mathematical Society 44 (2012), 1285-1291.

21. G.O. Jones, J. Kirby, T. Servi, `Local interdefinability of Weierstrass elliptic functions’, Jussieu J. Math., to appear.

22. J. Koenigsmann, `Defining Z in Q’, Ann. Math. 183 (2016), no.1, 73—93.

23. R. Laking, M. Prest and G. Puninski, `Krull-Gabriel dimension of domestic string algebras’ arXiv:1506.07703.

24. J. Paris and A. Vencovska, Pure inductive logic, ASL series 'Perspectives in Mathematical Logic', CUP, 2015.

25. J. Pila, J. Tsimmerman, `Ax-Lindemann for A_g’, Annals of Math. 179 (2014), 659- 681. 26. J. Pila, J. Tsimmerman, `Ax-Schanuel for the j-function’, Duke J. Math., to appear.

27. M. Prest , G. Puninski, `Ringel's conjecture for domestic string algebras’, Math, Zeit. 282 (2016), 61—77.

28. M. Rathjen, `Relativized ordinal analysis: The case of Power Kripke–Platek set theory’, Ann. Pure Appl. Logic 165 (2014) 316--339.

29. M. Rathjen, `Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman’, J. Symb. Logic, to appear.

30. I. Sharpe, P. Welch, `Greatly Erdos cardinals and some generalisations to the Chang and Ramsey properties’, Ann. Pure Appl. Logic 162 (2011), 863-902.

31. A.J. Wilkie, `Complex continuations of R_an-definable unary functions with a diophantine application’, Journal of the Lond. Math. Soc. (2016), doi:10.1112/jlms/jdw007.

32. B. Zilber, `Model theory of special subvarieties and Schanuel-type conjectures’ Ann. Pure Appl. Logic, to appear. INDUSTRIAL MATHEMATICS

Lead Author: Prof Colin P Please Contributors/Consulted: Listed at the end of the document

1. Introduction

Industrial Mathematics is defined in a European context [1] as the mathematical sciences considered in their broadest sense, when applied to any activity of economic or social value regardless of whether it is in the public or private sector. Hence research in this area overlaps with and interacts with many more tightly focussed interdisciplinary research areas including statistics, operational research, scientific computing, computer science, data science, as well as mathematical biology, mathematical medicine and mathematical finance. As a consequence, the borders of the area are necessarily fuzzy and highly porous to ideas and researchers. This document therefore takes a broad view of the topic, but emphasises those aspects not covered by other landscape documents (particularly those in Statistics, Mathematical Aspects of Operational Research, Mathematics in Biology and Medicine, and Financial Mathematics). Furthermore Industrial Mathematics should be viewed as an approach to mathematical research involving motivation from industrial problems, identifying and modelling relevant practical aspects into a mathematical framework, exploration of such models through diverse high-quality mathematical research, and interpretation of mathematical results back to the original situation, rather than viewing Industrial Mathematics as simply a technical discipline. Since this area is closely related to industry it is critical that companies are closely integrated into the research. However, independence of the academics, the necessity to explore abstract mathematical concepts motivated by the interactions, and the speculative nature of underpinning technical concepts which such interactions entail mean that industry typically see the research as very high risk and this makes underpinning government funding crucial in leveraging industrial support. Critically, industrial mathematics must retain a high level of quality mathematical research and avoid being simply a mathematical service to industry.

The main aim of this document is to set out the current state of industrial mathematics in the United Kingdom and help address the three key elements of the EPSRC strategy of Balancing Capability, namely: i) Quality, ii) National Importance and iii) Capacity.

Regarding Quality, within the 2014 Research Excellence Framework a large fraction of the impact case studies submitted in UOA10 demonstrated their quality by exploiting the connections between mathematics and industrial applications. A very accessible overview of 38 examples of these case studies covering a huge range of mathematical topics is given in “UK Success in Industrial Mathematics” [2]. Similarly the mathematical-science-based CDTs announced by EPSRC in 2014 all exploit industrial connections for motivation, exploitation and support of high quality mathematical research, across the breadth of mathematical disciplines. There has however been very little explicit support from the EPSRC Mathematical Sciences Theme, in the form of responsive mode research grants or fellowships, to support mathematical research directly connected to industry. The UK remains at the forefront of industrial mathematics leading innovations such as Study Groups with Industry to enable strong interactions between mathematicians and industrialists both internationally and nationally. However, there has been a serious decline in EPSRC support for such innovative connections in recent years and, outside the CDTs or recently instigated activities by the Issac Newton Institute (INI) and International Centre for Mathematical Sciences (ICMS), the core infrastructure for creating and nurturing the academic-industry interface has been, and continues to be, supported primarily on an ad hoc basis by individual universities.

1 Regarding National Importance, the use of mathematical models to help streamline and optimise business activities, regardless of sector, has never been more necessary. EPSRC’s 2010 IRM [3] action plan says “In addition to existing programmes that connect industry and mathematical sciences research, long-term collaborations focusing on basic research driven by industrial challenges should be explicitly encouraged,” and the EPSRC-Deloitte 2012 [4] report says “The quantified contribution of mathematical science research to the UK economy in 2010 is estimated to be 2.8 million in employment terms (around 10% of all jobs in the UK) and £208bn in terms of GVA contribution (around 16% total UK GVA).” Regarding Capability, there are long-term issues related to the key need to supply world- class modellers to retain the UK’s national competitiveness as indicated in the 2011 McKinsey report [5], and which has been recognised in a speech by Willetts (Department for Business, Innovation & Skills) in 2013. At the postgraduate level, the 2012 House of Lords report on STEM subjects [6] indicates the need to address the demand by employers for graduates with analytical thinking and problem solving skills. The growth of CDTs involving industrial connections, the health of existing MSc courses and the existing schemes such as EPSRC's CASE and Innovate UK's KTPs indicate there is a huge appetite by students to pursue research in this area but there are currently few stepping stones past the postgraduate level to ensure the career pipeline is available to replace the senior academics in this area. Numerous universities have identified Industrial Mathematics as an area of future growth and a significant number of new permanent positions (currently more than six) have been created since the last IRM in 2010 and more are expected soon, so there is a need to make provision for connecting the very early career and mid career expectations. This requires support for adaptive and flexible mathematicians who are at ease with a range of modelling techniques including continuum, discrete, scientific computing and other mathematical frameworks, and can readily identify where transferring ideas from across mathematics might be beneficial. Providing the infrastructure to nurture these highly talented mathematicians is currently achieved primarily by the actions of individual universities on a one-off basis and there needs to be greater support for a coherent strategy to address the growing needs.

2. Statistical Overview

Data from Universities on the amount of industrial mathematics activity must be interpreted carefully due to the precise definition of the subject area and the fact that many academics undertake industrially motivated research while remaining identified with more conventional discipline groups. Taking a sample of a dozen Universities with significant industrial mathematical activity, approximately 20% of the academic staff in UK mathematics departments are involved in Industrial Mathematics. Looking at the individual subjects, the largest percentage of staff involved with industry is in Operational Research and Statistics, then Applied Mathematics and Computer Science and a small fraction in Pure Mathematics.

3. Discussion of research area

Industrial Mathematics has continued to experience growth in activity both in the UK and internationally since the International Review in 2010 [3]. This expansion of mathematical research and industrial interaction has been driven by changes in the needs of industry, the demands of the impact agenda, and significant alterations in government support for academic-industrial activity. Overall the primary response to these challenges and opportunities has been a vast number of relatively uncoordinated innovative activities at separate Universities. Exceptions to this disparate response have been EPSRC CDTs, activities by Innovate UK and the Knowledge Transfer Network (KTN) in Industrial Mathematics, and initiatives from ICMS and INI. The funding of the large number of much less coordinated activities have been from within Universities, through MSc courses, EPSRC

2 CASE awards, sKTPs and internships, by Innovate UK, and various methods of funding PDRAs each of which has engaged significant industrial input.

The landscape of postgraduate provision in mathematics has been positively transformed by the twelve mathematical-science based CDTs : SAMBa (Bath), Analysis (Cambridge), Risk (Liverpool), MathSys (Warwick), MIGSAA (Edinburgh, Heriot Watt), PDE (Oxford), STOR-I (Lancaster), SABS (Oxford), Geometry (UCL, Imperial, Kings), Planet Earth (Reading), InFoMM (Oxford), and Fluids (Imperial). The effect on Industrial Mathematics has been significant with the proposals for these CDTs indicating that over 148 companies were engaged in supporting these efforts with an average of 12 company partners per CDT. The engagement of these companies takes many different forms, and includes workshops, soft skill courses, technical courses and funding for projects. One of the main outcomes of these CDTs is a large group of postgraduates who see industrial engagement as a part of their mathematical research activities. The effect on mathematical research is not yet possible to quantify, but as a possible example, in 2016 Oxford indicates that 24 mathematics staff are involved in supervising projects that are both motivated and supported by industry. For those departments not closely involved in CDTs, funding of PhD students has become much more difficult. EPSRC CASE awards continue to be an important source of support, either through industrial CASE or those managed by the Smith Institute. In addition some departments find KTPs a very effective source of postgraduate funding. For many universities, the lack of central funding has created a situation where the main source of PhD funding is now internal university or departmental funds in the form of studentships. Such localised activity is supporting high quality research by postgraduate students but in a manner that has no national strategy or supporting infrastructure.

Activity at MSc level in Industrial Mathematics remains buoyant, in spite of removal of all EPSRC funding, with support now provided through self-funding by students and some inventive local university activities. Such MSc courses include those run at Bath, Birmingham, Edinburgh, Greenwich, Lancaster, Loughborough, Manchester, Oxford, Queen Mary, Southampton, and UCL. A significant number of these MScs include periods where each student is actively involved with mathematical research topics directly engaging with a company either through a secondment or internship. There has also been significant growth in other short term postgraduate connections between universities and industry which have proved highly popular with postgraduate students and companies and created very vibrant and productive collaborations that have enabled longer-term deeper interactions to become viable. Such interactions have been supported by Knowledge Transfer Partnerships (sKTP), Smith Institute internships, CDTs and a vast array of various internship and secondment activities organised at individual Universities. Outstanding examples of such wide engagement can be seen at Bath and Lancaster.

There have been recent initiatives by INI, via the Turing Gateway to Mathematics (TGM), to promote connection between industry and academics. The INI has also introduced regular “Industry Days” which have proved highly successful at drawing in a selection of companies and engaging them with cutting-edge mathematical concepts. The ICMS has also continued to expand its workshops and conferences with several having strong industrial focus and participation, including several dedicated knowledge exchange workshops each year. These national activities show the depth of interest in creating strong interactions and there is significant scope for expanding these and creating a national infrastructure to complement the currently much larger localised efforts at individual universities. Note the key conclusion of the 2015 EPSRC review of mathematical infrastructure was that “Mathematical Science infrastructure has been shown to be of great importance to UK Mathematics..... This is the core mechanism for linking mathematicians across subfields of mathematical science and also for connecting mathematical scientists with those working in other fields including government and industry.” and it also found that “The evidence showing the flow of industrial problems influencing mathematical sciences was patchy...... There needs to be deeper and

3 wider community involvement, coordination and governance across this area”. The report concentrated on the INI and ICMS, comparing to other international centres, with comments about their role in creating industrial interactions. There was recognition of the role of Study Groups with Industry in creating the interactions but it did not suggest any support and the report had little to indicate the substantial long-term role of the individual universities in creating the necessary environment for high quality mathematics to emerge from industrial engagement.

The developments in industrial mathematics rely on three pillars namely i) leadership ii) academic intellectual capacity, iii) effective and efficient industrial-academic interaction mechanisms. The UK academic base in this area has great strength in all these pillars and has the capability to retain its internationally leading position.

Leadership: The international and national growth of this area has been led by key researchers from many universities commonly acting through centres funded by various mechanisms with the top seven being Bath, Bristol, Liverpool, Manchester, Nottingham, Oxford, Southampton and Strathclyde. There is significant growth in the area with mathematics departments, such as Birmingham, Durham and Portsmouth creating new permanent positions in industrial mathematics to grow groups in this subject area, and these will add to existing centres of leadership. The common philosophy central to the success of all these centres is to use industrial problems and interaction to motivate high quality mathematical research. Continuing support for these focal points of leadership is critical to growing and sustaining the current level of research activity. Several key leaders in the area, including Ockendon, Hinch, McKee, Blake and Davies, have retired but there are a reasonable number of senior academics in the area who have stepped up to keep the momentum of the expansion in the subject area.

Recent evidence of the international status of UK industrial mathematics is the central role this played in convincing KAUST to set up OCCAM, a $25million research centre at Oxford. The rapid expansion of this centre created a unique opportunity for industrial mathematics drawing together an array of international talent, exploiting new industrial connections, and creating a critical mass of activity. The amount of research being undertaken can be understood by noting that OCCAM has had 278 visitors, supported 31 doctoral students and supported 45 postdoctoral projects since its creation.

Academic intellectual capacity: Central to ensuring that industrial engagement occurs and that good mathematical ideas are produced from these interactions is the quality of the academics who participate in the process (faculty, graduate students but especially postdocs). For traditional mathematicians to become industrial mathematicians requires them become, to a certain extent, polymaths. Such a breadth of experience requires significant training and the strong base of MSc and doctoral courses in the UK provide this. The lack of underpinning government funding for the MSc element contrasts with continental Europe where the ECMI initiative continues to extend the number of European Masters degrees in industrial mathematics. This demonstrates Europe has recognised the need to train students to realise the societal value of their mathematical skills and to do this across country boundaries. The UK needs to address how to retain its strength at the MSc level and ensure prospective UK and European students can gain some or all of their qualifications here in the UK.

At the doctoral level the CDTs, CASE awards, KTPs and individual university scholarships are providing a steady flow of well-trained students. One highly successful mode of training is the use of “modelling camps”. These week-long sessions provide an environment where students are challenged with industrial problems and must explore the problem and develop mathematical ideas that can be applied to understand them. Since 2010 these have

4 occurred in Oxford and, since 2016, also at ICMS and the UK is very active in the European versions of these both as student participants and as mentors.

There is therefore a groundswell of interest from postgraduates, keen to engage in problems motivated by practical situations and involving diverse interdisciplinary activity with underlying high quality mathematics. However, the available postdoctoral positions in the area are very few and are the major log jam in the system and the point where key future leaders will be driven from the area or leave academia. There is a need to grow the academic capacity, specifically at postdoctoral level, to respond to these demands.

Effective and efficient industrial-academic interaction mechanisms: Creating and nurturing interdisciplinary collaboration requires time to develop relationships, skill to interpret the different styles and languages, and for both industrialists and academics to be motivated and resourced to bridge such gaps. Creating long term interactions requires significant effort to enable companies to see the benefit of the research, to react to changes in company focus and challenges, and to ensure that supporting high quality mathematics remains high on the company agenda. A major hurdle in instigating new collaborations is that companies, and particularly Small and Medium sized Enterprises (SMEs), often have little experience of engaging in academic research, particularly with mathematics. They perceive such activities as high-risk financially, and tend to want short term benefits. Overcoming such barriers, including those presented by academics reticent in engaging with industrial problems requires a range, or “menu” of, good interaction mechanisms. Since 2010 there have been several institutions who have appointed “facilitators” to help create and manage such interactions. Some exemplars of such activity are Bath, Bristol, Oxford and Manchester who have developed and exploited many of the effective mechanisms mentioned previously.

At a European level, coordination of such activities has been led through the European Consortium for Mathematics in Industry (ECMI) who have created a network system to exchange best-practice and connect industrial mathematicians (http://www.ecmi- indmath.org/). This includes the EU-funded COST network MI-NET (Mathematics for Industry Network) which is led by the UK and comprising members from 31 countries (https://mi-network.org/), and EU-MATHS-IN (http://www.eu-maths-in.eu/) where the UK plays a pivotal role. On the national level the key network is centred on the Smith Institute. The role of the Smith Institute has changed substantially from 2010 due to major changes in government structures. Previously the KTN in Industrial Mathematics, run by the Smith Institute, was a focal point for the industrial mathematical community and its main connection to government. It organising workshops and study groups, distributed CASE awards and coordinated production of policy documents. With the consolidation of the KTNs into a single company supported by InnovateUK, the Smith Institute is now independent of government but continues to develop activities and play a key focal role.

Study Groups with Industry have proved to be one of the most successful mechanisms for initiating industry-academic interactions in mathematics and use of these has grown from its UK base to a major international activity as well as being adopted as a successful mechanism by other interdisciplinary mathematical areas including Biology (eg: Mathematics in Plant Science) and Medicine (eg: the very successful MMSG). Evidence for the UK's high international profile is the extensive number of international requests for UK academics to participate in workshops and study groups as mentors and leaders.

Since the 2010, besides annual UK meetings, there have been a total of 65 study groups worldwide with central involvement of UK researchers in these. These include approximately 5-7 European Study Groups with Industry (ESGI) each year and other study groups in USA, Australia, Canada, China, Malaysia and others. As a centralised resource on these activities

5 details, including reports on the mathematics that was undertaken, are held in repositories run by Oxford (www.maths-in-industry.org) and by the Smith Institute (www.smithinst.ac.uk).

One of the measures of the expanding interest in the UK in industrial mathematics is the attendance at the annual UK Study Groups with Industry where there has been major growth with the recent meeting attendance being Durham (100), Manchester (180) and Oxford (220). There is no underpinning support for these meetings which is making long term planning extremely difficult. The demographics of these meetings has significantly changed from 2010 with the recent meeting having large numbers of postgraduates (primarily PhD students), a significant number of staff from many institutions but far fewer postdoctoral attendees. This uneven spread is a major challenge that needs addressing to ensure the long term viability of this area.

Creating effective collaborative research interactions with industry requires full engagement between the different groups and this is best enabled by movement of people. Getting senior mathematicians to transfer from academia to industry is very difficult and almost impossible from industry into academia, although there are some successful examples through industrial fellowships and industrially funded positions. The mechanisms that use the more mobile postgraduates are however highly effective and relatively inexpensive. Support for these, including the mechanisms outlined above, should continue but there is a need to consider how to enable processes that can provide fast-track support to extend those engagements that reveal the possibility of long-term collaboration.

4. Discussion of research community

There has been good support for industrial mathematicians at senior levels with fully-funded academic positions. The main issue is the small numbers of high-quality candidates available for such posts. Many of these positions are in Mathematics departments but some industrial mathematicians have moved into departments of other disciplines as their interests specialise. At postdoctoral level the funding is much more dispersed with direct contribution from EPSRC mathematics being very small. In contrast to the significant funding EPSRC has put into CDTs an assessment of the responsive mode grant and fellowship funding shows that aligning research with industrial activities is not a good career move. Reviewing the outcomes of the Mathematics Prioritisation Panels [7] since 2010 it appears that, of the successful grants (the unsuccessful rates remain unknown), only 10% of fellowships, 10% of standard grants and 7% of first grants (figures based on grant value are similar) have ANY industrial partner and only one grant to build an academic-industrial network was successful. What support any partner gives to each grant is not specified, it may range from some in- kind effort to be associated with the project to actual financial commitment but it is obvious that the effort required by researchers to identify such partners, make a strong research connection with the company, and agree their support has little, if any, effect on the ability of researchers to gain recognition by the mathematical peer assessment community and EPSRC support.

A significant fraction of industrial mathematicians gain their postdoctoral experience within other disciplines. From limited direct data it appears that approximately half of graduating PhD industrial mathematical students pursue postdoctoral research and follow an academic career while most of the other half go into a very wide range of industries. This movement of well-trained talented mathematicians into the wider industrial and academic community is a great strength but there needs to be sufficient opportunities for some staff to remain as academic mathematicians with the unique benefits gained by engagements that draw on the breadth of mathematical approaches, ideas and conceptual frameworks.

6 Central to ensuring that such industrial collaborations thrive is to ensure that new bright mathematicians, at an early stage in their career, are inspired by the collaborative atmosphere of the area and trained in effective interaction mechanisms. Such training requires experience of a vast range of interdisciplinary engagements. Recognition and appreciation from the wider community of the quality of the research resulting from these activities has been difficult as it spans so many different topics.

5. Cross-disciplinary/Outreach activities

UK industrial mathematics is at the cutting edge of dynamic and efficient quantitative cross- disciplinary research. It has explored and identified unique mechanisms that are particularly effective at the interface of mathematics with other disciplines. This activity uncovers and pursues new interfaces where mathematical research can have a major impact through more general cross-disciplinary collaborations. It has shown how to bring a focus where there was previously little coherence within the mathematical community or to the impact it could make. Examples of such previously identified new interfaces include mathematical finance, environmental mathematics, mathematical medicine and the growing area of data analytics. In such areas many mathematical methodologies, such as stochastic differential equations, exponential asymptotics, homogenisation, and methods associated with free boundary problems and thin film models have transferred across. There have also been a significant number of researchers who have moved from industrial mathematics to concentrate their efforts in these areas.

There has been significant change in the industrial sectors that mathematicians have been connecting with since 2010. In particular there has been a continuing reduction in the UK research and development base in the conventional manufacturing sectors. There remains significant effort in defence, pharmaceuticals, production and other areas. However, the huge growth has been in information-driven companies who have a growing need to provide a mathematical framework which gives insight into areas where there are substantial amounts of data or where network interactions are important. The industrial mathematics community has embraced these changes and new challenges and has moved to broaden its research interests and to gain greater interdisciplinary connections. At the postgraduate level there is great interest in exploring mathematics motivated by these new sectors. There is however, a more difficult problem at the senior and postdoctoral level where the number of high quality mathematicians is extremely limited.

6. Directions for the future

The UK industrial mathematics community has identified its strengths and sees the opportunity to exploit these, while also addressing possible areas of weakness. It has identified that its young researchers are a cohort of talented industrial mathematicians trained in an interdisciplinary environment who actively participate in mechanisms to connect mathematics to industrial challenges. There is a significant difficulty for industrial mathematics groups who are not in institutions with relevant CDTs to connect to and the innovative methods that are therefore created to provide PhD scholarships are uncoordinated and primarily driven by local strategic issues and priorities. This disparity needs to be addressed. The UK’s high reputation in industrial mathematics will continue but there are serious issues with the career pipeline at the postdoctoral level where the lack of opportunities make industrial mathematicians either move into industry or move into a different discipline. Support for this stage of career needs to be addressed. Identifying, creating and nurturing interactions between industry and industrial mathematicians requires significant effort. Currently the numerous excellent mechanisms being used are almost entirely provided on a local university basis. There is a need for underpinning support for more centralised activities that can leverage industrial funding and run alongside the existing individual activities in order to allow initiation of industrial engagement with mathematicians

7 to be more strategic, effective and efficient. The resources required are relatively modest and should provide infrastructure for workshops, study groups and other well-developed mechanisms.

In summary, the community of industrial mathematicians is growing rapidly. The mathematical research questions that arise motivated by industrial challenges are expanding as the industrial landscape changes. There is a large group of talented postgraduates motivated to work in this area but the career pipeline is throttled at the postdoctoral level. To ensure the UK retains its world-leading position in industrial mathematics, there is a real need for underpinning resource for coordinated infrastructure to facilitate activity at the industrial-academic interface.

References 1) 2010, ESF Forward Look on Mathematics and Industry, http://www.esf.org/index.php?id=6264 2) 2016, UK Success Stories in Industrial Mathematics, Eds: Aston, P.J., Mulholland, A.J., Tant, K.M.M. ISBN 978-3-319-25454-8 3) 2010, EPSRC International Review of Mathematical Sciences, http://www.cms.ac.uk/irm/irm.pdf 4) 2012, Measuring the Economic Benefits of Mathematical Science Research in the UK, https://www.epsrc.ac.uk/newsevents/pubs/deloitte-measuring-the-economic-benefits-of- mathematical-science-research-in-the-uk/ 5) 2011, Big Data: The Next Frontier for Innovation, Competition, and Productivity, http://www.mckinsey.com/business-functions/business-technology/our-insights/big-data-the- next-frontier-for-innovation 6) 2012, Science and Technology Committee - Second Report - Higher Education in Science, Technology, Engineering and Mathematics (STEM) subjects, http://www.publications.parliament.uk/pa/ld201213/ldselect/ldsctech/37/37.pdf 7) EPSRC Grants on the Web, http://gow.epsrc.ac.uk

People who were consulted or contributed to this document

David Abrahams - University of Manchester John Billingham - University of Nottingham Chris Breward - University of Oxford Chris Budd - University of Bath Alan Champneys – University of Bristol Jon Chapman - University of Oxford John Chapman – Keele University Stephen Cowley - University of Cambridge Russell Davies - Cardiff University Chris Howls - University of Southampton John King - University of Nottingham Andrew Lacey - Heriot-Watt University Joanna Jordan – University of Bath Robert Leese - Smith Institute Daniel Lesnic - University of Leeds Alexander Movchan - Liverpool University Mary McAlinden – University of Greenwich John Ockendon FRS - University of Oxford Kevin Parrott - University of Greenwich Nigel Peake - University of Cambridge Richard Purvis - University of East Anglia David Smith – University of Birmingham Peter Sweby – University of Reading Robert Whittaker – University of East Anglia Eddie Wilson - University of Bristol Stephen Wilson - University of Strathclyde Dave Wood - University of Warwick

8 NONLINEAR DYNAMICAL SYSTEMS AND COMPLEXITY

Lead Author: A.R. Champneys Contributors/Consulted: participants at the EPSRC Applied Mathematics Theme Day Leeds February 2016

1. Statistical Overview: The scope of this overview is restricted mainly to those aspects of Nonlinear Dynamical Systems and Complexity which were presented to the Mathematics panel of REF2014. Researchers in more applied areas of Nonlinear Dynamics and Complexity were also submitted to other panels, for example Physics and General Engineering.

The REF2014 data classifies 203 researchers submitted to the Mathematics panel as having primary affiliation to the area of nonlinear dynamics and complexity, with a further 22 having this as their secondary affiliation. Among the four main Applied Mathematics areas in EPSRC’s portfolio (Nonlinear, Math Bio, Continuum, Numerical Analysis) this is second only in size to Continuum Mechanics (222 primary affiliation and 12 secondary).

Comparison with RAE date from 2008 shows that this is a growing area. According to RAE data, 140 researchers were submitted to subpanel 21 (Applied Mathematics) with primary affiliation to nonlinear dynamics. This does not allow for those researchers who might have been submitted to the Pure Mathematics panel, but the majority of those would probably have had a primary classification of Topology or Analysis

This latter observation reflects the fact that this is a cross-cutting area. Dynamical Systems theory within Pure Maths can be expected to appear under Analysis or Topology and Geometry Landscape documents. Furthermore, applications of dynamical systems theory to areas that have their own landscape documents have not been fully covered; Mathematical Biology and Medicine, much of integrable systems (under Mathematical Physics), quantum dynamics (under Mathematical Physics), Mathematical Finance (under Probability and Statistics) fluid and solid mechanics (under Continuum Mathematics), and stochastic processes (under Probability). On the other hand, mathematical aspects of Control Theory is included here (although it clearly has overlaps with Operations Research and indeed Control has a Landscape of its own under Engineering) because it is not clear it would be covered elsewhere.

2. Subject breakdown:

A useful breakdown of subject areas might be as follows:

1. Bifurcations and low-dimensional dynamics 2. Pure dynamical systems and ergodic theory 3. Pattern formation and nonlinear waves 4. Hamiltonian dynamics and integrable systems 5. Control theory 6. Network science 7. Computational dynamics 8. Stochastic dynamics including dynamical systems aspects of fluid mechanics and climate science 9. Applied dynamical systems (e.g. neuroscience, population biology, engineering mechanics, socio-technical systems) This breakdown should be taken with a pinch of salt because the areas are neither disjoint nor exhaustive and some researchers are active in more than one area or mainly active in some other landscape area (as mathematicians we should be able to devise a more appropriate breakdown method, perhaps analogous to a partition of unity for a manifold). It might be interesting to note how this breakdown has changed since the last Landscape document put together in 2010 for the International Review of Mathematics. Notes on comparison with previous landscape document: "Bifurcations" and "Pattern formation" have been split, and pattern formation has been extended to include nonlinear waves in line with the SIAM Activity Group in this area. "Piecewise smooth dynamics" is popular but hardly deserves a classification in its own right, it is better thought of us as a subset of "Bifurcations and low- dimensional dynamics". "Hamiltonian dynamics" and "integrable systems" have been merged as they are so closely related, both have a strong interface with mathematical physics. "Pure dynamical systems" has specifically had ergodic theory added. "Applied dynamical systems" and "Other" have been made more explicit to be split into "Stochastic dynamics" and "Applied dynamical systems" to each of which some further qualifying aspects have been added.

Perhaps the biggest change is that the specific phrase "Complexity Science" has been removed as this phrase does not seem to be strongly recognized internationally as a separate discipline. Aspects of complexity science appear in “Network Science” and “Applied Dynamical Systems” as well as in statistical physics, engineering and the social sciences. However "Network science" has emerged as a brand new topic area, as witnessed by the huge growth in this area internationally, for example in the hugely influential conference NetSci. This is an interface with Statistics, Algebra (graph theory) and the area is often confused with Big Data, but those areas don't usually consider the important aspects of the dynamics of and on networks is an important part of network science. There is also a strong interface of this area with statistical physics and with Communications technologies. There remains a significant “pull” for applications of NDS&C to other EPSRC areas outside of mathematics, such as the Physics Grand Challenge: “Emergence and Physics Far From Equilibrium”, the Engineering Grand Challenge: “Identifying risk and building resilience into engineered systems”, and the Healthcare Technologies Grand Challenge: “Patient-specific predictive models”.

3. Discussion of research areas: By subject area, please provide a frank assessment (both positive and negative) of past and current international standing of the field, supported by reference to highlights as appropriate.

1. Bifurcations and low-dimensional dynamics: This topic is now quite mature, but continues to be a major strength within the UK, represented principally at Bath, Bristol, Cambridge, Exeter, Imperial, Leeds, Manchester, Nottingham, Oxford, Southampton, Surrey, Warwick, and Heriot-Watt. There is strong overlap between this activity and work in Mathematical Analysis. A particular growth area has been Piecewise smooth dynamics in which the UK is recognised as internationally leading. e.g. key book di Bernardo et al (Bristol/Bath) several large international workshops organised by Jeffrey (Bristol), Glendinning (Manchester) and others There is also leading work on bifurcations with symmetry (Exeter, Warwick, Surrey) but this area is now quite mature.

2. “Pure” dynamical systems theory (including Ergodic theory): There is also significant research at Bristol, Exeter, Imperial, Manchester, QMUL, Swansea, Surrey and Kent. Particularly significant among these are results on rates of mixing for flows and on statistical limit laws for chaotic systems, developed principally in Surrey, and applications of ergodic theory to Number Theory. Other areas of interest include applications of topological methods to low-dimensional maps and existence theory from Mathematical Analysis as well as the interface with quantum and classical billiards and related topics in Mathematical Physics, notable highlights being Markloff (Bristol) and van Strien (Imperial) gaving talks at ICM2014.

3. Pattern formation and nonlinear waves has become a more widely recognised topic internationally since the advent of the SIAM Conference on Nonlinear Waves and Coherent structures. The UK is well represented, principally at Bath, Bristol, Cambridge, East Anglia, Herriot-Watt, Leeds, Oxford, Reading, Southampton, Surrey, Sussex. The topic area has emerged as a coherent theme out of traditional areas of strengths in the UK in dynamics with symmetry, and mathematical aspects of water waves. The UK's strong position in this area is evidenced by the hosting of the SIAM conference in Cambridge in 2014.

4. Hamiltonian and Integrable systems: Most of the activity in this area is best classed as Mathematical Physics, so comments here are on only those aspects for which dynamics is at the centre. It is represented broadly, e.g. Bath, Cambridge, Imperial, Kent, Heriot-Watt, Leeds, Loughborough, QMUL, Reading and Warwick. The first wave of the integrable systems and solitons community has matured and it is no longer clear how strong the UK is within this international community. A notable exception is Fokas (Cambridge) who is one of the most highly cited scholars in mathematics.

5. Control theory: Although active in Bath and Exeter and many engineering departments, traditional control theory is disappearing from most mathematics departments. Within Mathematics there appears to be scope to further develop the interface between traditional control theory and dynamical systems on the one hand and Operational Research and Probability and Statistics on the other.

6. Network science is a fresh and emerging area internationally. It is a cross- disciplinary area, but the dynamics of and on networks is an important theme within mathematics. The UK is well represented but this is a growing area and new appointees are in high demand. A particular highlight is Porter (Oxford) who won the 2014 Erdos-Renyi and 2015 LMS Whitehead prizes. It is represented at Bath, Bristol, Durham, Exeter, QMUL, Manchester, Oxford and many other places as new hires are made.

7. Computational dynamics. There is significant overlap between dynamical systems and computation/simulation. There is a large overlap here with Numerical Analysis. On the one hand, numerical methods are now designed to capture the long-term ergodic behaviour of systems, and ideas from dynamical systems have found a powerful application in this design; the resulting methods are of extreme importance in molecular dynamics, meteorology, computational chemistry and celestial mechanics. On the other hand, techniques from dynamical systems theory are also used heavily in the error analysis. The UK is a world leader in this field, notably Bath, Cambridge, Edinburgh, Warwick and Dundee. Large-scale simulations of the dynamics of complex systems are often being made without sufficient theoretical understanding, and it is important that more mathematicians work in these areas. In addition, there is a significant expertise in numerical continuation and bifurcation analysis of attractors and manifolds, led by groups at Bristol, Surrey and Nottingham. However it could be said that computational mathematics generally, not just computational dynamical systems has moved from being a separate topic within applied mathematics, to be a core part of every applied mathematician’s toolkit.

8. Stochastic dynamics is an area that the UK appears to lead. Particular strengths are in climate mathematics, e.g. at Bath, Cambridge, Exeter, Imperial, Oxford, Reading and Warwick. Here there is a large overlap with Continuum Mathematics and Numerical Analysis. Another strength is in many body systems and physics far from equilibrium which has links to Pure Dynamical Systems and to Mathematical Physics. Highlights here include Stuart (Warwick) who gave a lecture at ICM2014 and Hairer (Warwick) who was awarded a Fields Medal. There are also many applications of mathematics to infinite-dimensional systems such as dynamo equations (St Andrews, Leeds), which has overlap with Mathematical Physics and Continuum Mechanics.

9. Applied Dynamical Systems. There is a wide range of research in applications of dynamical systems in the UK, much of which is left to other landscape documents (notably mathematical biology and medicine). So under this topic are covered just a selection of areas or approaches that might not feature elsewhere. Problems in dynamical systems arising in applications (for example in photonics or mechanics) often include noise, delay, hysteresis, and involve a mixture of both discrete and continuous dynamics. Various universities such as Bath, Bristol, Exeter, Manchester, Southampton and Strathclyde are active in pursuing research into such applications, leading to industrial collaborations in for example the electricity supply industry (Bath), the electronics sector (Exeter, Southampton), aerospace and automotive sectors (Bristol), and robotics (Warwick). Another major success is the use of dynamical systems methods in biology, being led by Exeter through major funding initiatives from EPSRC, Wellcome and MRC.

4. Discussion of the research community Please give a brief description of demographic trends, e.g. composition, origin and volume of research student pipeline, age distribution of academic staff, etc.

Overall the research community is healthy and vibrant, with a good number of centres of excellence. It is especially encouraging to see that there has been a steady flow since the 1990s of talented mathematicians getting new positions in the areas of dynamical systems and complexity. This upward trend is evidenced in the increased numbers in REF2016 compared with RAE2008.

However, very recently there have been some worrying trends; the first wave of appointees in this topic area are now reaching the age of retirement, but there is a significant core of academics appointed in the 1990s and early 2000s who are now in mid-career. A number of those are now leaving the UK (e.g. Stuart from Warwick and Porter from Oxford) as this is a vibrant topic internationally too. At the same time, the pipeline for academic appointments is beginning to wane, with less EPSRC funding going into PhDs, research grants and fellowships in this highly interdisciplinary area. There is a danger that since the financial downturn there has been a pincer movement of a retrenchment to the core (purer, more classical) areas of mathematics on the one hand, while on the other large amounts of funding going into new "big science" initiatives like “Big Data” and “post-genomics” and global challenges. It is certainly true though that NDS&C is often at the heart of many of the proposals that EPSRC funds in "Mathematics + X" (e.g. manufacturing, climate, healthcare).

The lack of renewal of the three CDTs in Complexity Science (Bristol, Southampton, Warwick) is a concern. The new one at Warwick on Mathematics Interdisciplinary Research appears to have less of a focus on NDS&C. The Bath CDT Samba also has some overlap. But the beauty of the Complexity Science CDTs was their strong interdisciplinary focus, which reflects the true nature of the cutting edge of NDS&C.

There have nevertheless been several centres that have grown recently. The most notable is Exeter where Terry in particular has received significant funding for use of dynamical systems and network science methods in physiology. There has also been emergence of the Mathematics of Climate, stimulated by the international Mathematics for Planet Earth initiative and in 2016 a new SIAM activity group on Mathematics for Planet Earth. This has attracted involvement of several NDS&C researchers, and funding from EPSRC and NERC. 5. Cross-disciplinary/Outreach activities: e.g. describe connections with other areas of mathematics, science, engineering, etc.

It is hard to think of a single subject classification area within EPSRC’s remit that has greater interdisciplinary overlap. Here are some of the topics that clearly have a large overlap (in no particular order):

- Continuum mathematics (both fluids, solids and soft matter generally)

- Mathematical biology

- Algebra and combinatorics, especially graph theory

- Numerical analysis and scientific computation

- Climate science

- Energy and Sustainability

- Engineering vibrations and dynamics

- Computational chemistry and Hamiltonian dynamics

- Differential geometry

- Control and systems engineering

- Complexity in socio-technical systems

The UK produces several of the leading journals in dynamical systems, e.g. Ergodic theory and dynamical systems, Dynamical systems, Nonlinearity. The UK has hosted some of the major international events in the area, e.g. Dynamics Days Europe: Loughborough, Bristol and Exeter, and European conference on complex systems: Oxford and Warwick. Dynamical systems feature very heavily on the main visitor programmes in the UK, e.g. at the Newton Institute and the ICMS.

UK academics are prominent in a number of international networks, bodies, journals and conferences involved with dynamical systems and complexity. Notable amongst these are the SIAM activity group in Dynamical Systems and the 2011 SIAM Conference in Dynamical Systems, both of which have chairs from the UK, and the Complex Systems Society. The SIAM Journal of Dynamical Systems (SIADS) currently has four associate editors from the UK.

Dynamical systems and complexity science also have many industrial applications, with Bath, Bristol, Exeter, Reading, Strathclyde and Southampton are particularly active in this area. There is both application pull and academic push. The situation seems very healthy. Areas of active engagement include meteorology, climate change, traffic dynamics, aerodynamics and handling, oil and gas, electronics, photonics and electricity supply. NDS&C contributed significantly to the mathematics impact case studies in REF2014

Numerical Analysis

1 Overview tures and the next generation of UK supercomput- Numerical analysis has been defined as “the study ers that will follow ARCHER. Indeed the US DOE of algorithms for solving the problems of continuous report “Applied Mathematics Research for Exas- mathematics—those involving real or complex vari- cale Computing” (March 2014) noted that “It is ables”. Important components are approximation widely recognized that, historically, numerical al- and computation. Computation plays an increas- gorithms and libraries have contributed as much to ingly large role in all STEM subjects and within increases in computational simulation capability as mathematics itself. Much of this is computation have improvements in hardware” and “Numerical with real numbers, often with very large and com- libraries will continue to play an important role at plex datasets. Numerical analysis therefore plays the exascale”. a vital underpinning role within engineering and Capacity is a concern, as is the resilience of es- the physical sciences. Its contributions are often tablished areas. The increasingly important areas hidden within widely used software: a statistician of uncertainty quantification and inverse problems analyzing data using R, or an engineer carrying out need to be built up, from the current relatively a stability analysis of a bridge using a finite element low numbers of researchers. All areas of numer- package, will not necessarily realize that the speed ical analysis can contribute to data science, and and accuracy of their computations rely very much appointments with a data focus are needed. Nu- on developments in numerical analysis. merical linear algebra is concentrated in three main Numerical analysis draws on pure mathematics groups, and needs investment in order to main- tools but fashions them for use in applications. Suc- tain its world leading position. Optimization is in cessful examples with important UK input in recent danger of decline—senior leaders have not been re- years include new theory and methods in compu- placed and there has been a dearth of junior ap- tational harmonic analysis (wavelets, compressed pointments. Numerical partial differential equa- sensing), convex analysis, and tropical algebra. tions (PDEs), while the largest area, and renewed significantly across the UK with early career perma- The UK numerical analysis community is inter- nent academic staff in the last few years, does not nationally leading and punches above its weight in have the strength in terms of impact of Germany, terms of citations, its involvement in the interna- for example, and needs to be grown—for example tional community, esteem indicators, journal and by exploiting its connections with areas such as bi- book editorial board work, and professional service ology and materials. In all these areas it is crucial to scholarly societies. to feed the people pipeline, especially in view of the The REF2014 UOA 10 (Mathematical Sciences) high employability of numerical analysis graduates overview report described numerical analysis as outside academia. being “high quality and found widely distributed throughout the submissions”. According to Math- SciNet, Acta Numerica (EiC Iserles) is the top-cited 2 Statistical Overview 1 journal for the last two years . It is difficult to estimate the size of the UK numer- The community has a few large centres ical analysis community. EPSRC’s own analysis (Bath, Edinburgh/Heriot-Watt, Imperial, Manch- of REF2014 submissions identifies 175 researchers ester, Oxford, Reading, Strathclyde, Warwick), as working in numerical analysis, with 142 having which work successfully together, and a positive numerical analysis as their primary research area. sign is that numerical analysis appointments have These numbers are similar to mathematical biol- recently been made in departments where there was ogy and somewhat lower than continuum mechan- previously little or no numerical analysis (e.g., Es- ics (235) and nonlinear systems (225). However, sex, Kent, UCL, UEA). some numerical analysis activity was submitted to Industrial involvement has increased greatly in panels other than mathematics, activity in STFC recent years, and led to a number of REF2014 im- and other labs may not have been submitted at all, pact case studies. Numerical analysis PhDs are in and the REF statistics are already 2.5 years out of great demand in industry, for example in software date. and data analysis companies. Indeed, numerical Since the last landscape document was prepared analysis is vital to industrial growth, which is one in 2010 it is felt that the number of researchers who important indicator of its national importance. would regard themselves as working fully or largely Numerical analysis is also key to the exploita- in numerical analysis has increased, as has the num- tion of both current manycore computer architec- ber of early career researchers. 1Source: http://www.ams.org/mathscinet/citations.html

1 For EPSRC Numerical Analysis April 15, 2016

EPSRC’s Grants on the Web lists 73 grants to- ear PDEs. The group led by Markowich at Cam- talling £68 million as of 17-3-16 under “EPSRC bridge is active in dispersive and highly oscilla- Support by Research Area in Numerical Analysis”. tory problems. He has contributed to the math- However, many of these grants, and in particular ematical analysis and numerical approximation of the top 12 or so by value, are CDTs and other cen- Schr¨odingerequations, classical and quantum ki- tres, and very few grants on the list are held by peo- netic equations, reaction-diffusion/convection sys- ple who would identify themselves as researchers tems, particularly the finite element approximation working in numerical analysis. Moreover, among of drift-diffusion problems that arise in semiconduc- the latter grants there are very few responsive mode tor device modelling. Elliott leads a group at War- grants, with a drop since 2010, as well as an under- wick working on surface PDEs and interfaces that representation in fellowships. has made important contributions to the field of computational geometric evolution equations. El- 3 Discussion of Research Areas liott has also contributed to the development and analysis of numerical algorithms in a range of prob- The UK has been internationally leading in numer- lems involving nonlinear diffusion. Barrett and ical analysis research since the early days of digi- N¨urnberg (Imperial) have contributed to the anal- tal computers. Linear algebra, optimization, and ysis of numerical algorithms for degenerate equa- approximation theory are long-standing areas of tions, phase separation, phase field approximations, strength. Research in (partial) differential equa- thin film flows, surfactants, superconductivity, crit- tions has grown in volume and strength consider- ical state problems, polymers, liquid crystals, har- ably over the last two decades and is the largest monic maps, geometric evolution equations, crys- subarea. The UK is internationally competitive on tal growth, and two-phase flows. S¨uli(Oxford) all the major fronts within numerical analysis. is active in several areas of nonlinear PDEs in- Several worldwide trends in numerical analy- cluding coupled Navier–Stokes/Fokker–Planck sys- sis and computational mathematics over the last tems arising in non-Newtonian fluid mechanics, 15 years have also been seen in the UK. First, the numerical analysis of high-dimensional PDEs, the area has become more integrated with core ap- and in nonlinear solid mechanics and materials plied mathematics, especially through the organi- science where he is engaged in the Oxford Cen- zational structures of some of the larger groups. tre for Nonlinear PDEs, directed by Ball. Car- Second, research straddling the different subar- rillo (Imperial) has contributed to the mathemat- eas has grown significantly, for example at the ical and numerical analysis of nonlinear PDEs linear algebra/optimization, linear algebra/PDE for nonlinear diffusion, hydrodynamic, and kinetic and (most recently) PDE/optimization interfaces. models of collective behaviour of many-body sys- Third, numerical analysts are increasingly engag- tems, including rarefied gases, granular media, ing in interdisciplinary research, for example in bi- charge particle transport in semiconductors, and ology, bioinformatics, data assimilation, engineer- cell movement by chemotaxis. Makridakis (Sus- ing, materials science, and networks. Fourth, un- sex) has contributed to the development and anal- certainty/inference has become an important as- ysis of numerical methods for nonlinear hyper- pect that is incorporated into models and numeri- bolic conservation laws and nonlinear parabolic cal methods. The hiring in the UK of researchers PDEs. He is coordinator and chair of the steering trained abroad has had a noticeable influence on committee of the European Network ModComp- some of these trends. Nevertheless, the UK numeri- Shock (http://modcompshock.eu). Other people cal analysis community remains less integrated with active in the broad area of computational nonlin- the broader mathematical and scientific research ear PDEs include Barrenechea (Strathclyde), Bur- landscape than in other research-leading countries. man (UCL), Farrell (Oxford), Hubbard (Notting- The following overview of the current research ham), Jensen (Sussex), Kyza and Lin (Dundee), landscape, and opportunities for the future, cannot Phillips (Cardiff), Ortner (Warwick), Pryer (Read- cover all areas of activity. We mention selected key ing), Stinner (Warwick), Styles (Sussex), as well institutions, but note that in many cases there is as those mentioned below. The field as a whole related UK-based work being conducted at other is growing within the UK, but remains small com- institutions. pared with the activity in the rest of Europe.

3.1 Differential Equations 3.1.2 Adaptivity 3.1.1 Nonlinear PDEs The construction and mathematical analysis of The UK has several world-leading research groups adaptive algorithms for nonlinear PDEs continues working in various aspects of computational nonlin- to be an area of strength in the UK. Houston

2 For EPSRC Numerical Analysis April 15, 2016

(Nottingham) has made important contributions to decade, primarily because of the increasing use of the development and analysis of adaptive finite el- stochastic modelling in many realms of science and ement methods, particularly hp-version discontinu- technology. The UK is well-represented in this ac- ous Galerkin finite element algorithms for nonlinear tivity, with several internationally leading research PDEs that arise in compressible and incompress- teams. The group of Stuart (Warwick) has studied ible fluid flow problems. Other important contrib- design and analysis of numerical methods for SDEs utors to this area include Budd (Bath), in con- and SPDEs, focusing particularly on long-term be- nection with the numerical approximation of sin- haviour, ergodicity, and links with Markov chain- gularity formation, Dedner (Warwick, hyperbolic Monte Carlo. More recently this group has been ac- PDEs), Georgoulis (Leicester, convection-reaction- tive in the Bayesian approach to inverse problems. diffusion problems), Giani (Durham, hp-adaptive The multi-level Monte Carlo (MLMC) method of discontinuous Galerkin methods), Jimack (Leeds, Giles (Oxford) has provided a step-change in the space-time adaptivity for solidification), MacKen- computational efficiency of certain calculations re- zie (Strathclyde, Stefan problems), and Lakkis and quired in financial mathematics, and the basic idea Makridakis (Sussex, norm-based a posteriori meth- appears to have far wider applicability: it is be- ods). At the more applied end, Sherwin (Impe- ing extended by researchers in Germany, Switzer- rial) has contributed to the development and ap- land, the UK, and the USA to other stochastic plication of the parallel spectral/hp element soft- settings. At Strathclyde the research groups of ware Nektar++ for problems in vortex flows of rel- D. J. Higham and Mao are active in the numer- evance to offshore engineering, vehicle aerodynam- ical approximation of SDEs, continuous Markov ics and associated biomedical problems, problems, chains and switching systems and time, jump dif- and Budd (Bath) and Weller (Reading) work on fusions, and the chemical master equation. There optimal transport methods for mesh adaptation on is also significant effort in the numerical solution the sphere with applications in weather prediction. of (primarily elliptic) PDEs with random coeffi- The field of adaptivity is therefore well-represented cients, where the key issues are often related to within the UK. The departure of Ainsworth (pre- linear algebra and preconditioning (see the refer- viously at Strathclyde) to Brown University, USA, ences to Graham, Powell, Scheichl, and Silvester in 2012 is likely to impact on UK strength in this in the previous subsection). Tretyakov (Notting- important area in the long-run. ham) is also very active in the numerical solution of SDEs and (with Milstein) has written an im- 3.1.3 Solution of Linear Systems Arising in portant research monograph in this area. Others PDEs active in this broad area include Szpruch (Edin- burgh), Bespalov (Birmingham) S. Cotter (Manch- The UK is strong in this important field, with ester), Lakkis (Sussex), Lord (Heriot-Watt), Lythe research leadership provided by Graham (Bath), (Leeds), Shardlow (Bath), and Voss (Leeds). The Wathen (Oxford) and Silvester (Manchester), with field is very healthy and growing in tandem with many contributions aimed at saddle-point problems broader trends in modelling, for example the work and related systems arising in fluid mechanics and of Leimkuhler (Edinburgh) on invariant measure in optimization. A. Spence and Freitag (Bath) approximation in molecular dynamics. have developed new iterative methods for large- The departure of Stuart (currently at Warwick) scale eigenvalue problems arising from discretisa- to a chair position at Caltech in 2016 is likely to tions of PDEs. impact on UK strength in this important area in Other active researchers in this and related ar- the long-run. eas include Kay (Oxford), Loghin (Birmingham), Pearson (Kent), Pestana and Ramage (Strath- 3.1.5 Geometric Numerical Integration clyde), Powell (Manchester), Scheichl (Bath), and Duff, Rees, and Thorne ((Rutherford Appleton Ordinary differential equations lie at the heart of Laboratory, RAL). many science and engineering challenges. Geo- An EPSRC network in the area would give the metric numerical integration (loosely, the design community a greater UK cohesiveness, and an INI of numerical methods that respect geometric and programme would enhance its international iden- structural properties of ODEs and, increasingly, of tity. PDEs) remains an area of strength within the UK. Members of the UK GNI community have pro- 3.1.4 Stochastic Numerics vided theoretical foundations for the subject. Iser- les (Cambridge) has been a leader in the gen- This is an area of computational mathematics that eral area of Lie-group methods and Magnus expan- has grown in importance significantly over the last sions, Budd (Bath) used Lie symmetries to develop

3 For EPSRC Numerical Analysis April 15, 2016 novel adaptive grids, Leimkuhler provided princi- scale context. Additional work not yet mentioned ples for the design of constrained symplectic in- includes efforts by the groups of Chandler-Wilde tegrators and time-reversible methods, while Hy- (Reading) and Graham (Bath) on high-frequency don and Mansfield (Kent) contributed to the un- wave scattering and propagation, in collaboration derstanding of of discrete symmetries and the lat- with T. Betcke (UCL), Hewett (Oxford/UCL), ter to the analysis of moving-frame methods. Most Moiola (Reading), Smyshlyaev (UCL), E. Spence of current GNI work in the UK reflects a broad (Bath). There is also research activity in the use of acceptance of this approach in computation and lattice Boltzmann models of fluids at Oxford (Del- seeks to advance GNI applications in a wide range lar) and Leicester (Gorban, Levesley), on stochas- of areas. Iserles plays a major role in the develop- tic simulation algorithms and multiscale dynam- ment of techniques for highly oscillatory problems, ics with applications in Biology at Oxford (Erban) from integrals to differential equations. Lately this and Edinburgh (Zygalakis), in statistical physics has led to algorithmic breakthroughs for computa- at Imperial (Carrillo, Degond, Pavliotis), and on tional quantum mechanics and the master equation atomistic-continuum coupling techniques, molecu- of chemistry. Sch¨onlieb(Cambridge) has employed lar simulation, quasi-continuum methods with ap- volume-preserving discrete-gradient algorithms in plications in materials science at Warwick (Ort- image processing. GNI techniques have also led ner). Makridakis (Sussex) has contributed to the to improvements in weather forecasting, for exam- development of atomistic-continuum coupling tech- ple the numerical treatment of the Monge-Amp`ere niques and the construction of atomistic-continuum equations for semi-geostrophic flows by Budd and energies that are free of ghost-forces. Cangiani Mansfield, and the mimetic discretization work of (Leicester) has made important contributions to S. Cotter (Imperial) for applications to fluid dy- the theory of generalized finite element methods, namics. mimetic finite difference schemes and virtual ele- Geometric integration is the foundation of algo- ment methods. rithms that conserve volume, an important part of The community of researchers engaged in multi- constructing effective sampling procedures for large scale computation has thus emerged naturally from scale applications. For example, Leimkuhler is ad- within existing research groups present in the UK. vancing the state-of-the-art in molecular modelling by designing fast stochastic sampling algorithms 3.1.7 Inverse Problems on a GNI foundation. Another emerging impor- tant application area for GNI is in relation to data Inverse problems is a growing area worldwide. This science, specifically in the design of procedures for is driven by many factors, including medical and parameter inference to describe large data sets; see security applications, design and optimization, and recent work of Leimkuhler and Zyagalakis (Edin- the integration of large data sets with mathemat- burgh) and Girolami and Stuart (Warwick). ical models. Computational challenges within the field are significant. The British Inverse Problems Society provides a coherence to the community as 3.1.6 Multiscale Problems a whole and has played a role in a number of inter- There is a growing trend worldwide to study a va- national meetings held within the UK over the last riety of problems which possess two or more dif- decade. ferent scales under a single broad umbrella, and The UCL Centre for Inverse Problems was es- to look for common themes across different ap- tablished in 2013. Led by Arridge, it brings to- plication areas, as evidenced, for example, by the gether ten staff across mathematics, computer sci- SIAM journal Multiscale Modeling and Simulation ence, and statistics, including Burman, M. Betcke, introduced in 2003, whose editorial board includes T. Betcke, and Jin. At Leeds, Lesnic applies var- Markowich (Cambridge), Pavliotis (Imperial) and ious computational techniques, including meshless Tanner (Oxford). This has led to an interest in and boundary element methods. developing and implementing numerical methods The inverse problems group at Manchester (S. which account for, or exploit, multiple scales. The Cotter, Dorn, Holman, Lionheart) has particu- UK has developed a range of expertise in this broad lar interests in electrical impedance tomography area, and was host in 2010 to an LMS-EPSRC (EIT) and industrial process tomography, with Durham Symposium in the field (led by Graham applications including airport security screening. and Scheichl from Bath, together with Hou). Some The Manchester-based Find A Better Way Inter- of the research in this area has appeared under national Research Centre announced in 2016 will previous items; in particular some of the work of provide a focus for research into land mine detec- Markowich, S¨uli,Barrett, Graham, Stuart, Iser- tion and eradication. Inverse problems and uncer- les, and Leimkuhler has ramifications in the multi- tainty quantification are at the heart of this work,

4 For EPSRC Numerical Analysis April 15, 2016 specifically three-dimensional inversion for ground methods in inverse problems, including by Chen penetrating radar and inductive (metal detector) (Liverpool), Lesnic (Leeds), and Chandler-Wilde problems. and Potthast (Reading), and significant develop- Driven by scientific challenges in the geosciences ment of integral equation methods for the wave and associated large-scale funding opportunities, equation and time-dependent Maxwell equations especially applications to numerical weather fore- by Davies (Strathclyde), Banjai and Duncan (both casting and reanalysis of climate models, there has Heriot-Watt). Iserles (with Brunner and Nørsett, been significant and growing activity in the last five Trondheim) has developed methods for the com- years across the UK in data assimilation, drawing putation of spectra of non-normal highly oscilla- on a range of computational mathematics from op- tory integral operators. Maischak (Brunel) works timization, control theory, numerical linear algebra, on the numerical analysis of boundary element and statistics, and tackling very high-dimensional methods. Chandler-Wilde and Potthast (Reading) applied Bayesian inverse problems. Notably, Read- have developed methods and analysis for scatter- ing hosts the large-scale, interdisciplinary Data As- ing by diffraction gratings and unbounded rough similation Research Centre (DARC, 20+ academic surfaces. Arridge and T. Betcke are developers of and research staff, including van Leeuwen, the Di- the BEM++ software package for solving bound- rector, recently awarded an ERC Advanced Grant, ary integral equations in acoustics, electrostatics Broecker, Dance, Lawless, Nichols, Potthast, Re- and computational electromagnetic, supported by ich), which forms the core of the Data Assimilation EPSRC Software Infrastructure grants. Division of the NERC National Centre for Earth Observation (NCEO), also led by van Leeuwen. 3.1.9 Imaging There are strong collaborations in data assimi- lation involving Reading, Bath (Budd, Freitag), PDE-based imaging has become an active field of Imperial (S. Cotter, Crisan), and Warwick (Stu- research worldwide during the last two decades. art), with strong applied links, in particular from The Centre for Mathematical Imaging Techniques Reading, Imperial, and Bath, to the Met Office, at Liverpool, led by Chen, studies all the ma- the Reading-based European Centre for Medium- jor imaging models including segmentation and co- Range Weather Forecasts (ECMWF), and interna- registration, each type of model addressing a class tionally, to the Deutscher Wetterdienst (DWD). of problems proposed directly by industrial and The work of Stuart (see item on Stochas- clinical partners. The group of Sch¨onlieb(Cam- tic Computation) is also directed more widely at bridge), focuses on image processing via advanced computational aspects of inverse problems, and PDE models. Other UK researchers active in in particular at using the Bayesian approach to this area include Drobnjak (UCL) and van Gennip quantify uncertainty. Others working in inverse (Nottingham). Crooks (Swansea) and Zhang (Not- problems include Chandler-Wilde (Reading), Chen tingham) have been granted the UK patent “Image (Liverpool), Iglesias (Nottingham), and Marletta Processing” (GB2488294, October 2015), which is (Cardiff). The community engaged in the numeri- concerned with a flexible toolbox of robust meth- cal analysis of inverse problems is relatively small ods for image, data-processing and computational when compared with the fairly broad research goals geometry tasks. New convexity-based theory devel- represented, and compared with international com- oped by the group is used, via a numerical imple- petitors, except in the specific sub-field of data as- mentation, to detect features in images or data, re- similation. move noise from images, and identify intersections. This important field is still underrepresented in the UK, although the forthcoming Isaac New- 3.1.8 Integral Equations ton Institute programme on “Variational Methods Research in this area is well-represented in the UK, and Effective Algorithms for Imaging and Vision”, especially in relation to previously mentioned work whose organisers include Sch¨onlieband Chen, will on multiscale and inverse problems. There has help to strengthen the position of the subject in the been large development in numerical analysis for UK. highly oscillatory integrals and integral equations, not least for boundary integral equations arising 3.2 Uncertainty Quantification in high frequency acoustics and electromagnetics, including ongoing work (mentioned above under Uncertainty quantification (UQ) uses computa- “multiscale” and “geometric integration”) involv- tional models, observational data, and theoretical ing T. Betcke, Chandler-Wilde, Graham, Hewett, analysis, to quantify uncertainty in a broad range Iserles, Langdon, Smyshlyaev, E. Spence. There of applications. It has seen rapid growth in re- is also significant application of integral equation cent years. For example, a SIAM Activity Group

5 For EPSRC Numerical Analysis April 15, 2016 was founded in 2010, SIAM inaugurated a biennial proximation, interpolation, and quadrature by Tre- Conference on Uncertainty Quantification in 2012, fethen and his group. and the the SIAM/ASA Journal on Uncertainty Quantification was founded in 2013, whose edito- 3.4 Numerical Linear Algebra rial board includes Farmer (Oxford), Lord (Heriot- Watt), Powell (Manchester), Scheichl (Bath), and The UK continues to have a small but world-leading Stuart (Warwick). UK numerical analysts have an effort in this area that provides international lead- important role to play in UQ. ership across the spectrum of theory, algorithms Important contributions in the UK include and software. those from researchers mentioned under “Stochas- Tisseur (Manchester), supported by an EPSRC tic Problems” and “Inverse Problems”. UQ groups Leadership Fellowship, has made significant ad- with significant numerical analysis involvement are vances in nonlinear eigenvalue problems, including at Bath, Manchester, Reading, and Warwick. the use of max-plus algebra to design preprocess- A Programme Grant in this area (EQUIP, 2013– ing steps to improve the problem conditioning and 2018) is led by Stuart (Warwick). The forthcom- the numerical stability of solvers. Supported by an ing Isaac Newton Institute program “Uncertainty ERC Advanced grant, the work of N. J. Higham Quantification for Complex Systems: Theory and (Manchester) on theory and algorithms for matrix Methodologies” (2018) is jointly organized by nu- functions has had strong influence on software (e.g., merical analysts and statisticians. MATLAB, NAG Library). G¨uttel(Manchester) has made advances in rational Krylov methods for large-scale eigenvalue problems. Dongarra (Manch- 3.3 Computational Harmonic Anal- ester, part-time) has made major contributions to ysis and Approximation Theory parallel numerical algorithms in linear algebra and to open source software packages and systems. In recent years the UK has developed a strong pres- At RAL, Duff, Hogg, Rees, and Scott work ence in computational harmonic analysis, includ- on both direct and iterative methods for large, ing compressed sensing. This “sparsity based” re- sparse linear systems, developing theory as well search is inherently interdisciplinary and has strong as production-quality software made available links with signal processing, imaging, and optimiza- through the HSL Library. Because this group is tion. The main centres of activity are at Cambridge outside the University sector there is a consequent (Hansen), Oxford (Cartis and Tanner), and Edin- lack of graduate throughput. Research on precondi- burgh (M. Davies). tioning for iterative methods is represented at RAL Approximation theory related to kernel based and in several universities (see “Solution of Linear approximation methods in Euclidean space and on Systems Arising in PDEs” above). compact manifolds continues with Levesley (Leices- The UK continues to play a significant role ter) and Baxter and Hubbert (Birkbeck). These in the study of spectra, pseudospectra, and more methods have been applied in engineering for sur- general operator pencils of non-normal matrices rogate models, and in finance. The interface with and operators, an area at the boundary of nu- data science is developing via kriging and kernel merical linear algebra, functional analysis, and ap- learning. RBF methods have been successfully ap- plications. Ongoing activity at this interface in- plied by Giesl (Sussex) for computing basins of at- cludes that of Freitag, A. Spence (Bath), Hansen traction in dynamical systems, developed in col- (Cambridge), Brown, Marletta (Cardiff), Boul- laboration with Wendland (formerly of Sussex and ton (Heriot-Watt), Shargorodsky (KCL), G¨uttel, Oxford). Solver of the De Boor conjecture, Alexei Higham, Tisseur (Manchester), Chandler-Wilde, Shadrin (Cambridge) computes sharp constants in Levitin, Pelloni (Reading), and T. Betcke (UCL), approximation inequalities. many of whom participated in the EPSRC MOP- A notable UK development on the computa- NET Network (2009–2012), led by Levitin (Read- tional side of approximation theory has been the ing) and Marletta (Cardiff), that coordinated ac- growth of the open source Chebfun software system tivity and links to engineering. led by Trefethen (Oxford), currently funded by an ERC Advanced grant. Chebfun exploits a mix of 3.5 Network Science old and new algorithms of polynomial and rational interpolation and approximation and quadrature to The emerging theme of network science offers a produce a system that feels like MATLAB but com- common framework in which to model, analyse and putes with functions instead of numbers or vectors, summarize data sets from a diverse range of ar- including the solution of differential equations. Un- eas in science, technology and even crime. Graph derlying Chebfun are many recent advances in ap- theory and linear algebra come together to offer

6 For EPSRC Numerical Analysis April 15, 2016 tools for analyzing the type of sparse, unstructured mentarity” constraints (Birmingham, Cambridge, but non-random connectivity patterns that arise. Dundee), structural optimization (Birmingham, Currently there is a limited amount of UK activity Southampton), second-derivative SQP methods at the intersection between applied/computational (Dundee, RAL), methods for least-squares prob- mathematics and network science: for example, lems (Oxford, RAL, Reading), the efficient approxi- by Estrada and D. J. Higham (Strathclyde), A. mate solution of (NP) hard combinatorial problems Spence and Rogers (Bath), Grindrod and Porter (Edinburgh, Imperial), multiobjective optimization (Oxford), and Vukadinovic Greetham (Reading). (Birmingham, Southampton), support vector ma- There is an excellent opportunity for the UK to chines, (Dundee, Edinburgh), the complexity of build on existing computational strengths and com- nonconvex optimization (Edinburgh, Manchester, pete with a highly active and data-driven interna- Oxford, RAL), derivative-free optimization (Cam- tional movement that is currently dealing with is- bridge, Oxford, Southampton), global optimization sues of high-dimensional, heterogeneous and time- (Oxford, RAL), financial optimization (Imperial, dependent network data. Oxford), and algorithms for infinite-dimensional problems for which (some of) the constraints are 3.6 Optimization differential equations or variational inequalities (Birmingham, Cambridge, Dundee, Imperial, Kent, There are sizable optimization groups in Birm- Oxford, RAL, Warwick). Upcoming areas such ingham (Kocvara, Butkovic, Nemeth, Zhao), Ed- as optimization over semi-algebraic sets, structure- inburgh (Gondzio, Grothey, Hall, McKinnon, based regularization and robust optimization are Richt´arik),Oxford (Cartis, Hauser, Tanner) and less well represented, but work is underway here Southampton (Fliege, Keane, Qi, Xu), as well too. as individuals scattered elsewhere, notably, Ralph A continuing trend in the UK is to match and Scholtes (Cambridge), Fletcher (Dundee), Lotz theoretical developments in optimization with the (Manchester), Misener and Rustem (Imperial), design and implementation of software. Gener- Gould (RAL), and Lawless and Nichols (Read- ically this is available in the HSL and NAG ing). Cross-area collaboration between optimiza- libraries, and in general-purpose packages such tion, linear algebra, and differential equations, par- as FilterSQP, HOPDM, LANCELOT/GALAHAD ticularly in the area of simulation-based (or PDE- and PENOPT. Many of the key developments– constrained) optimization, is commonplace. There the DFP and BFGS secant updates, augmented is also an established biennial series of Birming- Lagrangian and SQP algorithms and the non- ham conferences on optimization and linear algebra linear conjugate-gradient method—occurred either organized by the IMA (fifth conference September entirely or partly in the UK and as a direct con- 2016). sequence of the software needs of commercial, gov- While optimization has been a UK strength ernment and academic sectors. More recent key since the 1960s, the area is in need of bolstering. research—the trust-region and cubic regulariza- Powell, FRS, died in 2015, and Fletcher, FRS, is tion paradigms and their extensions, methods for retired but still active. Relatively few new appoint- large-scale linear, quadratic and conic program- ments have been made in the last five years, and all ming, large-scale interior-point and SQP methods, of those at a junior level. The importance of opti- the filter and funnel approaches, and randomized mization is only increasing, for example due to its coordinate descent methods—has continued this applications in model fitting, machine learning, and trend. Current computing developments are hav- data science, so it is essential for the UK to have a ing profound implications for optimization, and strong and vibrant optimization community. UK researchers are evaluating and exploiting both Current research in the UK has focused massively-parallel HPC and more modest multicore on theoretical and practical investigations into and GPU systems. interior-point methods for linear, semi-definite, conic, convex and nonconvex problems (Birming- 3.7 Data Science ham, Cambridge, Edinburgh, Manchester, Ox- ford, Southampton, RAL), the solution of (large) Problems involving large or complex data sets are problems for which some or all unknowns are increasingly important in many applications. Nu- required to take integer values (Birmingham, merical analysts are making major contributions to Dundee, Edinburgh, RAL), stochastic co-ordinate this area, and will continue to do so. Indeed many descent and first-order methods for huge data ap- of the research areas in the document are strongly plications (Dundee, Edinburgh, Oxford), meth- relevant to big data, including optimization, data ods for stochastic optimization (Edinburgh, Ox- assimilation, inverse problems, compressed sensing, ford, Southampton) problems involving “comple- and numerical linear algebra.

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In data science the data is already discrete and those numerical analysts who have a Google Scholar any practical way of extracting information is com- profile, putational, and overwhelmingly numerical. The full spectrum of numerical analysis is applicable • Dongarra, Duff, Hammarling, N. J. Higham, here as, for example, PDEs can be imposed on top Mao, Trefethen, have in excess of 15k cita- of unstructured data, as in imaging. Many prob- tions; lems in machine learning relate to the exploration of • Elliott, Giles, Gould, D. J. Higham, Iserles, parameter spaces to model large data sets; the UK Stuart, S¨uli,have in excess of 6k citations, is providing crucial numerical methods for these and types of sampling problems through its talent in stochastic numerics. • Barrett, Burman, Graham, Leimkuhler, The Alan Turing Institute (ATI) provides a na- Nichols, Reich, Silvester, Sweby, A. Spence, tional focus for data science and numerical ana- Tisseur, van Leeuwen have in excess of 3k ci- lysts will play a major role in the institute. The tations. joint venture partnership (Cambridge, Edinburgh, Oxford, UCL, and Warwick) includes some of the UK’s leading groups in numerical analysis. Tan- 4.2 Esteem ner (Oxford) is one of the five University Liaison • Stuart delivered invited lectures at the ICM Directors of the ATI, and Leimkuhler (Edinburgh) (Seoul, 2014) and the European Mathemati- and Sch¨onlieb (Cambridge) serve on its commit- cal Society 25th Anniversary Meeting (Paris, tees. Moreover, many of those named in this report 2015). Markowich gave an invited lecture at have been chosen as inaugural ATI Faculty Fellows. the ICM (Heiderabad 2010). S¨uliwas the 2015 London Mathematical Society and New 4 Discussion of Research Commu- Zealand Mathematical Society Forder Lec- turer. N. J. Higham, Iserles, and Stuart were nity invited speakers at 6th European Congress of Some major changes have taken place among se- Mathematics (Krakow, 2012). Invited speak- nior researchers. Ainsworth (Strathclyde), Hesse ers at the biennial ENUMATH conference in- (Sussex), Johansson (Birmingham), and Wend- clude D. J. Higham, (2015), N. J. Higham land (Oxford) left the UK; Arioli (RAL), Iserles (2011), and Elliott (2009). (Cambridge), Fletcher (Dundee), and Goodman (Dundee) retired; Cliffe (Nottingham) and Powell • ERC Advanced grants were held by N. J. (Cambridge) died; Porter and Stuart will leave the Higham (2011–2016), Stuart (2008–2014), UK in 2016. and currently Trefethen (2013–2017) and A good number of new appointments have been van Leeuwen (Reading, 2016–2021). D. made, almost exclusively at a junior level, and the J. Higham holds an EPSRC Digital Econ- number of institutions with a research presence omy Established Career Fellow in Data An- in numerical analysis has increased, most recently alytics/Internet of Things (2015–2019). T. through appointments at Essex, Kent, UCL, and Betcke held an EPSRC Career Accelera- UEA. tion Fellowship (2009–2014). Pearson holds Ortner (Warwick), Scheichl (Bath), Tisseur an EPSRC Early Career Fellowship (2015– (Manchester), and Wathen (Oxford) have been 2018). Tisseur (Manchester) held an EPSRC promoted to professorships in numerical analysis. Leadership Fellowship (2011–2016). Five Makridakis was appointed to a chair at Sussex, and year NERC fellowships are currently held by Carrillo and Degond, who were appointed to senior Weller (Reading, NERC Advanced Research positions at Imperial College London in applied Fellowship) and Ham (Imperial, NERC Inde- analysis, also have research interests in computa- pendent Research Fellowship). tional mathematics. Overall, however, appoint- • Hairer (Warwick), whose work has connec- ments of international leaders to senior positions tions with numerical analysis, was awarded in the field of numerical analysis have been lacking. a 2014 Fields Medal. Trefethen was awarded the 2011 IMA Gold Medal and the 2013 LMS 4.1 Citations Naylor Prize and Lectureship. Iserles (Cam- bridge) was awarded the 2014 SIAM Prize for The UK numerical analysis community is extremely Distinguished Service to the Profession. Far- well cited by international norms, both within and rell (Oxford) and Ham (Imperial) were co- outside mathematics. For example, from among winners of the 2015 James H. Wilkinson Prize

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for Numerical Software. Tisseur (Manch- 4.4 Other professional service ester) won a 2012 Adams Prize. The UK numerical community is very heavily in- • Davies (Strathclyde) and Budd (Bath) were volved in SIAM, the leading applied mathematics appointed OBE in 2014 and 2015, respec- organization. Recent election positions are Tre- tively. fethen (President, 2011–2012), N. J. Higham (Pres- ident, 2017–2018), N. J. Higham (Vice President at • Morton (Oxford), Chen (Oxford), El- Large, 2010–2013), Duff (Chairman of the Board liott (Warwick), Hammarling (Manchester), 2010–2013), Freitag (SIAG/LA Program Director, Nichols (Reading), S¨uli(Oxford), and Tis- 2016–2018), Pestana (SIAG/LA Secretary, 2016– seur (Manchester) were elected SIAM Fel- 2018), Ramage (SIAG/LA Vice Chair, 2016–2018), lows since 2010, joining Fellows Duff, Gould, Stuart (SIAG/UQ President, 2015–2016), Tisseur D. J. Higham, N. J. Higham, Stuart, Tre- (SIAG/LA Program Director, 2012–2015). fethen elected before 2010. Wathen co-chaired the SIAM Annual Meeting 2009, Trefethen co-chaired the SIAM Annual Meet- 4.3 Editorial service ing 2010, and D. J. Higham will co-chair the 2017 SIAM Annual Meeting. UK numerical analysts are heavily represented The SIAM United Kingdom and Republic of among the boards of the leading journals in nu- Ireland Section is one of SIAM’s largest and most merical analysis and related areas. Highlights not active regional sections and holds a one-day meet- mentioned above include: ing every January. Guettel (Manchester) is Co-Chair of the • 5 members of the SIAM Journal on Scientific GAMM Activity Group on Applied and Numeri- Computing board. cal Linear Algebra. • 8 members of the SIAM Review board (in- cluding Survey and Review Section Editor D. J. Higham). 4.5 Students and Training

• 4 members of the board of ACM Transactions There is no CDT in numerical analysis, but a good on Mathematical Software. education in applied mathematics requires a knowl- edge and appreciation of numerical analysis. • EiCs of IMA Journal of Numerical Analysis (Iserles, S¨uli). This UK-based journal (est. A number of Centres for Doctoral Training have 1981) continues to thrive, with an Impact strong connections with numerical analysis: those Factor regularly placing it in the top 10% of at Cambridge and Warwick, which cover analy- Applied Mathematics journals. sis, computation, probability, and statistics; that jointly held by Edinburgh and Heriot-Watt, which • EiC of Acta Numerica (Iserles). covers pure, applied, numerical, and stochastic analysis; that at Liverpool covering risk and un- • Tanner (Oxford) is founding editor-in-chief of certainty; and the newer CDTs, at Bath (inverse Information and Inference: A Journal of the problems, data assimilation, uncertainty quantifi- IMA. cation), Cambridge (differential equations, math- ematics of information), Imperial/Reading (math- UK numerical analysts are also heavily involved ematics of planet earth, including PDEs, numeri- in book editorial boards. For example (a very in- cal weather prediction, data assimilation), Imperial complete list): N. J. Higham (EiC of SIAM Fun- College (computational fluid dynamics), and Ox- damentals of Algorithms), S¨uli(Oxford University ford (partial differential equations, industrial math- Press Monograph Series in Numerical Mathemat- ematics, optimization). ics and Scientific Computation, Princeton Series Student involvement in activities is very in Applied Mathematics, Springer Monographs in healthy. Since 2010, SIAM Student Chapters have Mathematics, Handbook of Numerical Analysis), been formed at Strathclyde, Warwick, Reading, Iserles (Cambridge Texts in Applied Mathematics, Cardiff, Bath, Imperial, and Cambridge, making a Cambridge Monographs on Applied and Computa- total of eleven student chapters. UK students con- tional Mathematics). tinue to win international prizes, including SIAM N. J. Higham was EiC of the 1000-page Prince- Student Prizes and the SET For Britain competi- ton Companion to Applied Mathematics (2015). tion (bronze medal, 2016).

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4.6 Conferences, Networks and Gung Ho project, which is developing a new dy- Summer Schools namical core for Met Office weather and climate simulation on the massively parallel computer ar- The Biennial Conference on Numerical Analysis chitectures of the next 20 years, with funding for continues to run successfully at the University of partners including Scheichl (Bath), Ham, C. Cotter Strathclyde, having turned 50 years old in 2015. (Imperial), Thuburn (Exeter), Weller (Readin The Leslie Fox Prize continues biennially, and a positive recent trend is the greater number of UK- trained finalists, with both 2015 first prize winners 6 Future Direction/Opportunities being UK PhDs. While UK research in numerical analysis is strong, The EPSRC networks MOPNET (Matrix and it does not play the central role in applied mathe- Operator Pencils Network) and Numerical Algo- matics that it does in major research competitors. rithms and High Performance Computing are men- A strategy aimed at ensuring that the best numeri- tioned in sections 3.4 and 5, respectively. cal analysis informs large-scale scientific computing Among efforts across institutions the long- and mathematical modelling will have long-term standing annual Bath/RAL numerical analysis benefits to the science and engineering communi- days are notable. ties as a whole. A significant opportunity exists to grow inter- faces with other areas of mathematics, and to grow 5 Inter/Intra-Disciplinary Activities interfaces between different parts of numerical anal- and Engagement Activities ysis. Research in PDE analysis has grown in the Numerical analysts have been involved in organiz- last decade and developing this further to integrate ing several scoping workshops for the Alan Turing with, and build upon, the potential in the numerical Institute, and have widely engaged in the work- analysis PDE community is a key to competing in- shops. ternationally. The interface between computational The interface with high performance computing PDEs and optimization is a rich research area, has benefited from the EPSRC Network “Numeri- worldwide; both communities are strong within the cal Algorithms and High Performance Computing” UK but links between them are at an early stage. (2011–2014), led from Manchester, as well as the Research at the interface of statistics and nu- large “Numerical Algorithms and Intelligent Soft- merical analysis, including uncertainty quantifica- ware for the Evolving HPC Platform” centre in- tion, is an area of growing importance worldwide volving Edinburgh, Heriot-Watt, and Strathclyde where the UK has world-leading strengths which (2009–2014). Manchester and RAL are partners need to be further built upon, both on the theoret- in a large Horizon 2020 grant “NLAFET—Parallel ical side and in applications such as data assimila- Numerical Linear Algebra for Future Extreme- tion. Scale Systems” at the numerical linear algebra– Other areas of opportunity are numerical ap- HPC interface (2015–2018). plications in chemistry, mixed continuous-discrete D. J. Higham is an EPSRC Digital Economy Es- problems linked to networks, modern PDE ap- tablished Career Fellow in Data Analytics/Internet proaches to cell biology, and high-dimensional of Things (2015–2019), and is active in establishing problems. mathematics involvement in smart/digital/future The UK has been very successful in building cities. library software on top of research in numerical lin- Among the five recently-established EPSRC ear algebra and optimization over several decades centres for mathematics in healthcare, two (EPSRC (NAG, LAPACK, HSL, LANCELOT/GALAHAD Centre for New Mathematical Sciences Capabilities etc.). There is also heavy UK involvement in fi- for Healthcare Technologies, Liverpool and EPSRC nite element software (DUNE, FEniCS) and fur- Centre for Mathematical and Statistical Analysis of ther opportunities exist to build on the substantial Multimodal Clinical Imaging, Cambridge) have put research output in differential equations. numerical analysis at the centre of their activities. With the increased importance of machine Numerical analysis is central to the newly- learning (in particular, deep learning) an opportu- established Cantab Capital Institute for Mathe- nity exists to contribute to the topic through ap- matics of Information (Cambridge). proximation theory, numerical linear algebra, and Many interdisciplinary links in inverse problems optimization, as these are three of its underlying were mentioned in section 3.1.7. There have been components. related, ongoing and substantial contributions by Industrial involvement has great potential to in- UK numerical analysts and computational scien- crease, because of the need for the numerical, algo- tists to the large-scale NERC/STFC/Met Office rithmic, and software skills that numerical analysts

10 For EPSRC Numerical Analysis April 15, 2016 possess. Finally, numerical analysis has an important role to play in the exploitation of the next genera- tion of computers (one example being the ongoing project with the Met Office cited in the previous section). This includes both manycore architec- tures (accelerators, hybrid, distributed), and the top-end exascale machines. As noted by the report US DOE report “Applied Mathematics Research for Exascale Computing” (March 2014): “Numerical algorithms are the core of all physical simulation codes. To ful- fill the promise of extreme scale science enabled by exascale computers, exist- ing numerical algorithms must be re- designed and new numerical algorithms must be developed so that a significant percentage of the potential performance of these machines can be reached.”

11 Number Theory.

Kevin Buzzard, Andrew Granville, Samir Siksek Statistical Overview.

The UK is a clear international leader in number theory, with UK-based num-ber theorists prominent in most of the current major number theory research themes. There are 89 permanent academic staff in the UK who work mainly in number theory (based on data obtained by a direct approach to all depart-ment heads as well as an analysis of each departmental website), including those who identify themselves as working in more than one area. Among these there are researchers active in one area, and those whose interests encompass several.(For example, researchers in Diophantine approximation need techniques from analytic number theory, whereas those interested in arithmetic algebraic geom-etry inevitably work also in algebraic number theory. These two fields thrive in a state of symbiotic harmony). More than half of the UK number theorists work in the algebraic and arithmetic-geometric side of the subject. Many are motivated by the Lang- lands philosophy, but there are several other well-represented topics of interest, for example, arithmetic questions related to L-functions, especially by p-adic methods, and also rational points. Analytic number theory is the other main research area, with researchers applying combinatorial, classical analytical and computational techniques across the UK. Other UK number theorists focus more on explicit and computational number theory to facilitate better understanding of theoretical questions in both algebraic and analytic number theory. In all three areas, L-functions are one of the key objects of study, providing synergy between research groups.

Subject breakdown.

Research Area Number of researchers % of total in 2015 Algebraic Number Theory 25 28 Arithmetic Algebraic Geometry 28 31 Analytic Number Theory 28 31 Representation Theory 10 11 Diophantine Approximation 9 10 Computational Number Theory 11 12

1 As indicated above, several number theorists identify as working in more than one of the areas highlighted here – this is unsurprising given the nature of mathematical research – with the result that the percentages add up to well over 100. It should be noted that “Computational Number Theory” is a convenient but misleading title for an area, which uses explicit methods to throw light on traditional number theory questions, the thrust of the subject being the theory developed, not the calculations themselves.

International Research Themes.

Number theory in the UK has a long and extraordinary tradition of excellence; many of today’s most important research themes have their origins in the work of UK mathematicians. The school of Hardy and Littlewood in Cambridge and Oxford, from the early twentieth century, and Davenport at Cambridge and Roth at UCL and Imperial, introduced many of the central notions of analytic number theory. For example, the circle method, introduced by Hardy and Ramanujan in the early 1900s, has gone in and out of fashion, but has seen extraordinary developments in the last decade, led by the work of Wooley (Bristol), Gowers (Cambridge) and Green (Oxford). Today’s arithmetic geometry owes much to the brilliant calculations and conjectures of Birch (Oxford) and Swinnerton-Dyer (Cambridge), which have inspired many of the most important developments in pure mathematics of modern times. Indeed these questions are central to the explosion of worldwide interest in arithmetic algebraic geometry; they lie behind Wiles’ (Oxford) proof of Fermat’s Last Theorem. It is a great coup that we have attracted Wiles (winner of the 2016 Abel prize, among many other recognitions) back to the UK. Some of the strongest young researchers in the area (Gee, Loeffler, Thorne) are UK-based and direct mathematical descendants of Wiles, as are other more established members of the community (Diamond, Buzzard). But the UK’s influence on the area goes beyond this. The parity results of Tim and Vladimir Dokchitser show that there is still great power in ingenious use of “classical” methods. Euler systems provide a tool for proving the “main conjectures” of Iwasawa theory: UK researchers play an important part in the development of modern Iwasawa theory (Kakde, Bouganis, Ardakov), as well as Loeffler and Zerbes constructing a powerful new Euler System, consequences of which are being actively developed worldwide. The UK has several key figures internationally in “explicit methods” in num- ber theory (Cremona, the Dokchitsers, Booker, Demb´el´e,Siksek). There is also a great deal of expertise in the UK in number theoretic cryptography, with faculty members in both mathematics and computer science departments. It is unsur- prising that considerable resources are being invested in the subject by GCHQ; more than 40 researchers have been funded through the Bristol-based Heilbronn Institute for Mathematical Research, with another site recently opened in Lon- don.

2 Compared to other leading nations, there continues to be relatively little expertise in the UK in the analytic theory of modular and automorphic rep- resentations, and in the analytic aspects of automorphic forms. There is also little activity in transcendental number theory in the UK, perhaps reflecting a worldwide trend. Everything changes, yet everything stays the same: Today’s UK academic landscape continues to reflect and lead many of the themes of greatest inter- national interest in number theory, arithmetic geometry and related areas. At the same time that there are exciting new techniques to assimilate, we also find that (the right) important traditional techniques continue to contribute to the latest understandings. There are an increasing number of departments with substantial groups of highly active researchers in number theory, and this has been reflected in the deservedly high level of support that number theory has received from EPSRC and other funding bodies (e.g. ERC). For example, there are three number-theory related Programme Grants, the consequences of which can be seen by their effect on the number-theoretic landscape laid out in the rest of this document.

Research Community.

Oxford continues to have a world leading group of researchers in many of the important areas of number theory. Their group has expanded in the last few years, and currently boasts one of the most exciting groups of number theory postdocs anywhere in the world. Bristol, partly because of the Heilbronn Institute’s support, has a dozen faculty who research in number theory and closely related areas. They also have a good number of postdocs, including many who work half-time at the Heilbronn Institute. London has fifteen or so active researchers, spread amongst its universities. Most of these work in arithmetic geometry with a strong focus in modularity (and its relationship to the Langlands program), though there is a new thrust in analytic number theory, with an increasing number of students and postdocs. The EPSRC funded London School of Geometry and Number Theory, helps to make London an international magnet for number theory, training 15 well- qualified PhD students each year. In the last ten years, Warwick has gone from having no number theory to having half-a-dozen very active faculty in diverse areas of the subject, and an important hub of activity in computational number theory, as well as explicit Diophantine problems. The discovery of a new Euler System by Loeffler (War- wick) and Zerbes (UCL) has opened the door to deep new results on special values of L-functions. Siksek’s recent work on modularity of elliptic curves over totally real fields is another important, internationally recognized, development. York has a focus on metric Diophantine approximation and Schmidt games, led by Velani and Beresnevich. Several other departments have smaller but no less active groups (with three

3 or more members), representing a diversity of research interests, namely Cam- bridge, Durham, Exeter, Nottingham, Sheffield. There are further active number theorists scattered around the UK. In terms of a UK-wide view it is currently disappointing that there seems to be a large weighting in number theory towards the south (which may be due to hiring strategies by individual departments). The lack of women in number theory in the UK is also a cause for concern. It is extremely pleasing to note that, coupled with retirements of older num- ber theorists, there is an influx of young researchers into the area, keeping it extremely lively. The number theory seminars in Warwick, London, Oxford and Bristol are extremely well-attended and there are several other lively weekly seminars across the UK. The market for high level pure mathematics research is truly international: less than half of the recent appointments in number theory within the UK are UK nationals, while many recent UK PhD’s in number theory are finding post- docs and tenure jobs both in the UK and further afield. Our main concern here is the absence of a coherent system of postdoctoral fellowships which puts the UK at a relative disadvantage in comparison with North America. Although it has been difficult to get precise information, our best guess (based on partial responses by heads of departments) is that there are at least 70 PhD students working in number theory at this time within the UK – though this may well be an underestimate. The Geometry and Number theory CDT in London alone is producing 15 new PhD students per year, around half in number theory.

Inter/intra-disciplinary activities and Engagement activities.

The largest research area in UK number theory is arithmetic geometry (which nowadays has many links to algebraic number theory). This reflects the large growth in the subject worldwide in the last 25 years, spurred on by the enor- mous progress in the Langlands program, with its origins in the work of Taylor and Wiles on modularity. Both Taylor and Wiles are British, and the UK has a number of the world experts in various aspects of the Langlands program (Tay- lor’s students and grand-students continue to flourish in the UK – Buzzard, Gee, Loeffler, and Thorne for example), and there is a corresponding growth of in- terest in representations of p-adic groups and automorphic theory. For example Helm (Imperial), Stevens (UEA) and Ardakov (Oxford) do research straddling the boundaries between number theory and representation theory. The UK also has exciting researchers in the arithmetic of elliptic curves, and Thorne (Cam- bridge) is expert in the Bhargava program, which has been responsible for some of the greatest developments of the last few years. There is currently extensive synergy between these areas. Analytic number theory has greatly flourished in the last decade in the UK.

4 Heath-Brown at Oxford continues to do world-class research and has produced some extraordinary students recently, like Maynard and Irving. Gowers and Green both work on the border of analytic number theory, combinatorics and harmonic analysis, and have also produced several very strong students, includ- ing Sanders and Conlon (Oxford), Wolf (Bristol) and Harper (Cambridge), all doing top quality research. The return to the UK of experienced researchers Wooley (to Bristol) and Granville (to UCL) have helped those universities to develop substantial groups. It should be noted that another of the hottest new stars, worldwide, in analytic number theory, Matomaki, recently obtained her PhD at Royal Holloway. There can be little question that, at this moment, the UK and the US are the two leading places in the world for analytic number theory. Analytic arithmetic geometry is an important subject in the UK (with its links to the circle method) led by Heath-Brown (Oxford), Browning and Wooley (Bristol), and Dietmann (RHUL). Links between number theory and several areas of geometry are currently being exploited by Fesenko (Nottingham) and a team at Oxford, including Min- hyong Kim (Oxford) with possible applications to arithmetic. They, and Saidi (Exeter), are experts in the recent developments on the abc-conjecture, due to Mochizuki. If his approach does become more widely accepted then the UK is well positioned to benefit from what would surely lead to dramatic changes in many parts of number theory. Another area of great interest is on the boundary of logic and arithmetic geometry. The exciting developments on the Andre-Oort conjecture, which have received a lot of international attention, stem from ideas of Pila (Oxford) and Yafaev (UCL). The interface of number theory with algebra is increasingly relevant thanks to the work of Bartel (Warwick) and Dokchitser (Bristol), classifying Brauer relations with consequences for Selmer groups of elliptic curves and for Galois module structure. Another notable achievement at the interface of algebra and number theory is the work Bartel (Warwick) and Lenstra (Leiden) giving a conceptual reformulation and correction of the Cohen-Lenstra-Martinet conjec- tures. Galois module structure is another prominent subject, with applications to the Iwasawa main conjecture, and the equivariant Tamagawa number con- jecture (Burns at KCL, Johnston at Exeter). The subject of multiple zeta values, at the crossroads of arithmetic geometry and algebraic K-theory, is well-represented in the work of Gangl (Durham) and Brown (Oxford). Brown, in a sensational breakthrough, recently proved the linear independence of certain motivic multiple zeta values, thereby settling the conjectures of Goncharov, Hoffman and Deligne–Ihara. With the retirement of Baker (Cambridge), there is little activity in tran- scendental number theory in the UK, reflecting a worldwide trend, though the subject is currently re-emerging with exciting new ideas. On the other hand the UK has several leading researchers in the related area of Diophantine approx- imation, mainly concentrated in York (Velani and Beresnevich) and Durham (Ward, Vishe and Badzihan), their activities funded by an EPSRC program

5 grant. The exciting ideas of people like Lindenstrauss and Venkatesh using er- godic theory to answer questions in Diophantine approximation, are represented by Gorodnik and Marklof at Bristol. There are several key figures internationally in explicit methods in the UK. The UK remains a world leader in using computers to experiment and develop number-theoretic conjectures, and compiling comprehensive databases of num- ber theoretic objects. A great example of this is the L-Functions and Modular Forms Database (LMFDB), a huge database of number-theoretic data which will be of interest to many working number theorists, which has been a major beneficiary of the Warwick–Bristol number theory EPSRC Programme Grant. Among those who continue to develop new algorithms and push the boundaries of feasible computation are Cremona (Warwick) and Fisher (Cambridge) (algo- rithms for elliptic curves and descent), and Booker (Bristol), Tim Dokchitser (Bristol), Vladimir Dokchitser (Warwick), Stromberg (Nottingham), Demb´el´e (L-functions, automorphic forms). This area has close connections, both math- ematically and geographically, with cryptography; the very active group in Bris- tol having extensive links with the Heilbronn Institute (which also funds their seminar). The connections between quantum chaos and analytic number theory, via random matrix theory, continue to flourish, with a fantastic group of researchers at Bristol (Snaith, Keating) and Hughes at York, and these ideas have certainly entered the mainstream of thinking throughout analytic number theory. People working in the theory of modular and automorphic representations are spread out across the UK. On the algebraic side we have Cremona and Sik- sek (Warwick), Walling (Bristol), Hill, Diamond, Buzzard, Gee and Bushnell (London), Wiles (Oxford), Funke (Durham) and the large group at Sheffield (Dummigan, Jarvis, Manoharmayum, Berger, S¸eng¨un). This group has both strength and breadth, youth and experience, and has always been strongly rep- resented in the UK. In the analytic aspects of automorphic forms there is Booker and Saha (Bristol), Petridis (UCL) and Diamantis (Nottingham). This is a low number of researchers in this important area by international standards. The geometric Langlands program is currently reaching a new level of ma- turity (with Gaitsgory and V. Lafforgue leading the approach from different directions); there is currently very little activity of this type in the UK. Pe- ter Scholze has lectured on a program to translate some of these ideas to the arithmetic setting, and there is a huge amount of potential work to do be done here. This ties in with the research of Gee, Newton and Pal (London), Thorne (Cambridge), and with the p-adic arithmetic geometry group at Exeter led by Andreas Langer. Finally the emerging p-adic and mod p Langlands program are well-represented in London, with Diamond, Buzzard, Gee and their post-docs (EPSRC-funded). This area is sure to grow in the future.

6 Future Direction/Opportunities.

Additive combinatorics has been one of the most exciting new areas in analytic number theory in recent years and this is well-represented in the UK (Gowers at Cambridge, Green, Sanders and Conlon at Oxford, Wooley and Wolf at Bristol, etc). The alternative approach to analytic number theory is also well represented in the UK by Granville (UCL) and Harper (Cambridge). A potentially explo- sive area lies on the boundary between analytic number theory and probability theory, and this is represented by Keating (Bristol), Green (Oxford), Harper (Cambridge) and Wigman (Kings). Sieve theory is going through somewhat of a revolution at the moment, and these developments are again well-represented with Maynard and Heath-Brown (Oxford), Harman and Granville (London) and others. In arithmetic geometry, the work of Bhargava and its links with algebra and sieve theory is a challenging new approach to many of the important questions, and several people in the UK (e.g. Fisher and Thorne (Cambridge), Skoroboga- tov (Imperial) and Cremona (Warwick)) are involved. Torsion in the homology of arithmetic manifolds is currently a very hot topic in the Langlands program. Berger and S¸eng¨un (Sheffield) are UK leaders in this field, especially in the setting of Bianchi manifolds. The subject of rational points on varieties has been important throughout the history of number theory and continues to present challenges and opportunities. The most exciting recent development is the use by Kim (Oxford) of non-abelian fundamental groups to define higher reciprocity maps on the adelic points of certain varieties that cut out the rational points. This programme (through the work of a team at Oxford including Balakrishnan, Dogra and others) is yielding explicit results that are beyond the scope traditional approaches (Chabauty, Brauer–Manin), with clearly much more to come. Links between number theory and cyber-security have strongly influenced GCHQ’s decision to establish the Heilbronn Institute and to use it to fund various non-classified research activities in the UK. Protection of data is also important in industry and many new challenges arise as society needs to protect large quantities of data. The newly founded Alan Turing Institute is interested in funding number theorists, and number theoretic meetings, with this end in sight, especially to help develop foundational theories that specifically apply to “big data”.

7 Annex: Main Research Groups in Number The- ory.

Currently, the largest number theory research groups in the UK are in the following areas. London – arithmetic algebraic geometry. Oxford – analytic number theory, arithmetic geometry Bristol – analytic number theory, computational number theory Warwick – computational number theory, arithmetic geometry, Diophantine equations York – Diophantine Approximation Nottingham – Langlands program Sheffield – Langlands program

Annex: Non-EPSRC funding in Number Theory.

The Heilbronn Institute funds several (open) meetings in number theory in the UK each year, as well as the London-Paris number theory seminar. UK number theorists has received several grants from the European Research Council Tim Browning: Frontiers of Analytic Number Theory And Selected Topics Toby Gee: Automorphic Forms and Moduli Spaces of Galois Representations Andrew Granville: An alternative development of analytic number theory and applications Ben Green: Approximate algebraic structure and applications Andrei Yafaev: Some Problems in Geometry of Shimura Varieties Sarah Zerbes: Euler systems and the Birch-Swinnerton-Dyer conjecture as well as researchers whose research crosses disciplinary lines Jens Marklof: Homogeneous Flows and their Application in Kinetic Theory Igor Wigman: Nodal Lines

8 MATHEMATICAL PHYSICS Lead authors: P.E. Dorey, L.J. Mason, J.A. Vickers with E. Corrigan and A.C. Davis

Mathematical Physics is the study and exploration of mathematical structures that arise in physical theories. Traditionally one of the strongest areas of UK mathematics, it ranges over all branches of mathematics and physics to solve physical problems and uncover new theories, mechanisms and phenomena. Mathematical Physics draws on some of the deepest ideas of pure and applied mathematics, and feeds back generating new examples, techniques, structures and conjectures that underlie some of the most important current programmes in mathematics. It is, by definition, both inter- and intra-disciplinary. This country has been a key player across the breadth of this field for generations but the area is now under threat in the UK. Mathematical Physics was the only major1 area of Mathematical Sciences to be designated for reduction under shaping capability. UK strength in this area has meant that the field has to a certain extent been able to weather the effects of this reduction in the short term and it remains internationally leading across a remarkably broad spectrum. Nevertheless, significant damage has already been done and it is urgent that the policy now be reversed. Mathematical Physics lies at the crossroads between the mathematical and physical sciences and underpins much of the strength of other parts of the mathematical sciences, computer science, physics, materials science and their broader impacts. A continued reduction of this area will inevitably undermine future impacts from these wider fields.

1 Statistical Overview Data from the EPSRC shows about 300 academic staff including postdocs with significant activity in the EPSRC Mathematical Sciences oriented parts of Mathematical Physics in 2014 (excluding STFC and EPSRC Physics areas). About 200 of them had Mathematical Physics as their main focus when submitted to the REF, versus around 2000 for Mathematical Sciences as a whole. Data from heads of departments and departmental websites indicate around 260 permanent academic staff, 80 postdocs and temporary staff and 180 PhD students in the UK whose work involves Mathematical Physics. Of these, about 90% are in Departments of Mathematical Sciences and 10% in Physics Departments (counting DAMTP Cambridge as Mathematics).2 Figures are not directly comparable with the 2010 Mathematical Physics landscape because that document emphasised the interface with theoretical physics, whereas this document emphasizes the role of Mathematical Physics as part of the EPSRC Mathematical Sciences programme.3 A detailed like for like comparison with the data that went into the 2010 document shows that there has been overall decline in numbers of academic staff of about 7% since 2010, with some areas down by 10%.4 EPSRC and REF data shows that Mathematical Physics represents about 10% of the current mathematical sciences community (and about 11% in 2010). The following table gives current EPSRC funding levels and against those recorded in the 2010 landscape document before reduction, given also as percentages of total EPSRC commitments to mathematical sciences.5

total EPSRC funding % of EPSRC maths budget # EPSRC Fellows % of EPSRC maths Fellows 2010 £12.7 m 6.4% 17.05 17% 2016 £6.8m 3.4% 1.66 2.2%

Table 1: EPSRC support for Mathematical Physics

The total funding includes the cost of Fellows and the drop in funding reflects the current absence of Fellows. According to EPSRC data, only one Fellowship has been awarded to someone who is primarily a mathematical physicist since 2010; see appendix A on the distribution and counting of Fellows.

1 2 Subject Breakdown Mathematical Physics is a very broad field, with different branches for different parts of physics; it can be further subdivided by the type of mathematics that is applied. Although the numbers below are of people who work primarily in Mathematical Physics, many also work in geometry, analysis or algebra.

Research Areas Number of faculty 2016 % of total Quantum field theory 87 33% Statistical physics and condensed matter 81 31% The differential equations of Mathematical Physics 60 26% Quantum foundations, computing and information 26 10%

Table 2: Table of subdisciplines.

String theory/M-theory and Integrability are cross-cutting themes but are also identifiable as sub-communities, making up about 20% and 12% of the total respectively. See appendix C for further details and statistics.

3 Discussion of research areas The UK is unusual in housing much of its Mathematical Physics capability in mathematics departments and this has led to many striking intra-disciplinary impacts in which it has led the world. Mathematical Physics is a field of widespread worldwide activity with increasing regard and impact across the mathematical sciences. Eight (Avila, Hairer, Lindenstrauss, Okounkov, Smirnov, Tao, Villani and Werner) of the last eleven Fields medals relate to Mathematical Physics, reflecting the exciting progress being made on many different fronts across the field. Here we give an executive summary leaving a detailed discussion of individual topics to appendix §B.

3.1 Quantum field theory Quantum Field Theory (QFT) is a major worldwide activity that dominates Theoretical and Mathematical Physics groups from high energy physics and string theory through to axiomatic frameworks and integrable and topological quantum field theories. There is significant interaction with condensed matter theory both through conventional field theory and via AdS/CFT. The international importance of this field is recognised, for example, in the Clay Millennium prize problem on proving the existence of Yang-Mills gauge QFT with mass gap. The UK has major centres of excellence in this area but also significant gaps in coverage of some of the most topical recent developments. Supersymmetric gauge theories and conformal field theories are often constructed from string theory directly, from AdS/CFT or via axiomatic or traditional methods. Their study has emerged as an incredibly rich field, with deep connections and impacts across mathematics, particularly with algebraic and differential geometry, topology, algebra and integrable systems. This has been an area of intense worldwide activity with top quality UK contributions from groups in Cambridge, City, Durham, Imperial, Heriott-Watt, KCL and Oxford. However, the UK is not well represented in many of the major movements that have recently swept this field such as the development of Gaiotto theories, N = 2 theories and AGT, the conformal boostrap, entanglement entropy and resurgence. There have been recent strategic appointments in each of Durham, Oxford and Surrey, but the retirements of Cardy and Osborn, and the departures of Hubeny and Rangamani from Durham, Wecht from QMUL and Pasquetti from Surrey (Pasquetti and Rangamani being ERC grant holders) leave the UK considerably under-represented in these rapidly developing areas. Scattering amplitudes, correlation functions and spectra for QFTs and string theories has been another area of major worldwide activity, again with many rapid advances and connections across mathematics including surprising interactions with number theory in work of Broadhurst, Brown and others. The UK was able to play a leading role in these developments through its core strengths in twistor theory, AdS/CFT and integrability with groups in Cambridge, Durham, KCL, Oxford, QMUL and

2 Southampton. The groups in Cambridge, Durham, KCL and Southampton have been considerably strengthened by recent appointments. This field is still developing rapidly. Quantum gravity is a traditional area of UK strength that motivated the rise in string theory (now a widespread tool across Mathematical Physics) and quantum foundations and information, now a source of much wider impact. This has its own distinct character and, because of its particular conceptual issues, often seeks to extend its scope beyond traditional QFT. Different approaches bring in topological QFT, stochastic analysis in the form of random geometries and causal dynamical triangulations, and noncommutative geometry connecting widely with other parts of the EPSRC Mathematics portfolio. The UK has high quality groups in these areas in Cambridge, Imperial, Nottingham, Oxford, QMUL, St Andrews, Swansea, Warwick and York. Axiomatic, algebraic and functional analytic approaches to QFT have been pursued in the UK for many years on a relatively small scale. With recent appointments in Cardiff, Lancaster, Loughborough and particularly York, the UK now has several leading figures in the area. Interest in QFT from pure mathematicians grows worldwide and there are opportunities to capitalise on renewed UK strength.

3.2 Statistical physics and condensed matter This broad field includes condensed matter theory, random matrices, statistical mechanics, interacting particle systems, random geometries and quantum chaos. In addition to strong links with quantum field theory there are deep connections with other areas of mathematics and physics including probability theory, analysis, algebra and phenomenological condensed matter physics. Many recent Fields medals have been associated with this area, including those of Hairer, Lindenstrauss, Okounkov, Smirnov, Werner and Tao. A common thread is the interaction of large numbers of (classical or quantum) degrees of freedom in many-body systems, and the novel emergent phenomena that arise. In statistical physics a highlight has been Hairer’s development of a rigorous theory of the nonlinear stochastic PDEs necessary for describing the crossover regimes between certain universality classes (Warwick). Another important area is the study of classical and quantum spin chains that are integrable, or exactly solvable, and of their related integrable QFTs. This field has driven developments in quantum groups, Yangians and the study of functional relations. More recently it has led to the solution of the spectral problem for planar gauge theories in AdS/CFT and has new applications in condensed matter. The UK continues to have a strong presence in this area with active groups in City, Durham, Glasgow, Heriot-Watt, Kent, KCL, Leeds, Loughborough, Oxford and York. Conversely, the power of AdS/CFT to provide fully nonlinear insights into quantum systems at strong coupling have been systematically developed worldwide to apply to systems in condensed matter theory with a leading role played by UK groups in Cambridge, Imperial and Oxford. Random matrices have a long history and continue to be of great interest with many applications both within and outwith Mathematical Physics, with recent proofs of long-standing conjectures and deepening links with integrable systems, algebraic combinatorics and the Riemann-Hilbert programme. These results have had significant impacts on many aspects of pure mathematics, with UK mathematical physicists particularly prominent in developing links with number theory. There are also strong connections with quantum chaos, quantum information theory and matrix models. UK academics are active in this area in Bristol, Brunel, Imperial College, KCL, QMUL and Warwick. Quantum chaos itself is a further area pioneered in the UK, by Berry and his school in the 1970s. The UK used to be the best in the world in this area, but UK activity is now reduced, despite the strong groups in Bristol, Brunel, Loughborough, Nottingham, KCL and QMUL, and a new hire in Cardiff.

3.3 The differential equations of Mathematical Physics The study of classical and quantum field theories in physics often reduces to problems in partial or ordinary differential equations, leading directly to issues in geometry and analysis. Work in this area has

3 much overlap with nonlinear and dynamical systems, spectral theory, the analysis of PDEs and geometry. There is a continuum of work through these fields depending on whether the equations arise from Mathematical Physics or elsewhere, but Mathematical Physics often provides the benchmark problems. Recent prize winning Mathematical Physics work in this area include the Fields medals of Avila and Villani and Tao, or, in the UK, the Adams prize shared by Dafermos and Stuart in Cambridge. General relativity has diversified from the use of traditional applied mathematics combined with geometry and topology to PDE analysis on the one hand and detailed physical modelling and numerical analysis on the other. The more traditional community has moved to the study of higher dimensional black holes and the construction of geometries for application in AdS/CFT and supergravity. This is an area where the UK has been internationally leading but the shift leaves a gap in the former core. The PDE analysis community in general relativity has grown on the back of the countrywide growth of PDE analysis with around 8 recent appointments countrywide (Cambridge, Edinburgh, Imperial, Oxford, Surrey). This community is of top quality, as seen in plenary talks at major international meetings. Much of the modelling and numerical analysis is now in the STFC remit, although there are problems with people falling between the two stools. UK strength has been depleted by retirements of major figures such as Gibbons and Hawking in Cambridge. The geometry of strings and M-theory has been a continuing story of vigorous interaction between (algebraic) geometry and physics. The original motivation was model building in particle physics but early on the remarkable mathematical structures uncovered led to some of the deepest programmes in algebraic geometry via topological strings (and Donaldson-Thomas invariants) and homological mirror symmetry. More recently in the form of AdS/CMT these have been supplemented by broad ranging applications to condensed matter physics. These areas remain a major area both of worldwide activity and of UK strength in Cambridge, City, Durham, Edinburgh, Heriott Watt, Imperial, KCL, Oxford, Southampton, Surrey and UCL. Topological solitons and integrable systems is another traditional area of UK strength ranging from the study of monopoles and instantons though to classical and quantum integrability. It interacts vigorously with algebra, differential geometry, geometric analysis and quantum field theory, as well as with applied mathematics and applied analysis. UK strength has attracted a number of leaders from overseas and there are major groups doing leading work in Cambridge, Durham, Edinburgh, Glasgow, Heriot-Watt, Kent, Leeds, Loughborough, Oxford, York and Surrey.

3.4 Quantum foundations, information, computing and cryptography Quantum foundations, information and computing now impacts widely across disciplines including philosophy, mathematics, computer science, physics and material science. The introduction of quantum computing by David Deutsch as an SERC Advanced Fellow in Mathematics in the late 1980s gave the field a new mathematical focus that continues to develop rapidly worldwide. Quantum information, computing and cryptography continues to thrive in the UK and UK groups continue to set the worldwide agenda. This field now spreads into computer science and materials departments as well as interacting with topologists and functional analysts. There are groups across a variety of mathematics departments in Bristol, Cambridge, Leeds, Oxford, York, UCL. Although reduced under the EPSRC Mathematics programme this is an area of growth for other parts of EPSRC. This has led to an unfortunate incentive to deplete mathematical input into this activity, despite the fact that its origins and early development were in mathematics and mathematics continues to drive new progress.

4 Research Community UK Mathematical Physics has been an area of great strength for generations and continues to punch above its weight internationally. UK communities were established by (a) the pioneers of the standard model of particle physics: Goldstone, Higgs, Kibble, Salam, Taylor and Ward;

4 (b) Bondi, Hawking and Penrose, who set out the modern understanding of the global structure of space-times including cosmology, gravitational radiation and the theory of black holes; (c) founders of string theory including Goddard and Olive; Green, a leader of the ‘first string revolution’; Hull and Townsend, leaders of the second; Candelas leading Mirror symmetry; and Tseytlin leading AdS/CFT integrability; (c) Deutsch, who developed quantum computation/information/cryptography in the UK while on an SERC Advanced Fellowship, with Ekert, Jozsa, Popescu and Steane; (d) Berry with his work on geometric phases, caustics, and quantum chaos. Their successors in the UK continue to set the agendas of these fields worldwide. UK Mathematical Physics continues to be recognized for its excellence: Breakthrough prizes: Michael Green and Stephen Hawking. Fields Medal: Martin Hairer. Copley medals: Stephen Hawking, Roger Penrose. Shaw prize: Nigel Hitchin Recent FRSs: Candelas, Duff, Hairer, Hull, Keating, Marklof, Sklyanin, Ward, West. Whitehead prizes since 2008: Alday, Dafermos, Marklof, Mouhot, Snaith. ICM talks were given by Dafermos, Gibbons, Hairer and Marklof. European Networks: GAuge Theory as an Integrable System (includes Durham, KCL and QMUL). UK Mathematical Physics has been rewarded by 22 ERC awards of about £1.2m each: Alday (Oxford Maths), Barack & Drummond (Southampton), Gauntlett, Holzegel & Tseytlin (Imperial), Dafermos, Green, Mouhot, Reall & Tong (Cambridge), Hairer & O’Connell (Warwick), Krasnov & Sotiriou (Nottingham), Marklof (Bristol), Martelli, Murthy & Schafer-Nameki (Kings), Pasquetti (Surrey), Rangamani (Durham), Starinets (Oxford Physics). Nevertheless, this current quality is under serious threat. Many names on the first list above are close to retiring or have already done so, while of the ERC award holders two (Pasquetti and Rangamani) have recently left the UK. Although some Universities are replacing like with like and KCL and Surrey have expanded, many have seen attrition or significant movement to less mathematical areas that are funded by the STFC or the EPSRC theoretical physics portfolio where funding is not reduced. This has led to a total contraction of around 7% in a period when the Mathematical Sciences community has been stable. This is particularly apparent in Cambridge, which has seen the retirements or resignations of Gibbons, Green, Hawking, Osborn and Turok. Their successors have largely been appointed within the STFC or EPSRC Theoretical Physics remits, despite, in the case of the Lucasian chair, there being mathematical physicists available who excited the department. Cambridge Mathematical Physics has not successfully applied for a single grant from the EPSRC Mathematics programme since reduction. The 2010 Mathematical Physics Landscape document gave 11 permanent faculty in formal theory (i.e. string theory and QFT) and 11 in GR, the current respective figures are now 6 and 9 (including new PDE analysts and geometers). Perry and Townsend are close to retirement and there is a danger that further retrenchment will occur. This represents major lasting damage to one of the UK’s iconic Mathematical Physics groups, led successively by Dirac, Sciama, Hawking and Green, that for years set the agenda for the field in the UK and worldwide. This damage will be felt in the field countrywide. A similar move to STFC-facing research has taken place in Southampton and in the recent replacement of Wecht in QMUL. Others report difficulties with their administrations in replacing retiring mathematical physicists because of the reduce label. The leadership of other major UK groups are susceptible: for example, Corrigan and Sklyanin, Ward, and West are close to retirement in York, Durham, and KCL respectively. The lack of Fellowships since 2010 is a particular problem for the people pipeline: since reduction, Mathematical Physics has been unable to hire the best young international stars on their own independent Research Fellowships to retain or attract them into the UK system. Fellowships are also a key tool for improving the gender balance and age profile. A distinctive feature in the UK is that Mathematical Physics groups are placed in departments of

5 mathematics rather than physics. This has been a great strength, leading to more cross-fertilization with pure mathematics than arises elsewhere. It has also been a strength for hiring new faculty. With the Large Hadron Collider and LIGO coming on stream, physics departments worldwide have focussed on phenomenology and cut back on Mathematical Physics, leading to a pool of outstanding candidates who might have hoped to find positions in the US. Groups such as KCL and Surrey have exploited this to expand significantly. Mathematical Physics faculty positions often have at least 70-80 applicants, many of them internationally outstanding. The groups that have expanded have been well rewarded with ERC grants and strong performances in the REF, but less so in terms of EPSRC support. Mathematical Physics continues to fascinate young mathematicians and physicists and applications for PhD positions greatly outnumber the available funding or supervision (about 8:1). Indeed the new MMath-Phys in Oxford had 150 external applications for 5 places last year. The success in achieving ERC funding has mitigated what would otherwise have been a catastrophic drop in the number of postdocs in the system, but this source is dangerously unstable. There are substantial bottlenecks from PhD to postdoc, and from postdoc to faculty. This however ensures quality and mathematical physicists have transferable skills that are widely in demand in other areas of research both in industry and academia (e.g. the Mathematical Biologist Reidun Twarock or the famous biologist Lord May who both started in Mathematical Physics) and in the City and management.

5 Inter/intra-disciplinary activities and engagement activities Interdisciplinary impacts and engagement: Mathematical Physics clearly interfaces directly with those parts of physics with which it is concerned. One could not conceive of the international effort for the recent breakthrough measurements of gravitational waves from inspiraling black holes without the mathematical foundations provided by Bondi, Hawking and Penrose for gravitational waves and black holes 50 years ago (lead times from basic research to impact are long). Similarly, the impacts of the early UK pioneers Goldstone, Higgs, Kibble, Salam, Taylor and Ward of the standard model of particle physics have only just been fully realized with last year’s detection of the Higgs Boson. Penrose’s quasi-periodic tilings were observed in nature by chemists in the form of quasi-crystals, leading to the 2011 chemistry Nobel prize. Mathematical physicists made major contributions to understanding properties of graphene, which displays massless Dirac modes and a quantum Hall effect.12 UK cosmologists work closely with experimentalists to test modified gravity theories. The scalar force and analogue gravity are detectable in lab based Casimir force experiments (Cambridge, Nottingham). Quantum information has impacted on experimental physics. UK-based ion trap researchers presently hold the world record for the most well controlled qubits of any kind. For optics the UK has several large and leading teams (e.g., Bristol). Oxford has one of the four world-leading teams that simultaneously reported the first demonstration of Boson Sampling, a problem thought to be exponentially hard to solve classically.6 In technology, quantum cryptography is now being used by banks in Zurich.10 Four quantum hubs have been created as part of the £270M UK National Quantum Technologies Programme. These aim to realize the mathematical work from quantum information and related areas. Each Hub is tasked with developing a class of quantum technology towards commercial exploitation, e.g. the York Hub targets quantum communication8 and the Oxford-led Hub9 targets “Networked Quantum Technologies” using their optical and ion trap expertise to develop universal quantum computers. The mathematics developed in quantum chaos have found important applications in acoustics, radar scattering, elasto-mechanics and laser optics resulting in a number of REF2014 impact case studies. Pendry’s initial work on metamaterials/cloaking has had major impacts. Quantum foundations impacts on philosophy with quantum interpretation of probabilities (Wallace, Deutsch, Saunders), and physics on quantum thermodynamics, and studies of quantum phenomena in

6 biological systems, studies of fault tolerance of practical implementations (Oxford). It also naturally had a long standing interaction with computer science, see11 for a modern novel example. Mathematical Physics has a long tradition of public engagement, e.g. the REF public engagement impact case studies based on each of John Barrow and Stephen Hawking and Roger Penrose (using Tod’s work). Intra-disciplicary impacts: Mathematical Physics has had major and pervasive impacts on other areas of both pure and applied mathematics. Leading pure mathematicians worldwide, Atiyah, Borcherds, Connes, Drinfeld, Jones, Hitchin, Kontsevich, Manin, Okounkov, Tao etc. have long exploited developments in Mathematical Physics in their own disciplines, so much so that borderlines with many other parts of mathematics are now quite unclear. The many deep ongoing interactions with algebraic and differential geometry, algebra, analysis including PDE, nonlinear systems, continuum mechanics and number theory will already be clear from §B. In addition we mention also the following (many of which could easily have also been placed in §B). Quantum chaos interacts with many other parts of mathematics: semiclassical/microlocal analysis, probability theory, dynamical systems and ergodic theory number theory and automorphic forms. It now attracts world-leading researchers in other areas (Bourgain, Colin de Verdiere, Rudnick, Sarnak, Sinai, Soundararajan, Zelditch, Zworski). Mirror symmetry (Candelas) has led to the homological mirror symmetry programme in algebraic and symplectic geometry studied by the major UK groups in these fields (Cambridge, Imperial, Oxford, Sheffield, Warwick) and worldwide. String theory had a major impact on algebraic geometry, algebra and topology via topological string theory, topological quantum field theories and new invariants (Gromov-Witten, Donaldson-Thomas). A separate family of applications arose from the 4d:2d corespondences described in §B.1 that added a third pillar to the geometric Langlands programme & resolved major conjectures (Okounkov). Recent analogues of the original moonshine conjectures proved in fields medal winning work of Borcherds have emerged for the Matthieu groups whose representations arise mysteriously in the string calculations of gravitational entropy (Durham, KCL and worldwide). Quantum integrable systems have by now had many impacts on algebra via quantum groups, cluster algebras and representation theory. Similarly, noncommutative geometry has become a major topic in pure mathematics beyond its origins in quantum gravity and gauge theories. Tomita-Takesaki theory and the classification of von Neumann algebras provide further examples of interplay between rigorous QFT and pure mathematics. Topological QFTs have revolutionized parts of algebra leading to the ongoing programme of categorification in representation theory, and topology, both in the context of knot theory and in cobordism categories). They impact on quantum computing and computer science11 (Oxford, Leeds). Important interactions with number theory arise via quantum cryptography and the need to make cryptographic routines safe against quantum computers. The work of Broadhurst and Brown provides a distinct surprising link with scattering amplitudes with impact on conjectures in number theory.

6 Future directions, opportunities and threats Opportunities: Progress in Mathematical Physics is often fast and unpredictable and when identifying future research opportunities it becomes apparent that some have already been seized by UK groups in recent strategic appointments, such as those in Oxford and Durham in the area of the conformal bootstrap and resurgence. The most effective way to respond rapidly to such new developments is via the Fellowships programme: permanent faculty positions are infrequent and extremely competitive and PDRAs are not suitable when the proposed Fellow will lead a project for which there is no local expert. Other opportunities include:

• Major recent breakthroughs in QFT include the proof of Cardy’s a-theorem in higher dimensions, new applications of Berry-Howls resurgence for summing series, the classifications of higher

7 dimensional conformal field theories, the mathematical structures of N = 2 gauge theories, the new understanding of entanglement entropy, new relationships between BMS symmetries, amplitudes, ambitwistor strings and information. These are all areas where the UK has considerable expertise (reinforced by new appointments in Cambridge, Durham, Edinburgh and Oxford). • Important rigorous mathematical work being done outside the UK on systems out of equilibrium using C∗ algebras in the EU. The UK has the expertise to join this effort (City, KCL, Oxford). • Topological phases of matter and their classifications are a growing area with significant potential applications mostly outside the UK but with some representation in Oxford. TQFT and defects in topological insulators are not actively studied in the UK in mathematics, but ought to be, given the UK expertise in this area (Oxford, St Andrews, York Physics).

Threats: The biggest single threat to this field is continued reduction under balancing capability. Shaping Capability started about 5 years ago. The two main consequences have been: • Reputational damage and the associated loss of morale in the community. • The near-absence of Mathematical Physics from the EPSRC Fellowship programme.

The reputational damage is a deterrent to the retention of our most talented stars who have many opportunities worldwide, and to the attraction of new talent to the UK at a time when Mathematical Physics is designated a second class area for funding. It has led to attrition: • Department Heads and Vice Chancellors are moving posts from Mathematical Physics to STFC or EPSRC physics funded areas where funding is not reduced (cf. the Cambridge Lucasian Chair). • Leading external candidates for key chairs have declined citing the UK funding situation, as for example the Cambridge chair in Mathematical Physics vice Turok, that was filled by someone primarily supported by the STFC. • These effects have led to a reduction of the size of the community of about 7%, much of the shift being to STFC or EPSRC physics funded areas.

A widespread concern is that subjects that started in the UK and where UK groups had set the worldwide agenda are now falling behind, without proper funding and support. The loss of postdoctoral fellows is cited as a particular problem. The more than ten fold drop in Fellowships arises from (a) the deterrent effect of the reduce label both on committees and prospective applicants, (b) the intra-disciplinary requirement on a subject that is already inter-disciplinary requires a third plank that becomes excessive, and (c) applications falling between the different stools for funding.13 Fellowships used to be a key part of the Mathematical Physics pipeline for new academic appointments, allowing departments to attract rising international stars and retain the best UK talent. The effective exclusion from Fellowships has been partially mitigated by the community’s success in obtaining ERC funding but this source is unstable and does not provide for independent Fellowships. The lack of Fellowships serves to drive further gender imbalance and skew the age profile. Mathematical Physics is a field of widespread worldwide activity and increasing regard as demonstrated by the fact that eight of the last eleven Fields medals relate to Mathematical Physics. The absence of clear distinctions between mathematics and physics in our academic structure has helped the UK to produce world-leading cross-fertilization between physics and fields as diverse as algebra, geometry, PDE and stochastic analysis and, as apparent in the previous section, this has underpinned highest level achievements across the mathematical sciences, physics, chemistry and computer science. Continued lower levels of investment risks diminishing the community past a point of no return and destroying its capacity and potential for further widespread impacts. The reduction of Mathematical Physics should be reversed.

8 Acknowledgements:

We are grateful to John Cardy, Toby Cubitt, Benjamin Doyon, Clare Dunning, Artur Ekert, Chris Fewster, Martin Hairer, Jon Keating, Jens Marklov, James Sparks, Ian Strachan, Christopher Voyce for their inputs into this document.

Notes 1 By major, we mean more than 1% of the EPSRC Mathematical Sciences budget or size of the Mathematical Sciences community. 2 These figures and those below are necessarily very approximate because of the blurred borderlines between areas. 3 See https://www.epsrc.ac.uk/research/ourportfolio/themes/mathematics/introduction/mathphysics/. 4 This follows from a detailed like for like comparison of the major Mathematical Physics areas within the EPSRC remit with the data in the key areas III, IV and VI of the 2010 document which covers the major part of the research within the EPSRC Mathematical Sciences remit. 5 The 2016 funding figures are taken from the display on https://www.epsrc.ac.uk/research/ourportfolio/themes/mathematics/ which gives a total financial commitment of £197m for Mathematical Sciences. The figures for the 2010 EPSRC grants and Fellows in Mathematical Physics are taken from the 2010 landscape document, p64 of https://www.epsrc.ac.uk/newsevents/pubs/international-review-of-mathematical-sciences-in-the-uk- information-for-the-panel-part-1-3/ and the total spend of £190m on Mathematical Sciences in 2010 came direct from the EPSRC. See p43 of that document for the total figure of 100 Fellows in 2010 in Mathematical Sciences; the total number of EPSRC Fellows in Mathematical Physics came from the EPSRC directly. The figure of 74 current Fellows in Mathematical Sciences is taken from EPSRC data in grants on the web. 6 Science 339 798-801 (2013). 7 Newton Institute programme ‘Operator Algebras: Subfactors and their Applications’ 1-6/2017. 8 http://quantumcommshub.net/. 9 NQIT.org 10 See for example http://www.idquantique.com/news/colt-qkd-as-a-service.html Today’s, QKD is running commercially at several locations, e.g. in Switzerland and Luxembourg. An impressive example is a Geneva bank that has its back-up near Lausanne using QKD below lake Geneva to secure the link. 11 See http://www.qmac.ox.ac.uk/. 12 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature 438 (2005) 197 doi:10.1038/nature04233 [cond-mat/0509330 [cond-mat.mes-hall]]. 13 Although the EPSRC has a stated commitment to work with applicants and other UK research councils to avoid (c), different funding mechanisms make this impossible to implement.

A Fellowships

This data is taken from grants on the web http://gow.epsrc.ac.uk. Since many Fellowships are now intra-disciplinary, EPSRC ascribes a fraction to each theme. Six of the current 74 EPSRC Fellows include Mathematical Physics in their list of intra-disciplinary themes, but only one is over 1/3 in Mathematical Physics and the total of the fractions is 1.66 FTE according to EPSRC data. Certainly five of the six would not regard themselves as being primarily mathematical physicists.

9 Area total postdoc early established Algebra, Geometry & Topology 18 7 5 6 Number Theory 12 3 6 3 Logic & Combinatorics 2 1 1 0 Mathematical Analysis 13 4 5 4 Mathematical Physics 1 1 0 0 Nonlinear systems 3 2 0 1 Continuum Mechanics 3 2 0 1 Complexity 1 1 0 0 Operations Research 2 1 1 0 Statistics and applied probability 19 5 8 6

Table 3: Current distribution of Fellowships across Mathematical Sciences

B Detailed discussion of research areas

These subsections describe the more striking recent developments in Mathematical Physics worldwide and in the UK. There are omissions both for reasons of space and the ignorance of the authors. In §5 we focus on areas that interact with other areas of mathematics and physics but there are also many examples here. Recent new appointments are taken to be those in the last 10 years.

B.1 Quantum field theory and conformal field theories.

Work in Quantum Field Theory (QFT) aims to understand the fundamental principles underlying QFT, to elucidate their general consequences, and to produce and classify mathematically consistent QFTs. The international importance of this activity is recognised, for example, in the Clay Millennium problem on proving the existence of Yang-Mills gauge QFT with mass gap. Seiberg dualities led to the modern view of QFT in which quite different supersymmetric gauge theories are realized as different weak coupling limits of the same QFT. This has recently been made systematic by applying string dualities to gauge theories built from strings in different brane configurations. The vacuum moduli have particularly rich geometry, with links to integrable systems, spectral curves and hyperk¨ahlergeometries which can become noncommutative. Explicit constructions arise from the algebraic geometry of quiver varieties, cluster algebras, dimer models, twistors and geometric representation theory; wall-crossing in Bridgeland stability conditions determines the low energy theory. This is a major worldwide activity with leading work in the UK (Cambridge, Imperial, Heriot-Watt, Oxford including new appointments in Imperial and Oxford). Conformal field theories (CFT) in various dimensions have seen many recent breakthroughs worldwide. New examples come from the 4d:2d correspondences based on the putative six dimensional (0, 2) theory (Gaiotto theories and AGT). Observables in the 4d CFT are computable via 2 dimensional CFTs (Oxford and new appointments in Heriot-Watt, Surrey). AdS/CFT associates a supersymmetric CFT in d dimensions to a gravity theory in in d + 1-dimensions. The CFT is controlled by a compact Einstein manifold. Explicit realizations match localization on the CFT to the geometry of the Einstein manifold (Imperial, KCL, Oxford). The conformal bootstrap has led to a series of new breakthroughs outside the UK. Using symmetries efficiently to constrain the CFT spectrum and structure constants it gives nonperturbative bounds and leads to analytic insights (new appointments in Durham, Oxford). Axiomatic and Constructive QFT involves functional analysis, operator algebra theory, geometry and microlocal analysis. Recent deep mathematical and physical insights include new constructions of integrable and conformal QFTs in 1+1 dimensions, the rigorous construction of perturbative gauge theory and effective gravity, and locally covariant formalisms for axiomatic QFT in curved spacetimes. Leaders include UK mathematical physicists (Cardiff, Lancaster, Loughborough, York with new

10 appointments). Quantum Gravity: Quantum gravity has distinctive problems that sets it aside from other QFTs. They were a major motivation for the re-examination of quantum foundations that led to §B.4. Approaches to quantum gravity range from string theory and amplitudes to the stochastic analysis of causal dynamical triangulations and random geometries, loop quantum gravity, canonical methods and noncommutative geometry connecting with mainstreams in other branches of mathematics. Noncommutative geometry and loop quantum gravity are well represented in the UK (Nottingham, QMUL, Swansea). Dynamical triangulations were developed in the UK but most of the work is now carried out elsewhere (e.g. Horava-Lifshitz theory) except for nice work solving statistical mechanical models on random triangulations leading to Liouville quantum gravity (Oxford physics). Random geometries are studied extensively by stochastic analysts and have major new results on Liouville quantum gravity and its connection with the Brownian map by stochastic analysts (Cambridge and Warwick). Topological quantum field theory (TQFT) now intersects a number of areas such as diagram categories (Leeds, Nottingham and Oxford computer science). These are used for observables in quantum gravity, but also in statistical mechanics models and quantum information, are known as tensor networks in the physics literature, and also model defects in topological insulators (Oxford, St Andrews, York Physics). There has been a major development internationally in the study of black hole entanglement entropy outside the UK. Another major development is the interaction between the gravity amplitude story and asymptotic symmetries and black hole information that has been led by Strominger (Harvard) but with important input from UK workers connecting with amplitudes (Cambridge, Durham and Oxford). Amplitudes, spectra and correlation functions in gauge and gravity theories. These are the key outputs of QFTs that determine the physical and mathematical meaning of the theories and have recently been discovered to have many rich mathematical structures. AdS/CFT led to nonperturbative computations of correlation functions and scattering amplitudes for planar N = 4 supersymmetric Yang-Mills via integrable systems modelled on the AdS worldsheet and spin chains. These methods have matured into a nonperturbative framework ‘the quantum spectral curve’ for the spectrum with a parallel programme for amplitudes. The UK made key advances in this worldwide activity (Cambridge, Durham, Imperial, KCL, Oxford, Swansea & new appointment in Surrey). Witten’s twistor-string led to a renaissance in perturbative methods for amplitudes in the US that was quickly picked up in the UK generating new powerful tools, twistor actions, Grassmannian and polyhedral formulations that make contact with combinatoric algebraic geometry via the positive grassmannian and amplituhedron with extensions to correlation functions. In a breakthrough, twistor strings were recently generalized to ambitwistor strings that generate loop amplitudes for many theories (Cambridge, Durham, Oxford, QMUL, new appointments in Cambridge, Durham, QMUL, Southampton). Explicit formulae for scattering amplitudes both in QFT and string theory give polylogs, multizeta values (and modular forms for strings) that arise as periods of mixed Tate motives in algebraic geometry & number theory leading to Cartier’s ‘cosmic Galois group’ conjecture and motivic formulations of the S-matrix. The conjecture that this is the Grothendieck Teichmuller group is false for Φ4 theory (Francis Brown) but is open for maximally supersymmetric theories and strings, leading to a highly successful bootstrap with leading UK work (Cambridge, Open and new faculty in Durham, Oxford, Southampton). Integrable QFTs and spin chains: continue to be important with significant recent inputs into the spectral problem for higher-dimensional gauge theories, developing the Thermodynamic Bethe Ansatz (TBA) linking to cluster algebras, and the computation of correlation functions. Work on the TBA connects integrable QFTs to the spectral properties of non-Hermitian operators via the ODE/IM correspondence, and has significant links both with analysis, and with PT-symmetric quantum mechanics. Correlation functions have recently been used as a theoretical laboratory for exploring properties of entanglement, non-equilibrium statistical mechanics, the string/gauge theory correspondences, and their relationship to representation theory. Work continues in exploring relativistic and non-relativistic integrable QFTs on semi-infinite or finite domains, or in the presence of defects of

11 various kinds. This links in to algebra via representation theory and to analysis via odes and pdes. The UK makes significant contributions (City, Durham, Heriot-Watt, KCL, Kent, Oxford, York).

B.2 Condensed Matter/Statistical mechanics

Classical and quantum statistical mechanics is the mathematical study of many-body systems and their thermodynamics in and out of equilibrium. It includes criticality, many-body entanglement, non-equilibrium physics, large deviations, fundamental aspects of thermalization such as Fourier’s law and generalized Gibbs ensembles, disordered systems and glasses, and new states of matter such as topological states. There has been substantial mathematical progress in the last 10-15 years in quantum statistical mechanics using C∗ and vertex operator algebras and their connections to QFT and CFT7 (Cardiff, KCL Nottingham, Oxford, QMUL, UCL, Warwick). In certain crossover regimes between universality classes, the fluctuations of systems in statistical mechanics are formally described by nonlinear stochastic PDEs. These equations are often severely ill-posed, so that even their mathematical meaning was not clear until recently. In Fields medal winning work, Hairer has developed a general theory allowing to give, for the first time, a solution theory for many of these stochastic PDEs, and to link them rigorously to the large-scale behaviour of microscopic models (Warwick). Condensed matter theory concerns the mathematical study, using various advanced theoretical frameworks, of electrons in strong interaction, including magnetism, the Kondo effect, electron transport, high-temperature superconductivity and the quantum Hall effect, with applications to electronic devices, spintronic, quantum computing, etc (KCL, Oxford, UCL). Many such models of interacting particle systems such as quantum spin chains, which both have experimental realizations in solid state physics, and more recently in cold atoms, are integrable, satisfying the Yang-Baxter relations, and their spectrum may be found using the Bethe ansatz. There are also related integrable quantum field theories. The mathematics of these has led to the development of quantum groups, Yangians, and systems of functional relations. These link in to discrete integrable systems which have emerged relatively recently by investigating analogues of integrability structures from the continuous case. They are related to integrable dynamical systems and field theoretic models and bringing in cluster algebras and algebraic geometry. The study of these models has for many years been a thriving UK activity and these now also have applications to gauge and string theories (City, Durham, Glasgow, Heriot-Watt, Kent, KCL, Leeds, Loughborough, Oxford, York). Another current area of condensed matter is that of topological phases of matter, for example fractional quantum Hall states and topological insulators. One of the interesting problems is to classify such phases. This brings in such tools as K-theory and modular tensor categories and also bears on quantum computing (Oxford and outside the UK). There is a substantial UK component to the effort to use the AdS/CFT correspondence (here referred to as AdS/CMT) to understand strong coupling quantum field theories in general, and, to apply these to real condensed matter systems and statistical physics (Cambridge, City, Durham, Edinburgh, Heriott Watt, Imperial, KCL, Oxford, Southampton, Surrey, UCL). Random Matrix Theory, initially developed by mathematical physicists including E. Wigner and F. Dyson, has flourished in recent years. Major developments include (a) proving the long-standing universality conjecture going hand-in-hand with advances in analysis and combinatorics that were stimulated by the universality programme, and (b) deep links to integrable systems, algebraic combinatorics and the Riemann-Hilbert programme. These results have been influential across Pure Mathematics and Probability attracting mathematicians of the highest calibre (e.g. A. Borodin, P. Deift, A. Its, M. Kontsevich, A. Okounkov and T. Tao). UK mathematical physicists have taken a leading role in exploring connections between Random Matrix Theory and Number Theory which have attracted considerable attention (e.g. from leading non-UK number theorists including M. Bhargava, H. Iwaniec, N. Katz, P. Sarnak, K. Soundararajan, and T. Tao), and they have led in exploring connections with

12 random processes (e.g. with P. Diaconis, A. Guionnet and O. Zeitouni), including the topical problems of determining the extrema of correlated random surfaces, random growth models, the KPZ equation etc. There are close interactions between Random Matrix Theory and quantum chaos, quantum information theory and matrix models, and with other researchers in computer science, finance and information theory. UK mathematical physicists have played a substantial role in these developments (Bristol, Brunel, Imperial College, KCL, QMUL and Warwick). Quantum chaos was established as a field in the UK by Berry and his school in the 1970s. Recent advances include the use and justification of random matrix techniques in quantum transport, the analysis of spectral data of closed and open quantum systems, the value distribution of eigenstates, and results on quantum unique ergodicity. In particular the combination of ergodic-theoretic and semiclassical techniques have led to spectacular breakthroughs such as Lindenstrauss’ proof of quantum unique ergodicity in an arithmetic setting (Fields Medal 2010) and Anantharaman’s lower bounds on entropy of eigenstates (Salem Prize 2011, Poincare Prize 2011). Amongst the most exciting developments in recent years is the application these ideas to the spectra of discrete graphs and quantum networks. Quantum chaos remains highly topical, as indicated by the presence of dedicated quantum chaos sections in the leading journals in the subject (Comm Math Phys, Annals Henri Poincare). The UK is well represented (Bristol, Brunel, Loughborough, Nottingham, KCL, and QMUL with a recent appointment in Cardiff).

B.3 The differential equations of Mathematical Physics

The PDEs and dynamical systems that arise in physics range from classical field theories such as general relativity and Yang-Mills to equations with a quantum origin describing vacua and instantons or bound states. This interfaces with geometry, analysis, nonlinear systems and continuum mechanics. Gravity: UK mathematical physicists Bondi, Hawking and Penrose set out the modern understanding of global space-time structure in the 1960’s and 1970’s. As PDEs, gravitational theories have become a benchmark in the analysis of PDEs. Recently the drive has been to prove the stability for black hole space-times and theorems with minimal regularity and optimal decay. There are new theorems for linear and nonlinear fields on curved space-time backgrounds. UK workers make major contributions to this effort (Cambridge and QMUL, with new appointments in Edinburgh, Imperial, Oxford, and Surrey). Numerical analysis has led to significant advances in mathematical formulations of relativistic hydrodynamics and critical phenomena in general relativity. A significant new area of activity is the use of numerical methods to investigate higher-dimensional black holes and string theory. There are interfaces with geometry in work on extremal K¨ahlermetrics, and STFC funded work to produce accurate gravitational wave templates for black hole and neutron star mergers in support of their detection by LIGO (Cambridge, Cardiff, Edinburgh, Imperial, Southampton). Higher dimensional black holes have seen breakthroughs with the discovery of new families of examples analytically via PDE existence and via integrability, and numerically. These exhibit striking new topological phenomena with black rings, black lens’ and Saturns. Unlike four dimensions, instabilities for rapidly spinning black holes signal the bifurcation of solutions leading to rich phase diagrams that are still being investigated (Cambridge, Imperial, and new appointments in Edinburgh & Southampton). Mathematical Cosmology studies modifications of general relativity that account for the observed accelerated expansion of the universe. There is much work to establish these theories on a sound mathematical footing, for example to check singularity theorems and investigate black hole solutions and when the no hair theorem holds in these theories (Cambridge, Durham, Nottingham). Geometry of String/M-theory. String/M-theory vacua are geometries that provide candidates for unification of particle theories or for AdS/CFT constructions of gauge theories and strongly coupled condensed matter or fluids systems. They also exhibit an extraordinarily rich geometry often involving some of the deepest ideas from differential geometry, leading to new programmes in pure mathematics. Calabi-Yau 3-folds are key candidates but, despite the 5 × 108 explicit examples, only in the last 10 years

13 has there been a single example that nearly yields the standard model. Recent systematic classifications and constructions of Calabi-Yaus and heterotic geometries with low Hodge numbers give thousands of such examples. Other more general geometric structures with applications to AdS/CFT have been constructed. UK groups lead in this area (City, Imperial, KCL, Oxford, and new faculty in Surrey). Generalized complex structures recently introduced by Hitchin arose from string compactifications with extra supersymmetry. More recently these have been extended to exceptional group versions and further to T-folds with a local version of mirror symmetry with a new noncommutative geometry. The UK has led this area (Cambridge, Imperial & KCL and of course Hitchin in Oxford). Branes and near horizon geometries play a major role in string theory and its applications to gauge theories and AdS/CFT. Their classification has been an intensive worldwide activity with leading work in the UK (Cambridge, Durham, Imperial, KCL, Oxford, and new appointments in Edinburgh & Surrey). Integrable systems and topological solitons: Instantons, Monopoles, Skyrmions and Vortices are solutions to geometric PDEs often motivated quantum mechanically. UK mathematical physicists made seminal contributions and have been leading for decades and there are important recent developments that freely cross boundaries with differential geometry. They are studied using functional and numerical analysis together with studies of the moduli spaces of their solutions. Remarkable symmetry properties have been discovered and recently surprisingly accurate models of nuclear structure have emerged from Skyrmions. Integrability remains an important tool. There is a continuum of work in this area through to equations from fluids (KdV, KP and their relatives) and other applications in physics. This is an area where the UK continues to lead (Cambridge, Durham, Edinburgh, Glasgow, Heriot-Watt, Kent, Leeds, Loughborough, Oxford, York, Surrey). Spectral theory: the UK has several groups working in the spectral theory of differential operators: at Bristol, Cardiff, Imperial, KCL and UCL. Individual researchers are based at Bath, Heriot-Watt, Kent, Loughborough, QMUL and Reading. A major result obtained in recent years is the proof of the Bethe-Sommerfeld conjecture in the general setting (Parnovski and Sobolev, UCL).

B.4 Quantum foundations, computing and information

UK researchers made the seminal contributions that have set the world-wide agenda for this field. Deutsch set out the modern view of quantum computation while an SERC Mathematical Physics Advanced Fellow followed by Ekert, Jozsa and Popescu who have been prominent in this rapidly evolving area both in terms of discoveries made and in setting the agenda for world-wide research. Although it started in Mathematical Physics (an EPSRC reduce area), it now straddles computer science, physics and materials (an EPSRC growth area). Foundations work continues with rigorous work on uncertainty principles (York), both sheaf-theoretic and category theoretic understanding of contextuality and generalised probabilistic theories (Oxford computer science). Quantum cryptography has seen significiant developments (Cambridge). In recent years the field has been drawing on (and contributing to) increasingly sophisticated areas of mathematics. For example topological quantum field theory now plays a significant role in quantum and topological computation11 (Oxford, Leeds). A major breakthrough in quantum Shannon theory was the disproving of the additivity conjecture drawing in the interest of random matrix theorists (Bristol). Functional analysts have also moved into quantum information theory: Bell inequalities - one of the earliest pieces of quantum information theory - are directly related to operator spaces (part of modern Banach space theory). Some of the best Banach space theorists now work in quantum information theory. Functional analysis (in particular harmonic analysis of the boolean cube) plays a significant role in modern theoretical computer science. Researchers worldwide have been extending these techniques to the quantum case, using the more challenging non-commutative setting, Bristol, Cambridge, UCL. Recently, quantum information theory has led to a completely new direction to many body quantum systems. By applying quantum complexity theory, Hamiltonian complexity has been able to prove many exciting new results. Applying quantum Shannon theory to many-body quantum systems has proved to

14 be another very successful quantum-information-theory-inspired approach to this topic (Bristol, UCL).

C Main research groups in Mathematical Physics

The list below gives the composition of the main research groups in Mathematical Physics. The names have been obtained from a variety of sources including REF data and information from Heads of Departments. It includes researchers who have a substantial interest in areas of Mathematical Physics within the EPSRC remit (although many also have other research interests). There are 321 names below, but in counting for table 2, we have included only those who are primarily in the EPSRC Mathematical Sciences remit. In calculating mathematical physics as 10% of mathematical sciences, we only include those who work primarily in the mathematical physics part of the EPSRC Mathematical Sciences remit. It is not definitive and the criteria for inclusion have been interpreted differently by different departments. Despite our best efforts to make it complete we apologise in advance if it has omitted individuals or groups. Bath: Burstall, Mathies, Nordstrom, Zimmer. Bristol Dettmann, Grava, Keating, Linden, Maltsev, Marklov, Mezzadri, Montanaro, Muller, Robbins, Sawicki, Schubert, Sieber, Snaith, Tourigny, Wiesner. Brunel: Brody, Meier, Rodgers, Savin, Smolyarenko, Virmani. Cambridge (DAMTP): Adamo, Allanach, Barrow, Baumann, Challinor, Davis, N Dorey, Dunajski, Green, Hawking, Horgan, Jozsa, Kent, Manton, Matthews, Perry, Quevedo, Reall, Santos, Shellard, Skinner, Sperhake, Stuart, Thomas, Tong, Townsend, Williams, Wingate. (DPMMS): Batchelor, Dafermos, Grimmett, Luk, Miller, Mouhot, Cardiff: Behrend, Evans, Lechner, Logvinenko, Pugh. City: Alvarez, Castro Alvaredo, De Martino, Fring, He, Silvers, Stefanski. Coventry: Fytas, Kenna, Platini, Weigel. Durham: Abel, Bowcock, Donos, P Dorey, Dorigoni, Gregory, Heslop, Mansfield, Peeters, Piette, Ross, Smith, Sutcliffe, Taormina, van Rees, Ward, Zakrzewski, Zamaklar. Edinburgh: Blue, Braden, Figueroa-O’Farrill, Lucietti, Simon. Glasgow: Athorne, Feigin, Gilson, Korff, Nimmo, Strachan. Heriot-Watt: Doikou, Hollands, Johnston, Konechny, Saemann, Schroers, Szabo, Weston. Hertfordshire Young, Vicedo. Imperial (Maths): Barnett, Edwards, Graefe, Hewson, Holzegel, Pruessner, G West. (Theoretical Physics): Dowker, Gauntlett, Halliwell, Hanany, Hull, Stelle, Tseytlin, Waldram, Wisemen. Kent: Clarkson, Deano, Dunning, Hone, Hydon, Krusch, Manfield, Wang, Xenitidis. KCL: Cook, Doyon, Drukker, Gromov, Lambert, Martelli, Murthy, Papadopoulos, Recknagel, Rogers, Schafer-Nameki, Watts, P West. Lancaster: Belton, Braun, Elton, Grabowski, Hillier, Lazarev, Lindsay, Power, Leeds Caudrelier, Chalykh, Fordy, Harland, Martin, Mikhailov, Nijhoff, Ruijsenaars, Speight, Faraggi, Gorbahn, Gracey, Jack, Jones, Mohaupt, Rakow, Teubner, Vogts. Loughborough: Bartsch, Bolsinov, Ferapontov, Hallnas, Hunsicker, Lorinckzi, Mazzocco, Neishtadt, Novikov, Strohmaier, Veselov, Winn. Newcastle: Billam, Emary, Moss, Proukakis, Rigopoulos, Toms. Northumbria Angelova, Atkinson, Goussev Huard, Lombardo, Moro, Sommacal.

15 Nottingham: Adesso, JW Barrett, Fuentes, Gnutzmann, Guta, Krasnov, Louko, Oblezin, Ossipov, Sotiriou, Weinfurtner. Open University Broadhurst, Grimm. Oxford (Maths): Alday, Candelas, de la Ossa, Ekert, Hodges, Mason, Sparks, Tod. (Phys): Essler, Fendley, Lukas, Starinets, Wheater. (Comp Sci): Abramsky, J Barrett, Coecke. QMUL (Maths): Baule, Beck, Fyodorov, Goldshied, Just, Khoruzhenko, Klages, Majid, Prellberg, Valiente-Kroon. (Phys): Berman, Brandhuber, Rangoolam, Russo, Spence, Travaglini. Royal Holloway: Audenaert, Bolte, Essam, Kay, O’Mahoney, Schack. Sheffield: Ballai, Dolan, Jain, Ruderman, van de Bruck, Verth, Von Fay-Siebenburgen, Winstanley. Southampton (Maths): Andersson, Barack, Dias, Gundlach, Hawke, Howls, Jones, Lobo, Ruostekoski, Skenderis, Taylor, Vickers. (Phys): Drummond, Morris, O’Bannon. Strathclyde: Estrada, Grinfeld, Lamb. Surrey: Grant, Gutowski, McOrist, Pasquetti, Torrielli, Wolf. Sussex: Calmet, Melgaard, Tsagkarogiannis. Swansea: Kelbert, Neate. UEA: Proment, Salman. UCL: Boehmer, Halburd, Parnovski, Sobolev, Vassiliev. Warwick: Adams, Grosskinsky, Hairer, Kotecky, O’Connell, Ueltschi, Weber, Zaboronsky, York: Bostlemann, Busch, Colbeck, Corrigan, Daletskii, Fewster, Hawkins, Higuchi, Kay, MacKay, Rejzner, Sklyanin, Twarock, Weigart. (Comp Sci): Braunstein. Analysis of Staff by role and gender Although the above list is not definitive, we have used it to analyse the composition of the Mathematical Physics community by role and gender. The conclusions are as follows: Lecturers 21% of the total SL/Readers 27% of the total Professors 52% of the total Although the percentage of Professors is not out of line with Mathematics as a whole it is on the high side and is indicative of an age profile with fewer younger members of staff than one would expect in steady state. In terms of gender we find that 12% of the list above are female with 9% of the professors female. To put these figures in context the LMS report ‘Advancing women in mathematics: good practice in UK university departments’ (2013) records that 21% of HEI senior lecturers/lecturers and 6% of professors, in mathematical sciences in 2011 were female (excluding staff on teaching only contracts). Fellowships provide an excellent way of improving both the age profile and the proportion of females working in Mathematical Physics and it will be much harder to tackle these issue without them.

16