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# University of East Anglia

## UOA 20 - Pure Mathematics

### RA5a: Research environment and esteem

The Pure Mathematics Group at UEA has continued to develop and flourish during the assessment period. Its researchers in **Logic** (Dzamonja, Evans), **Number Theory and Dynamical Systems** (Everest, Stevens, Ward) and **Algebra and Combinatorics** (Lyle, Siemons, plus Category C staff Camina and Zalesski) are known internationally for the quality and originality of their work. The vitality of the group during the review period is confirmed by the publication by Category A staff of over 80 papers in high-quality journals, half with co-authors external to UEA. The Group has seen a dramatic rise in its research income and all Category A staff in post before 2007 have been PI on at least one large research grant (EPSRC, CEC, Leverhulme) since 2001. There have been two strong new appointments to lectureships in the review period: Dr Shaun Stevens in September 2002 and Dr Sinéad Lyle in January 2007, and their papers [Ste4] and [Lyl1] are among the high-points of this submission. The Group does not see its small size as a barrier to vigorous research activity. For example, Evans is the coordinator of Modnet, a Marie Curie Research Training Network with a total budget of 2.7 million Euros distributed amongst 14 partner institutions during 2005-08, and Dzamonja held an EPSRC 5-year Advanced Fellowship from February 2002 to May 2007.

**Research Structure:**

**Research activities and Achievements**

We describe highlights of research achievements in the three areas.

**Logic (Set Theory, Model Theory**):

A major theme in the work of **Dzamonja** has been the study of universal models, their appearance in model theory, the set theory behind the fact that their existence is often independent of ZFC, and applications to questions in other areas of mathematics. A universal member of a partially ordered class is one which dominates all others - a graph which embeds all graphs in a class or a topological space which maps onto all spaces in a class. The research has involved methods from fields such as functional analysis, measure theory, and mathematical logic. Dzamonja’s work on the subject has included the solution of a long-standing open problem in topology/functional analysis: that of when there can be a uniform Eberlein compact space mapping onto all others of the same weight. The solution was previously understood under the assumption of the generalised continuum hypothesis (GCH), but Dzamonja has recently solved the problem in circumstances when the GCH is not assumed [Dza1]. The paper [Dza4] has become a reference for constructing universal models by forcing. Another project was [Dza2] where the authors improved previous results of Woodin and Hauser about which large cardinals can fail the combinatorial principle Diamond. Recent work (submitted) marks the beginning of a long-term plan to solve the Maharam problem for finitely additive measures on Boolean algebras.

The interests of **Evans** are in model theory, applications of model theoretic methods (particularly those of geometric stability theory and its generalizations [Eva4]) in algebra, and aspects of infinite permutation groups connected with model theory (such as in [Eva3]). Much of his work in the assessment period has been on constructions exploring hierarchies of complexity of non-forking in stable and simple theories. The central problem is to understand the geometries arising from strongly minimal sets. This line of research was initiated by Zilber in the late 1970s when he proposed that such geometries were 1-based (‘vector-space like’) or involved an infinite field. This was shown not to be the case by some remarkable constructions invented by Hrushovski in 1988. These structures have a property called CM-triviality which limits the complexity of non-forking and it is a major open problem to decide whether there are strongly minimal sets which do not interpret an infinite field and are not CM-trivial. As a partial solution to this, Evans has produced [Eva2] a construction which gives stable structures not interpreting an infinite field in which non-forking is neither trivial nor CM-trivial: apart from non-abelian free groups (recent work of Sela and Pillay) these are the only known examples. A by-product of these constructions is a new viewpoint on the original Hrushovski constructions [Eva1], showing that they are reducts of ‘simpler’ structures.

There have been 5 PhD completions and 4 postdocs (Pourmahdian, 2000-01, EPSRC; Kamensky, 2006-07, CEC; Morgan, 1999-2001, Leverhulme; and Moore, 2000-01, EPSRC) in the Logic group since 2001.

**Number Theory and Dynamical Systems:**

The work of **Everest** in number theory has pushed ahead with primality and primitive divisor conjectures for elliptic divisibility sequences. In joint work [Eve3] with Miller (Princeton) and Stephens (London), then with Everest’s PhD student, King, many non-trivial cases of the main conjecture concerning prime appearance in elliptic divisibility sequences have been resolved. In joint work with **Ward** and PhD student McLaren, some highly effective bounds for the appearance of primitive divisors in these sequences for elliptic surfaces have been found [Eve1]. This work led Everest to become involved in efforts to resolve Hilbert’s Tenth Problem over the rational numbers using elliptic curves, and he has recently obtained a result furthering Poonen's work on this for large subrings of the rationals. In ergodic theory Everest (together with Ward and former EPSRC PDRA Einsiedler (Ohio)) has been instrumental in realising global and local canonical heights of rational points on elliptic curves as global and local entropies for sequential dynamical systems on the adeles [Eve4].

The main interest of **Stevens** is in the representation theory of *p*-adic groups. He has worked largely on construction problems for classical groups, when *p* is odd, and a major result here is that constructions found by Stevens give all supercuspidal representations for these groups [Ste1-4]. Joint work [JIMJ, to appear] with Sécherre (Marseille) shows that similar constructions give all supercuspidal representations of *p*-adic central simple algebras. There are also collaborations with: Paskunas (Bielefeld) on explicit Whittaker functions and epsilon factors of pairs for general linear groups [Amer. J. Math., to appear]; Goldberg (Purdue) and Kutzko (Iowa) constructing types and computing Hecke algebras for self-dual supercuspidals of the Siegel Levi subgroups in classical groups; and Broussous (Poitiers) on a concrete description of the building of classical groups. Much of this collaboration has been funded by grants from Nuffield and EPSRC (for a 3-year PDRA).

**Ward** has worked on rigidity problems in ergodic theory; in particular in [War3] with Einsiedler (Ohio) it is shown that many zero-dimensional algebraic dynamical systems are mutually disjoint, and in [War2] with Bhattacharya (TIFR, Bombay) finite entropy is shown to be the essential constraint for topological rigidity of higher-rank actions. In number theory Ward has continued to collaborate with Everest and others on arithmetic properties of integer sequences. Results here have included purely combinatorial constraints on dynamical zeta functions and finding highly effective Zsigmondy theorems for certain elliptic surfaces [Eve1]. In an EPSRC-funded project (PI Ward), Miles and Ward have shown that the (geometrical) expansive subdynamics studied with Einsiedler and Lind (Seattle) [War4] is visible in the (combinatorial) periodic point data for a large class of algebraic dynamical systems. A very detailed picture of orbit growth for the non-hyperbolic systems arising has been found [War1], and progress has been made on how the orbit growth changes in different directions. This project provides a good illustration of the group's collaborative energy: Ward proposed the programme and provided the algebraic dynamics machinery, Everest was able to push through the non-trivial analytic number theory problems that emerged, Stevens contributed the necessary ultra-metric analysis and combinatorics, and the post-doc Miles resolved some of the more subtle ring-theoretic problems that cropped up.

There have been 5 PhD completions and 7 postdocs (Stasinski, 2004-05; Sécherre, 2006; Minguez 2007; Miyauchi, 2007-08; Yayama, 2007; Miles, 2005-07; and Mahé, 2006-08; all EPSRC) in this area since 2001.

**Algebra and Combinatorics**:

**Lyle** uses combinatorial methods to study modular representations of finite dimensional algebras, in particular the representations of the symmetric groups and related algebras. She has worked extensively on the Hecke algebras of type *A*, where her work [Lyl1,2,4] helped to complete the classification of the irreducible Specht modules. In characteristic zero, she has shown that the LLT algorithm can be used to obtain explicit formulae for certain decomposition numbers by computing the composition factors of the Specht modules indexed by partitions with at most three parts. She has also worked with James (Imperial) and Mathas (Sydney) on the Rouquier blocks of the *q*-Schur algebras [Lyl2], generalizing results of Chuang / Kessar and Leclerc / Miyachi to describe the decomposition numbers. Together with Fayers (Queen Mary), Lyle has used constructive methods to study homomorphisms between Specht modules for the symmetric group algebras, and has extended these methods to the Hecke algebras of type *A*, proving an analogue of the Carter-Payne Theorem [Lyl1]. Other work involves the representations of the general linear groups in non-defining characteristic. This was used by Brandt, Dipper, James and Lyle to define a conjectured ‘standard basis’ for the Specht modules indexed by two-part partitions [Proc LMS, to appear]. Research with Mathas has focused on the representation theory of the Ariki-Koike algebras and the cyclotomic *q*-Schur algebras, where they have been able to describe the block structure [Lyl3].

**Siemons** has pursued his interests in three main areas: combinatorial reconstruction, permutation groups and representations, and simplicial and combinatorial geometries. Reconstruction problems are motivated by the need to retrieve information that is lost in transmission: such questions are of interest in combinatorics, but also appear in computer science and biology. Using a new invariant called the reconstruction index, Siemons has developed algebraic methods by which large classes of reconstruction problems can be analyzed uniformly, and many best-possible bounds on reconstructibility arise in this way [Sie1]. In permutation group theory, Siemons has shown, in [Sie3] with Zalesski (UEA), that all cyclic subgroups of a finite simple group have at least one regular orbit in any of its permutation representations. Other research includes the study of transitive groups whose proper subgroups are all intransitive, and work on representations of symmetric groups and plethysm. In his work on simplicial geometries Siemons has developed a new type of homology that arises only when coefficients come from a field of positive characteristic. This modular homology – also called rigid homology by Yoshiara - has features in common with ordinary homology, but is also quite distinct from it. While the homotopy properties of modular homology are poor, it behaves very well from the viewpoint of group representations. Interesting new representations akin to the Steinberg representations arise in this way for automorphism groups of the complex [Sie2,4]. By-products include new *p*-rank formulae for incidence matrices of shellable complexes and buildings [Sie2].

There have been 8 PhD completions and 2 postdocs (Maynard, 2001-05, Leverhulme; Mnukhin, 2005, Leverhulme) in this area since 2001.

**Mechanisms for promoting research**

The Pure Mathematics Group at UEA is part of the School of Mathematics. The School has 15.5 FTE faculty members (one joint with Environmental Sciences), all of whom are being returned in RAE 2008 (7 in UOA 20, 7 in UOA 21 and 1.5 in UOA 17). The School is part of the Faculty of Science; the other Schools being Biological Sciences (RAE 2001 grade 5), Chemistry and Pharmacy (grade 5), Computing Sciences (grade 4), and Environmental Sciences (grade 5**).

There is a regular research seminar programme in Pure Mathematics at UEA. For example in 2005 and 2006 there were 38 speakers external to UEA, 15 of whom were from outside the UK. Research groups also organise their own working seminars: examples of these in recent years are a working seminar on Logic, an internal Arithmetic seminar and a Permutation Groups and Combinatorics working seminar.

Visitors to the Group since 2001 who have stayed for at least one week have included: Bhattacharya (TIFR, Bombay), Cornelissen (Utrecht), Einsiedler (Ohio), Goldberg (Purdue), Ingram (Toronto), Kutzko (Iowa), Lalin (UBC), Larson (Gainesville), Paskunas (Bielefeld), Roettger (Iowa), Sharechevsky (Ohio), Shlapentokh (E. Carolina), Vaanaanen (Helsinki), van der Poorten (Macquarie).

Since 2001 the School has hosted two conferences in Pure Mathematics: a one-day Conference on Groups and Combinatorics, 16 June 2003, (LMS funded; 40 participants) and a Workshop on Pure Model Theory, 3-8 July 2005, (EPSRC/LMS funded; 80 participants). Research Networks involving members of the group include: MODNET, a Marie Curie Research Training Network in Model Theory and Applications (2005-2008), UEA is the coordinating partner; CAMELEON: an LMS-funded Network in Set Theory and Related Areas (2002-2006); Stevens is an ‘external’ member of AAG, a Marie Curie Research Training Network in Arithmetic Algebraic Geometry (2004-2007) and on the Steering Committee of the EPSRC-funded Network 'Representation Theory Across the Channel.'

**Research Infrastructure**

All faculty have individual offices with networked computing facilities supported by technicians within the Faculty. Graduate students have their own desk and computer in a shared office; postdocs, research-active retired faculty and visitors enjoy similar facilities. The School has a Common Room for staff and graduate students which houses its library of around 4000 graduate-level texts. All of this accommodation has recently been refurbished, funded by a SRIF3 grant. Further library and computer facilities are provided centrally, including access to MathSciNet, major journals, and computer packages such as GAP, Pari, Magma and Maple.

Research administrative support is provided at Faculty level in the areas of graduate recruitment and support, grant applications and research contract management. Further administrative support for research is available centrally in the areas of staff training and development.

**Research Students**

Research students are an integral part of the Pure Mathematics Group and 18 PhD’s were awarded during the reporting period. Almost all of these resulted in at least one individual or joint publication in a high-quality journal. Research students are initially registered for the MPhil degree and upgrading to PhD status is conditional on a progress report at the end of the first year, based on a dissertation and oral examination. All research students have either two co-supervisors or a first and second supervisor. Progress of each research student is reviewed annually by the School’s Research Committee with input from the student and the supervisors. Students participate in the weekly Research Seminars and working seminars, including giving talks. They are encouraged to attend advanced lecture courses within the School and take part in short courses, workshops and conferences elsewhere. For example, our students have recently attended LMS short courses on Algebraic Groups (Birmingham, 2004), Algebraic Topology (Swansea, 2005), Modnet summer schools and workshops on Model Theory and Applications (Leeds 2005, Freiburg 2006, Lyon 2006, Berlin 2007), and a Workshop on Diophantine Equations (Leiden, 2007). All research students participate in UEA’s ‘Transitions’ programme of personal and professional development. From October 2007, first and second-year students are attending courses at the EPSRC-funded London Taught Course Centre, funded by the School and by Roberts money.

**Arrangements for supporting interdisciplinary and collaborative research**

The School’s Strategic Research Fund regularly supports, or underwrites grant applications for, collaborative research visits and conference attendance. Appropriate joint PhD supervision and grant applications, and a culture of joint seminars, promote collaboration within the group. The seminar-programme of the School’s Centre for Interdisciplinary Mathematical Research provides opportunities for wider interactions. For example, it has highlighted links between Siemons’ work on reconstruction problems and work on phylogenetics being done in Computing Sciences.

**Relationship with research users**

As examples of interactions with research users, the general public and service to the mathematical community during the review period we mention the following. Dzamonja, Evans, Everest and Ward are members of the EPSRC Mathematics College; Camina (Category C) and Everest have visited the Heilbronn Institute (Bristol/GCHQ); Dzamonja is Secretary of the British Logic Colloquium; Siemons was on the Scientific Committee of PME 26, an international conference on Psychology of Mathematics Education, (600 participants, UEA, 2002); all Category A staff have given public lectures about Mathematics to audiences of local sixth-formers and the Mathematical Association.

**Staffing Policy**

**Developing and supporting staff in their research**

Research activity at UEA is supported through a system of sabbatical leave on full salary at the rate of one semester in seven. The School operates a system of allowances against teaching and administrative load for supervision of PhD students and postdocs. Early-career staff receive an additional allowance reducing their overall load by about one third. All staff are appraised every 2 years (every year for early-career staff). There is an annual ‘audit’ of research within the School overseen by the School’s Research Committee and this in turn reports to the Science Faculty Research Executive. Both School and Faculty routinely provide funds to support conference attendance and research visits.

**Early-career staff**

Early-career staff have a Mentor within the School and are enrolled on the University’s Postgraduate Certificate in Higher Education Practice. The Mentor is closely involved in evaluating progress and providing feedback and advice on lecturing, research and professional development. For example, Stevens attended a course on Development of Research Practice, part of which was to attend a mock assessment panel organised by EPSRC. Where appropriate, early-career staff have been co-investigator on a research grant, and acted as a co-supervisor of PhD students. All early-career staff within the School are strongly encouraged to apply at an appropriate time for a grant under the EPSRC New Lecturer scheme and have a very high success rate in this.

**Category C staff**

The two Category C staff are internationally-renowned active researchers who have retired from the Group in the assessment period. They regularly participate in research seminars and have office space and access to facilities within the School.

**Management of change of staff**

Since the last RAE, Camina and Zalesski have retired from the Pure Mathematics Group (in August 2001 and September 2004 respectively), though both remain part of the School of Mathematics as honorary/ emeritus professors. In replacing them, the School has adopted a strategy of securing leadership through internal promotions and making high quality new appointments of young lecturers. Ward and Evans were promoted to Chairs in 2004 and 2007; Dzamonja and Stevens to Readerships in 2002 and 2007. Ward was Head of School from 2002-07 and Evans is the current Head. Stevens was appointed in September 2002 and his interests in representation theory and number theory have enriched two research groups. The recent appointment of Lyle (January 2007) further strengthens Algebra and Combinatorics. These changes give the Group a healthy age profile and no retirements are anticipated in the 5 years after the end of the review period.

**Research Strategy**

Evidence of the Group's sustainability and vitality for the future is provided by its demonstrated ability to recruit and retain high quality staff with an international profile. There is a strong upward trend in research income and (crucially for sustainability) in the number of undergraduates admitted to the School.

**Future research plans:**

**Logic:**

In **Dzamonja**’s work an important theme will be to classify the Boolean algebras that support a finitely additive strictly positive measure. She wishes to produce a classification analogous to the classical theorem of Maharam for countably additive measures. Classifying the finitely additive measures seems delicate and it is likely to involve much set theory. The first difficult problem is to determine when an algebra supports a separable measure. Another theme will be the combinatorics of singular cardinals. Dzamonja already has a number of papers on this central topic of set theory which lies behind questions such as cardinal arithmetics and limitations of forcing. An instance is to understand the class of trees of size kappa with no kappa branches, when kappa is singular. As part of his project to construct new superstable structures which are not CM-trivial, **Evans** will try to understand further the connections between the Hrushovski constructions and other parts of mathematics: in particular the model theory of analytic functions and Zilber’s programme of providing a model-theoretic setting for non-commutative geometry. As a possible starting point for this, he has recently answered some questions of Zilber, showing how to expand the ‘fields with green points,’ obtained by Poizat via the Hrushovski constructions, by a ‘random’ N-th root function to obtain a simple theory with a probability measure on definable sets. Recent work of Hrushovski shows that there are interesting connections between Zilber’s ideas and and the study of finite covers. Evans has previously worked extensively on these, and intends to pursue the connections further.

**Number Theory and Dynamical Systems:**

Projects planned by the group include the following. Some of the mysteries of the growth behaviour of the non-hyperbolic maps studied suggest that there may be a way to recover a reasonable notion of meromorphic extensions for zeta functions and obtain powerful methods for orbit-counting by using an adelic zeta function. This is highly speculative and all we know is something about its desirable properties were it to exist. The theme of arithmetic of integer sequences, rational points on elliptic curves and iterates of rational maps will be pursued in various combinations. With Mahé (EPSRC PDRA) Everest and Stevens hope to prove the full primality conjecture for elliptic curves. Everest and Ward will try to push the effective results on primitive divisors to a larger collection of elliptic surfaces and to other sequences arising in a dynamical setting. **Everest** plans to work with Siksek (Warwick) to exploit methods arising from Wiles' proof of Fermat's Last Theorem. In particular they hope to make effective a recent generalisation of Siegel's Theorem obtained by Everest, Reynolds and Stevens. The full strength of the primality and primitive divisor conjecture for elliptic curves includes a uniformity statement which currently looks hard in full generality. Further collaboration with Ingram (Toronto) is anticipated as well as with other workers in elliptic transcendence theory to explore the possibilities for realising the primitive divisor conjecture. Collaboration is expected on the interface between Hilbert's 10th Problem and elliptic curves. **Stevens** will continue to work on the construction of types for *p*-adic groups: for the non-supercuspidal representations of classical groups; and for a general reductive group in the wild case. He also hopes to work on: genericity of the representations he has constructed, and explicit *L*-packets; making explicit the functorial correspondences between representations of different groups; and the constructions of *p*-adic representations of *p*-adic groups. In algebraic dynamical systems and ergodic theory, **Ward** will pursue some of the many open problems in measure rigidity: in particular the fundamental problem of joinings between pairs of such systems. In topological dynamics he hopes to extend understanding of topological rigidity and orbit-growth problems.

**Algebra and Combinatorics:**

The research of **Lyle** on finite dimensional algebras will focus on the Ariki-Koike algebras and the cyclotomic *q*-Schur algebras. One reason for the importance of these is that they simultaneously generalize the Hecke algebras of type *A* and type *B*. They have been shown to be cellular, and the study of homomorphisms between the cell modules gives a way of deriving information about their structure and composition factors. Lyle also plans to pursue her collaboration with Miyachi (Nagoya) which involves extending work of James on the general linear groups in order to construct modules for classical groups. **Siemons** will be active in all three of the areas outlined previously. Research on q-simplicial complexes will involve PhD students and other collaborators. In permutation group theory, work on plethysms of symmetric groups is ongoing and opportunities for collaboration with Lyle will arise. Research on reconstruction goes on in collaborative projects with Konstantinova and Levenshtein (Moscow).

**Esteem indicators**

** Dzamonja:**

- EPSRC Advanced Fellowship (February 2002 – May 2007)
- Selected paper for the 2006 Triennial Issue of Phil. Trans. of Royal Soc. A on Mathematics and Physics [Measure Recognition Problem, Philos. Trans. Royal Soc. A , vol. 364 (2006), 3171-3182]

**Evans:**

- Coordinator of MODNET, a Marie Curie Research Training Network in Model Theory and Applications with CEC FP6 funding of 2.7 million Euro, 2005- 2008.
- Organiser of an international meeting on Pure Model Theory at UEA, July 2005 (Satellite Meeting of a Newton Institute Programme; funded by EPSRC, LMS and Newton Institute).

**Everest:**

- Organiser of conference 'Elliptic Curves and Hilbert's 10th Problem', ICMS, Edinburgh, June 2007 (EPSRC and LMS funded).
- Heilbronn Lecture, Bristol University, January 2006.

**Lyle:**

- Sesqui Postdoctoral Fellowship at the University of Sydney, June 2004 - December 2006.
- In January 2007, awarded a 5-month Fellowship to participate in the programme ‘Representation Theory of Finite Groups and Related Topics’, MSRI, Berkeley, January-June 2008.

**Siemons:**

- PI on major research grant (2001-5) 'On the Reconstruction Index of Permutation Groups' Leverhulme Foundation (one 3-year PDRA and one 3-year PhD studentship)
- Invited lecture, EPSRC workshop on Modular Invariant Theory and Group Representations, Kent, 2005.

**Stevens:**

- PI on 125K EPSRC research grant (for 3-year PDRA)

- Invited talk at 'Representation Theory of p-adic Groups: A conference on the occasion of Phil Kutzko's 60th Birthday', Iowa City, USA, 2006.

**Ward:**

- Editorial adviser, London Mathematical Society journals, since 2001.
- Invited series of lectures at Austrian Science Fund National Research Network 'Summer School on Dynamical Systems and Number Theory', Graz 2007.

**Category C staff:**

**Camina:**

- Facilitator for LMS-EPSRC Short Instructional Courses

**Zalesski:**

- Leverhulme Emeritus Fellowship (May 2007 - April 2009)
- Member of organizing and scientific committees of an international conference in Algebra dedicated to the memory of Brian Hartley (Antalya Algebra Days IX, Antalya, Turkey, May 2007).
- Organiser of an international conference 'Locally Finite Lie Algebras' (Banff International Research Station, Canada, September 2003).

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