You are in: Submissions > Select institution > University of Nottingham > UOA 20 - Pure Mathematics > RA5a

# University of Nottingham

## UOA 20 - Pure Mathematics

### RA5a: Research environment and esteem

The RAE period has been highly successful and dynamic for the Unit, with existing strengths significantly enhanced and new focus areas vigorously developed. Among the highlights are:

- Strong appointments in each research group reflecting an overall growth in the number of permanent staff from 10 in 2001 to 15 currently;
- Creation of a world-leading research group in quadratic forms and areas related to Voevodsky's work, through appointments of a Professor, a Reader and a Lecturer. The creation of the first group in the UK in these internationally highly active areas of algebra addresses a shortcoming pinpointed in the 2004 "International Review of UK Research in Mathematics";
- Becoming a fund-holding node in two European RTN networks: GTEM (Galois theory and effective methods) and KTAGS (K-theory, algebraic groups and related structures);
- Organisation and hosting of four international conferences. In addition, colleagues have served on the scientific and organising committees of twelve major international conferences, some held at prestigious venues such as the Newton Institute, Oberwolfach and the Euler Institute;
- Publication of more than 145 peer-reviewed articles in international journals and proceedings, including Annals of Mathematics, Crelle, Inventiones, etc;
- Establishment of a new postgraduate training programme, enriched by participation in the forthcoming EPSRC-funded network MAGIC;
- Significant increases in doctoral degrees awarded compared to the previous RAE period (7 during 1996-2000, 28 since 2001), and in research income (£192K during 1996-2000, £689K since 2001)

Research in Pure Mathematics at Nottingham is carried out within the School of Mathematical Sciences (SMS), which consists of three Divisions: Pure Mathematics, Applied Mathematics and Statistics. The current Head of Pure Mathematics (**PM**) is **DW Hoffmann**.

Research activity in **PM** falls into three main areas:

**Number Theory and Arithmetic Geometry (NTAG)**:

Professor: **Fesenko**;

Lecturers: **Delbourgo**, **Diamantis**, **Trihan**, **Wuthrich**;

**Complex and Functional Analysis (CFA)**:

Professor: **Langley**;

Associate Professor: **Feinstein**;

Lecturers: **Winter**, **Zacharias**;

**Algebra and Discrete Mathematics (ADM)**:

Professor: **Hoffmann**;

Reader/Associate Professors: **Vishik**, **Woodall**,**Edjvet**;

Lecturers: **Ardakov** (Leverhulme Early Career Fellow), **Pumplün**.

**Research strategy and staffing activity since RAE 2001
**This has focused around building on existing strengths in number theory and analysis and the development of a strong algebra group. This strategy is best reflected in the staffing changes and the new appointments made since 2001.

**NTAG**saw the departures of Spiess (2003, to a Professorship in Germany) and Cremona (2007, to a Professorship in Warwick). These departures allowed an infusion of promising young talent with the appointments of lecturers

**Delbourgo**(2001, previously EPSRC Research Fellow at

**PM**),

**Diamantis**(2003),

**Wuthrich**and

**Trihan**(both 2007). These appointments have transformed

**NTAG**into one of the largest coherent research groups of its kind worldwide, producing internationally-leading research at the forefront of arithmetic geometry and analytic number theory.

In **CFA**, the focus has been on developing functional analysis. Following **Lindsay**'s departure in 2005 (to a Professorship in Lancaster), **Winter** joined **CFA** as lecturer in 2007, strengthening further Nottingham's leading role in the study of C*-algebras.

Following an externally informed review, a strategic initiative to create a research group in algebra resulted in the recruitment of **Hoffmann** and **Pumplün** in 2004, and **Vishik** in 2007, establishing **ADM** as an internationally-leading centre in quadratic forms and related areas. **Ardakov**'s appointment as Leverhulme Early Career Fellow in 2007 (to be followed by a lectureship from 2009) will foster links between algebra and number theory, in particular in Iwasawa theory.

In addition, five fixed-term lecturers covering staff on research leave have enriched the unit's research activities. All five continue their academic career as professor (Berhuy in Grenoble, from 2008 on), associate professor (Mattarei in Trento, Italy), or on temporary posts (Astier in Konstanz, Germany; Rosehr in Würzburg, Germany; Zsak in Leeds).

**Research support and staffing policies
**The School and the University provide extensive support for research in pure mathematics. Within the SMS, the Head of School holds a devolved budget, from which research funds for conferences, seminars and visitors are allocated to Divisions on a per capita basis; additional School funding is made available to individuals competitively. Staff are routinely successful in securing external travel support. Divisions can also bid against strategic funds to support larger-scale initiatives. Research support within the SMS is provided by thirteen support staff including three computing officers. A mathematically trained Research Facilitator has been appointed (starting 1/1/2008) to provide staff with support in the preparation and management of research grants. The School’s Research Committee ensures that promising funding calls and collaborative opportunities are identified and disseminated, and areas for strategic investment are prioritised. At University level, Research Innovation Services provide comprehensive research management support.

The University has strongly supported the development of mathematical sciences in Nottingham, enabling the SMS to establish fifteen new permanent posts since RAE 2001, including three Chairs. In **PM**, this investment is reflected by an increase from 10 permanent staff in 2001 to 15 currently.

Early career permanent staff in **PM** are offered research support through lighter teaching loads (at most four contact hours of teaching and tutorials per week) and limited administrative responsibilities. All newly appointed academic staff are assigned experienced mentors from within **PM**. The University offers staff training in generic research skills through its Staff & Educational Development Unit. Workloads in SMS and **PM** are distributed so as to protect successful researchers from demanding teaching or administrative loads. As a research-led institution, annual promotion and performance review processes reward research success based purely on merit. The review period saw one promotion from Reader to Professor (**Langley**) and two from Lecturer to Associate Professor (**Edjvet, Feinstein**).

Staff in SMS work collegially to enable their colleagues to attend conferences and seminars during term-time. The SMS operates a flexible research leave scheme, which (with School, University and external funding) relieves staff from all teaching and administrative responsibilities for a semester or more. This scheme enabled five members of **PM** to have semester-long sabbaticals during the review period.

**Research Infrastructure and Facilities
**The School ensures that staff and visitors in SMS are provided with full office facilities, including high-specification PCs and networking facilities, and secretarial and administrative support. Nottingham provides outstanding library facilities with access to the full range of relevant print and online journals in pure mathematics. There are also excellent audiovisual (including Access Grid) and high-performance computing facilities (a 1024-processor GRID facility, the second fastest academic machine in the EU at the time of procurement, with a satellite 10-processor clone dedicated to SMS; these are regularly used by

**PM**'s staff working in computational group theory and number theory).

**Management, training and supervision of research students
**The SMS offers 3, 3.5 or 4-year postgraduate degree programmes, depending on the individual training needs of each student and the relevant funding source. The School invests heavily in studentships to ensure the vitality of the research student body, and flexible recruitment policies enable the School to attract many outstanding postgraduates. Postgraduate and RA training is overseen by the School’s Research Committee. Research students are provided with shared office facilities, their own PC and full access to library, computing and secretarial facilities.

Each student’s personal training programme in **PM** is delivered through graduate-level courses (exploiting the School’s MMath programme and recently enhanced by the MAGIC programme), supplemented with workshop and project activities, and is formally assessed through a credit system. Monitoring of research student progression operates within strict quality assurance guidelines, involving regular recorded supervisory meetings, end-of-year assessments including a written report and an oral examination by independent assessors, and a compulsory formal research presentation.

The University’s Graduate School provides a comprehensive range of training courses covering all the requirements set out by the UK Research Councils in their Joint Skills Statement. More than 80 courses, designed in partnership with Schools, help develop discipline-specific and transferable skills; career development advice is also available. The Graduate School works with the SMS in securing studentship funding from research councils, for example through interdisciplinary doctoral training accounts, CASE studentships (one in **PM**), and from other agencies (such as Overseas Research Student Awards). In addition to travel and conference support from SMS, the Graduate School offers a programme of travel awards (including a Universitas 21 prize scheme).

**PM** also provides its own research-specific training in the form of semester- or year-long study groups comprising postgraduates, postdocs, visiting and permanent staff and guest speakers, on topics of current research interest. Since 2001, study groups in **NTAG** were held on, for example, Stark's and the Bloch-Kato conjectures, Heegner points, and p-adic and Hilbert modular forms, and in **ADM** on the Thompson group and on intersection theory and Chow groups. Frequent lecture series for postgraduates in discrete mathematics were offered. **CFA** has held study groups on Banach algebras and regularly organises postgraduate seminars on various topics, in part jointly with members of the Statistics Division.

**PM** attracts promising and highly qualified postgraduate candidates through various activities such as advertising (nationally and abroad), visits to and lectures at other institutions specifically for that purpose, for example at the Prospects in Mathematics symposium at Durham, and participation in postgraduate open days. The quality of **PM**'s postgraduates is reflected in the fact that of those students that have graduated since 2001, nine have continued in an academic career: five hold postdocs or lectureships abroad (Alotaibi, Paskunas, Popescu, Serbest, Taylor), two hold EPSRC postdoctoral fellowships (Skalski, Stasinski), one is a Heilbronn Fellow (Waters), and one holds a permanent lectureship in Cambridge (Birkar).

**International visitors and PDRAs
**Members of

**PM**maintain numerous international contacts. The number of postdoctoral and senior visitors has increased significantly since 2001, funded mainly through grants gained by members of

**PM**from the LMS, EPSRC, Royal Society, INTAS and European RTN networks.

Since 2001, three postdocs joined **NTAG** for 22 months combined, funded by the RTN network GTEM. Howson (Adams Prize winner in 2002) joined **NTAG** in 2001-2003 as Dorothy Hodgkin Fellow, and Breuning for 12 months in 2004-2005 as EPSRC Postdoctoral Fellow. **Fesenko**'s EPSRC grant supported six postdocs for a combined 39 months, and another two-year EPSRC postdoc is currently working with **Diamantis**. Among the many senior research visitors of **NTAG** staying for a longer term (at least 2 weeks), three have worked with **Diamantis** (EPSRC, LMS funding), and **Fesenko**'s numerous visitors include Kato, Kazhdan and Saito (mainly funded by EPSRC).

Two postdocs in **ADM** stayed for a combined 23 months funded by the RTN network KTAGS. Longer-term senior research visitors included Juhasz working with **Edjvet**, and Sivatski working with **Hoffmann** (Royal Society).

In **CFA**, EPSRC grants have supported one postdoc working with **Zacharias** (12 months), and one currently working with **Langley** (two years). **Langley** also supervises a DFG-funded two-year postdoc. Long-term senior research visitors include Kamowitz working with **Feinstein** (EPSRC) and Drasin working with **Langley** (EPSRC).

**Research seminars, meetings and conferences at PM
**All staff and postgraduates in

**PM**participate in weekly Pure Mathematics Seminars, where guest speakers give colloquium-style talks to a general pure mathematics audience. In addition, there are frequent specialised seminars in number theory, analysis and algebra featuring invited speakers, research visitors and local speakers, often attracting participants from neighbouring institutions. The analysis seminar continues to involve members from Mathematical Physics within the School with strong analytical interests. These seminars are supported from

**PM**'s devolved budget.

Four international workshops and conferences have been organised by members of **PM**:

- Workshop “Geometric Representation Theory and Number Theory” (organiser:
**Fesenko**), 2001; - LMS Midlands Regional Meeting and 3-day workshop on quadratic forms and related topics (organisers:
**Cremona, Hoffmann**), 2004. Invited speakers included Bayer-Fluckiger, Colliot-Thélène and Kahn; - Conference “Quadratic Forms, Linear Algebraic Groups and Related Topics” (organisers:
**Hoffmann, Pumplün**), 2005, funded by RTN network KTAGS, EPSRC, LMS. Speakers included Panin, Rost, Serre and Totaro; - Symposium “Quantum Probability, Information and Control” (co-organiser:
**Zacharias**), 2006, funded by RTN network "Quantum Probability with Applications to Physics, Information Theory and Biology", LMS, EPSRC.

Other activities hosted at **PM** include two one-day group theory conferences of the North Eastern Geometric Group Theory Seminar (organiser: **Edjvet**; 2004, 2006), and a special seminar series on higher arithmetic in 2003-2004 (organiser: **Fesenko**), with speakers including Kato, Kazhdan and Saito.

**Research achievements in Number Theory and Arithmetic Geometry
NTAG** has been involved in a variety of networks.

**NTAG**has been a fund-holding node in the RTN network GTEM (2000-2004) and its successor (since 2006).

**NTAG**also participates in the Durham node of the RTN network Arithmetic Algebraic Geometry (2004-2008) and its predecessor. Nationally,

**NTAG**has participated in the LMS Scheme 3 networks SECANTS (South of England Computational and Algorithmic Number Theory Seminars, 1995-2005) and NOMADS (North of England Algebraic Number Theory Group, since 2000), with several meetings having been held in Nottingham.

**NTAG** research covers a wide range of problems lying at the forefront of arithmetic geometry and analytic number theory, specifically the modern theory of automorphic forms and the major themes in the arithmetic of elliptic curves, including the main conjectures about them, using advanced methods such as Iwasawa theory and the new two-dimensional adelic analysis programme.

**Delbourgo** has pioneered the study of Euler characteristics over universal deformation rings. His theory deals with the deformation theory of Kato's zeta elements and related developments on the leading terms of algebraic L-functions. He introduced a new theory of height pairings over this deformation ring, and the constructed regulators appear in the dominant term of the improved p-adic L-functions. **Delbourgo** bounded the order of the Selmer group associated to a weight deformation via the study of Selmer families, and he discovered new nonabelian congruences between L-series. Most of his important results will form a monograph accepted for publication by CUP.

**Diamantis** has established himself as an internationally leading expert in the new theory of higher-order automorphic forms. He proved major results concerning the structure of spaces of higher-order forms, including explicit bases and dimensions (with Chinta, O'Sullivan, Sim and Sreekantan); functional equations and converse theorems for L-functions of higher-order modular forms (with Knopp, Mason and O'Sullivan); a geometric interpretation of higher-order forms through Chen's iterated integrals and mixed Hodge structures (with Sreekantan); and the Rankin-Cohen structure on the space of higher-order forms (with Choie). **Diamantis** also works on applications of higher-order forms on values of derivatives of L-functions of standard cusp forms towards the Beilinson conjectures.

**Fesenko** has started a major programme of two-dimensional adelic analysis which studies the zeta functions of elliptic curves over global fields using new geometric and analytic methods based on his recent generalized Haar measure and harmonic analysis on higher dimensional local fields. This programme includes a new study (different from Wiles's approach) of the functional equation and meromorphic continuation of the zeta function, and applications towards the Riemann hypothesis in dimension 2 and the Birch-Swinnerton-Dyer conjecture. His work is funded by two EPSRC research grants worth £524K in total, supporting visits of research assistants and of world-renowned collaborators such as Gaitsgory, Ihara, Kato, Kazhdan, Kim.

**Trihan** is one of the leading experts in p-adic cohomology theories and their applications to arithmetic. He proved the finiteness of p-adic cohomology and potential semistability of coefficients in the curve case. He studies relations between analytic and arithmetic properties of L-functions as in the Birch and Swinnerton-Dyer conjecture. In a joint paper with Kato (2003), they reduced this conjecture to the finiteness of the Tate-Shafarevich group in the geometric case. He actively works together with several leading Japanese mathematicians on equivariant versions of the noncommutative geometric Iwasawa main conjecture.

**Wuthrich** works on elliptic curves and Iwasawa theory. Since his PhD in 2004, he extended Kato's theorem on the main conjecture to curves admitting an isogeny of degree p over the rationals. He established important new properties of the p-adic analogue of the L-series of an elliptic curve E over the rationals for curves with good ordinary reduction at p. In collaboration with Stein, he implements methods from Iwasawa theory to obtain a powerful algorithm to compute upper bounds on the order of the p-primary part of the Tate-Shafarevich group, independent of the arithmetic rank of E. Partly in collaboration with Delauney, **Wuthrich** has been working on the theory of self-points with potentially many significant applications.

**Research achievements in Complex and Functional Analysis
**Regional activities with involvement of

**CFA**members include a joint Nottingham-Loughborough seminar on integrable systems and complex analysis, co-organised by

**Langley**(2003-2006), as well as participation in the North British Functional Analysis Seminar (NBFAS).

**Zacharias**has also participated in the European RTN network Quantum Probability (2003-2006).

The research interests of **CFA** cover complex analysis (meromorphic functions, value distribution, Nevanlinna theory) and functional analysis (Banach algebras, normed algebras, operator algebras, K-theory, C*-algebras). There have been regular interactions with the Mathematical Physics group in the Applied Mathematics Division.

**Langley** works in complex analysis. The most prominent among his papers published since 2001 is that with Bergweiler and Eremenko completing Wiman's 1911 conjecture on the zeros of real entire functions and their second derivatives. **Langley** extended this to higher derivatives which, combined with another theorem of Bergweiler-Eremenko, proves Polya's 1943 conjecture. **Langley** proved the strongest result towards Gol'dberg's conjecture that the frequency of poles is controlled by zeros of the second derivative, and a 1976 conjecture about entire functions integer-valued at the same points. Other joint work includes equilibrium points of potentials in the plane and space and a question of Douady concerning Newton's method.

**Feinstein** works in the area of Banach algebras. Among his most important contributions since 2001 are a counterexample to Morris's weak amenability conjecture (subsequently strengthened with Heath), characterizations (with Kamowitz) of (quasi-)compact endomorphisms of Banach function algebras and a characterization (with Kaniuth and Somerset) of spectral synthesis via closed ideals. Other significant work includes: (with Oliver) results on extensions of endomorphisms and characterization of algebraic closedness; (with Heath) compact derivations and compact cohomology; (with Bland and Dales) completeness, completions and approximation for algebras of differentiable functions; (with Dawson and Dales) topological stable rank (introduced in connection with K-theory of C*-algebras by Rieffel, and linked to **Winter**'s work).

**Winter** works mainly on Elliott's programme of classifying nuclear C*-algebras via K-theory. While Kirchberg-Phillips theory covers the simple purely infinite case, the simple finite case is incomplete. **Winter** (with Kirchberg and Ng) introduced the well-behaved decomposition rank, particularly relevant for approximately subhomogeneous algebras, and C*-algebras given by minimal diffeomorphisms. **Winter** (with Toms, Hirshberg, Hirshberg and Rordam) introduced strongly self-absorbing C*-algebras and D-stability, giving a unified treatment of the finite and purely infinite cases of the Elliott conjecture. Replacing the Cuntz algebra by the Jiang-Su algebra Z, this leads to classification up to Z-stability, and **Winter** proved the Elliott conjecture for certain Z-stable C*-algebras.

**Zacharias** works on generalised Cuntz algebras, K-theory and approximation properties for C*-algebras, with links to **Winter**'s work. **Zacharias** studied Tsirelson's product systems, obtaining important results on pure infiniteness of continuous analogues of Cuntz algebras (joint with Hirshberg). With Popescu and Skalski, **Zacharias** worked on higher rank generalisations of Cuntz algebras, KK-theory and noncommutative entropy. With Bhat and Dey, **Zacharias** applied Cuntz type algebras to dilation theory of operator tuples, obtaining even more general results with Skalski. **Zacharias** also worked on approximation properties of Pimsner type algebras and noncommutative entropy, and he partially solved a problem by Roe on approximation properties of groups concerning invariant translation.

**Research achievements in Algebra and Discrete Mathematics
**Nottingham has been a fund-holding node in the RTN network KTAGS following

**Hoffmann**and

**Pumplün**'s appointment in 2004.

**Woodall**participated in an INTAS project on graph colourings (1999-2002). Regional activities include participation in the North Eastern Geometric Group Theory Seminar (LMS Scheme 3, since 2003).

Research in **ADM** covers combinatorial and geometric group theory, quadratic and higher degree forms, motives, composition algebras, Milnor K-theory, algebraic Iwasawa theory, and graph theory and combinatorics.

Research in graph theory and combinatorics is undertaken by **Woodall**, mainly on graph colourings. The chromatic number and choosability (list chromatic number) are conjectured to be the same for certain classes of graphs. **Woodall** proved various cases of these conjectures in several long papers. He also found good upper bounds for the choosability when it cannot be determined exactly. **Woodall** obtained a major improvement in the best lower bound known for the average degree of a graph that is edge-minimal with respect to edge colouring. Other work during the RAE period concerns hypergraphs, the Tutte polynomial, and sufficient conditions for the existence of certain configurations in graphs.

Research in group theory is undertaken by **Edjvet**. Since 2001, he achieved significant progress in equations over groups, proving groups infinite and cyclic presentations. With Juhasz, **Edjvet** has proven many types of equations solvable. Using involved curvature arguments, he calculated the second homotopy module for generalised tetrahedron groups (joint with Howie, Rosenberg and Thomas) and for the Coxeter group G(m,n,p) (joint with Juhasz), leading to a near classification of all finite examples (an open problem from the 1940s). Other work includes building up a picture of when cyclically presented groups are trivial, joint work with Cremona involving number theory, and with Howie where embedding theorems were obtained.

Composition algebras are studied by **Pumplün** and **Hoffmann**. **Pumplün** made important contributions to the following aspects of composition algebras: tensor products (joint with Morandi and Pèrez-Izquierdo); involutions; zeros of polynomials (joint with Walcher); new Cayley-Dickson doublings; and new constructions of Albert algebras over rings and varieties. **Pumplün** classified vector bundles and bilinear forms over certain curves, thus classifying quaternion algebras over such curves, and she obtained new results on nonassociative quaternion algebras over rings (joint with Astier) and on (hermitian) levels of composition algebras (partly joint with Unger). **Hoffmann** answered a 1989 question by Lewis and Leep on levels of quaternion algebras.

Research in the algebraic theory of quadratic forms is the main focus of **Hoffmann**'s and **Vishik**'s work. **Hoffmann** (joint with Laghribi) developed the most comprehensive theory of function fields of quadrics in characteristic 2, generalizing many results from characteristic not 2. **Hoffmann** determined the splitting behaviour of bilinear forms in characteristic 2 for most algebraic field extensions. Joint with Becher, he solved a 1979 question by Gross on isotropy of infinite-dimensional quadratic spaces.

**Vishik** is the foremost expert in the geometric study of quadrics developed initially for the proof of the Milnor Conjecture. This includes joint work with Orlov and Voevodsky in which Knebusch's degree conjecture from 1976 was also proved. **Vishik** invented the powerful notion of discrete invariants of quadrics leading to the determination of the canonical dimension of orthogonal groups. **Vishik** also constructed new symmetric cohomological operations in algebraic cobordism and uses them to compute discrete invariants, and to study rationality of cycles. A spectacular application is **Vishik**'s uniform construction of fields of all known and many new values for the u-invariant, vastly generalising earlier results by Merkurjev and Izhboldin.

Higher degree forms are studied by **Pumplün**, who comprehensively investigated algebraic properties of such forms: indecomposability; multiplicativity; composition; norm principles (joint with Fitzgerald), and by **Hoffmann**, who developed a theory of forms of degree p in characteristic p, with applications to differential forms and Kato's cohomology. **Hoffmann** also studied symbol length in Milnor K-theory (partly joint with Becher), and, joint with Garibaldi, he answered affirmatively Totaro's question on the existence of closed étale points for certain G-torsors.

**Ardakov**'s research focuses on noncommutative Iwasawa theory, in particular the algebraic structure of Iwasawa algebras. In the Iwasawa Main Conjecture by Coates, Fukaya, Kato, Sujatha and Venjakob, characteristic elements play an important role. **Ardakov** (joint with Wadsley) contributed greatly to their understanding by computing the characteristic element of an arbitrary p-torsion Iwasawa module. Other significant contributions by **Ardakov** include: proof that the so-called Harris ideal is zero, thus solving a problem from 1980 arising from Harris's work; and determination of general bounds for the Krull dimension of Iwasawa algebras, thus answering a 1982 question by Brown, Hajarnavis and MacEacharn.

**Outlook
**Prospects for the next five years are excellent. The appointments in 2007 of five highly talented young staff across all three research groups will further focus developments in directions of great current interest (elliptic curves in number theory, Iwasawa theory in both algebra and number theory, quadratic forms and motivic theories in algebra, C*-algebras in analysis). The algebra group is now an excellent team of talented people working in coherent areas and well positioned to make influential contributions. The group's visitor programme will be expanded, partly through grants and international projects (some already in place such as Royal Society International Joint Projects with Russia and Chile), a proposal for participation in a future European network is currently being discussed with partners, and another major conference on quadratic forms is planned. Common research interests of

**NTAG**and

**CFA**in the internationally highly active area of noncommutative geometry are fostered by

**Winter**'s appointment in analysis, and further development of that research area is planned through cross-disciplinary study groups.

**Ardakov, Trihan**and

**Wuthrich**are all experts in Iwasawa theory, approaching it from different viewpoints. Their appointments will thus generate promising synergies between

**NTAG**and

**ADM**.

**Fesenko**'s recent EPSRC grant (£384K) will guarantee a steady stream of visits by world-leading scientists collaborating on some of the most timely and important problems in number theory. The Division is now exceptionally well placed to build on its strengths across a range of important research areas.

**Esteem Indicators**

**Ardakov**:

- Leverhulme Early Career Fellowship (since 2007)
- Invitation and lecture, workshop "Pro-p Extensions of Global Fields and pro-p Groups", Oberwolfach (2006)
- Invited speaker, conference "Trends in Noncommutative Geometry", Newton Institute, Cambridge (2006)

**Delbourgo**:

- Invited speaker, Congrès Iwasawa, Besançon, France (2004)
- Invitation and lecture, Oberwolfach Arbeitsgemeinschaft Arithmetic Geometry (2007)

**Diamantis:**

- Invitation, Oberwolfach workshop “Riemann Zeta and Allied Functions” (2004)
- Invited speaker, Conference on Modular Forms in honour of Marvin Knopp, Philadelphia (2006)

**Edjvet**:

- Invited Speaker, Group Theory Conference Trento, Italy (2001)

**Feinstein**:

- Professeur invité, Université de Metz, France, one month (2004)
- Plenary speaker, International Conferences on Function Spaces, Edwardsville, Illinois (2002, 2006)

**Fesenko:**

- Documenta Mathematica Special Volumes Editor (2003, dedicated to Kato and 2006, dedicated to Coates)
- Organiser (with Deninger, Kurokawa, Manin, Parshin, Vostokov) of regular conferences on arithmetic geometry at the Euler Institute, St Petersburg (2004, 2005, 2007)
- Two EPSRC standard grants (£524K in total) supporting his programme on adelic analysis on arithmetic schemes and its applications.
- Organiser (with Cremona, M Taylor), Symposium "Noncommutative Aspects of Number Theory", Durham (2003)

**Hoffmann**:

- Organiser (with Tignol, Merkurjev), Oberwolfach workshops “Quadratic Forms” (2002, 2006)
- 5-part invited lecture series, international Minicourse “Quadratic Forms”, Lens, France (2006)
- Invitation and lecture, Banff International Research Station conferences "Quadratic Forms and Algebraic Groups" (2003, 2006)
- Plenary speaker, conference "Linear algebraic groups and cohomology", Atlanta, USA (2007)

**Howson**:

- Adams Prize (2002)

**Langley**:

- Plenary lecture, 19th Nevanlinna Colloquium, Finland (2003)
- Invitation to Oberwolfach workshops on complex functions (2001, 2004, 2007)
- 2-month research stay at University of Kiel, Germany, on a Humboldt Foundation award (2005)

**Pumplün**:

- Research Fellowship from Deutsche Forschungsgemeinschaft for an 11-month research stay at University of Trento, Italy (2005-2006)
- Plenary speaker, Workshop “K-Theory and Algebraic Groups”, Dublin (2004)

**Trihan**:

- Research fellowship, Japan Society for the Promotion of Science, University of Tokyo (2001-2002)
- Invited speaker at the "Cohomological Week" in honour of P. Berthelot, Rennes (2003)

**Vishik**:

- Invited 3-part lecture series, conference “From Quadratic Forms to Algebraic Groups”, Ascona, Switzerland (2007)
- Member of Institute for Advanced Study, Princeton, 9 months (2004-2005)
- Visiting professor, IHES, Bures-sur-Yvette, France, 3 months (2005)
- Invitation and lecture, Banff International Research Station conferences "Quadratic Forms and Algebraic Groups" (2003, 2006)

**Winter**:

- Invited speaker, Oberwolfach workshops on C*-algebras (2001, 2003)
- Invited speaker, workshop “The Structure of Amenable Systems”, Banff, Canada (2004)
- Invited speaker, conference “Operator Algebras and Applications”, Odense, Denmark (2006)

**Wuthrich**:

- 1-year advanced research scholarship from the Swiss National Science Foundation funding visits at Keio University, Tokyo, and McGill University, Montreal (2006-2007)
- Invited speaker, conference “Algebraic Number Theory”, Kyoto (2006)
- Schläfli Prize of the Swiss Academy of Sciences for outstanding achievement by a young Swiss researcher (2007)

**Woodall**:

- Nash-Williams Memorial Lecture, 18th British Combinatorial Conference, Brighton (2001)
- Council Member, Institute of Combinatorics and its Applications (Winnipeg, Canada) (2004-2007)
- Editorial Board Member, “Journal of Graph Theory” (2001-2007)

**Zacharias**:

- Invitation to Fields Institute Program on Operator Algebras to co-organise its workshop “Noncommutative Dynamics and Applications” (2007)
- Plenary lecture, Joint AMS-IMS-SIAM Summer Research Conference “Advances in Quantum Dynamics”, Mount Holyoke, Massachusetts (2002)