PROFESSOR ANGUS MACINTYRE FRS of the University of Edinburgh, is awarded the Polya Prize for his widely influential contributions to model theory and its applications.
Over the last thirty years he has found many new applications of model theory in algebra, geometry, number theory, asymptotics and theoretical computer science. His work established a theory of p-adic semi-algebraic sets, and has since been widely used in p-adic contexts, for example to prove a conjecture of Serre on the rationality of Poincaré series. His papers on totally transcendental fields and on algebraic groups created new paradigms for research in these areas. His work on the first-order content of Weil cohomology has opened up new possibilities for applying model theory in algebraic geometry.
DR TOM BRIDGELAND of the University of Edinburgh is awarded the Berwick Prize for the paper: 'Equivalences of triangulated categories and Fourier-Mukai transforms' published in the Bulletin of the London Mathematical Society Volume 31 (1999) pages 25 - 34.
This paper introduced new methods and solved an important question in algebraic geometry. Subsequently these pioneering methods have been used to illuminate concepts and to solve problems across a number of areas of mathematics.
In 1981, Shigeru Mukai established the duality result that the derived category of coherent sheaves on an abelian variety X and the one arising from the dual X- are the same. Mukai conjectured that such a duality should hold more generally, in particular between a K3X surface X and a certain moduli space of vector bundles over X. Tom Bridgeland's paper gave a simple proof of Mukai's conjecture. In a joint paper with A. King and M. Reid, the methods were used to understand the generalised McKay correspondence between the geometry of crepant resolutions of a quotient singularity C3/G and the representation theory of the finite group G. Another application of these ideas gave rise to Bridgeland's proof of the conjecture that flops (which are transformations between certain 3-dimensional varieties) induce an equivalence between the relevant derived categories. A consequence is that birational Calabi-Yau three-folds have equivalent derived categories. In recent work he has applied his methods to problems of interest to mathematical physicists by providing a precise setting to study stabilities related to Dirichlet branes.
DR PETER NEUMANN of Oxford University is awarded the Senior Whitehead Prize in recognition of his contribution to and influence on research into diverse branches of group theory, and for his broad-ranging service to British mathematics over many years.
Peter Neumann’s research interests have included varieties of groups, finite permutation groups of degree p or 3p (where p is a prime), a classification of groups with a cofinite Jordan set, infinite Jordan groups, automorphism groups of ordered sets, Frobenius groups, and recognition algorithms for matrix groups.
In all these endeavours he has been instrumental in designing research programmes and setting up teams of experts to work on them. A list of his past research students is impressive not only for its length, but for the high proportion which are now prominent mathematicians in their own right.
Neumann takes a keen interest in the history of mathematics and especially in the history of group theory. As with his mathematical lectures, his history talks on Burnside, Galois and others are beautifully constructed and a delight to attend.
His contribution to the health of mathematics in this country is wide-ranging. He was until recently President of the British Society for the History of Mathematics. He was Chairman of the British Mathematical Olympiad Committee from 1995 to 1997, and chaired the UK Mathematics Trust from its inception in 1996 till now. He has been a long-term stalwart of the LMS, and served as Publications Officer for six years. As generations of Oxford graduates will confirm, he takes a keen and inspirational interest in undergraduate teaching and was a natural choice to produce the report on the structure of undergraduate degrees which bears his name.
He receives this accolade in recognition of his extensive contribution to a great many aspects of our discipline.
DR NICHOLAS DOREY of the University of Wales Swansea is awarded a Whitehead prize for his contributions to mathematical physics, specifically to the understanding of non-perturbative effects in gauge field theories.
In a remarkable series of papers produced over seven years in collaboration with Hollowood, Khoze and Mattis, he has developed powerful methods for the computation of non-perturbative (instanton) effects in supersymmetric gauge field theories. With Khoze and Mattis, he has provided remarkable and highly influential calculations of multi-instanton effects in N=2 supersymmetric gauge theory.
Dorey’s work, often in collaboration, is characterised by the ability to find beautiful ways to perform exact computations. He has been one of the world leaders in the very significant developments in understanding of supersymmetric gauge theories that have taken place in recent years.
DR TOBY HALL of the University of Liverpool is awarded a Whitehead prize for his work on the dynamics of surface homeomorphisms. Hall has obtained some beautifully detailed and informative results, with the structure he has uncovered developing from, and extending, the one-dimensional theory, especially the famous Sarkovskii Theorem for interval maps, sometimes in unexpected ways.
Hall's results on forcing relations for periodic orbits of horseshoe type started with his thesis, and in recent years these results have been very considerably extended, with a reasonable conjecture for the full picture, in an extraordinarily rich study of horseshoe-like families, together with his collaborator, André de Carvalho.
The project continues to run apace, already represents an extremely important advance in the study of families of maps in low dimension, and is a quite startling example of what can be achieved from the topological viewpoint.
DR MARC LACKENBY of St Catherine's College and the University of Oxford is awarded a Whitehead prize for his contributions to three dimensional topology and to combinatorial group theory.
He has proved unexpected results about Dehn surgery, which is a much used method to construct a three-dimensional manifold M2 from another one M1, based on a knot K Ì M1 and a twisting coefficient p/q . One is a uniqueness result and the other provides a bound on the type of surgery that can give an ‘exceptional’ manifold.
He has found an algorithm enhancing Thurston's famous result giving the existence of hyperbolic structures on a large class of three dimensional manifolds. It allows one to calculate (up to explicit bounds) the volume of the (hyperbolic) complement of a class of knots. Another result is related to the famous 2π; theorem of Gromov and Thurston that a Dehn filling of a cusped hyperbolic manifold M3 along a curve of length more than 2π always gives rise to a negatively curved manifold. Lackenby has shown that if 2π is replaced by 6 then the fundamental group of the resulting manifold is Gromov hyperbolic. A consequence is that at most 12 manifolds obtained by surgery on a hyperbolic knot can have non-negatively curved fundamental group. Conversely, it is known that the figure eight knot has ten exceptional surgeries.
His recent work on the Heegaard genus of coverings has opened up new relations with other areas of mathematics and there are exciting possible consequences for combinatorial group theory.
DR MAXIM NAZAROV of the University of York is awarded a Whitehead prize . He is famous for his work on the covering group of the symmetric group. He constructed the representations of the covering group of the symmetric group, thus solving a problem which had been open for 75 years. His work also opened the door for the construction of the irreducible modular representations of the covering group.
Nazarov has furthermore constructed Young symmetrizers for the covering group and, more recently, for Brauer centralizer algebras. He presented some of this work in an Invited Lecture at the 2002 International Congress of Mathematicians.
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