WAG07-08, Case for support 1 Track record ============== Track record (2 pp.): boast about WAG82-83, WAG95-96 and Nute HDG 2002 and their outstanding contribution to world science and to UK alg geom. Also info about the main co-organisers. 1.1. Boasting about previous AG programs This is an application for a year-long symposium in algebraic geometry at Warwick, following the distinguished precedent set by the 3 previous Warwick symposia (1970-71, 1982-83 and 1996-97) and the Jan-Jun 2002 Newton Institute program in this subject (alongside meetings abroad, of which the Aug 1991 Utah summer school is very pertinent). Each of these events represented important landmarks in the subject itself, initiating new areas of research work that still reverberate around the world; they also had a major impact on the development of UK and European mathematics, bringing in new specialists from overseas to leading positions at UK universities, and giving many UK geometers additional scope for research collaboration. Three research areas initiated and developed at the 1996-97 Warwick symposium and at the 2002 Newton Institute program have since grown to major subjects of world-wide significance: (1) the explicit birational geometry of [CPR] (for which both Corti and Reid have been awarded separate LMS prizes); (2) the McKay correspondence (that led to the ground-breaking paper of Bridgeland, King and Reid [BKR]); (3) the higher dimensional minimal model program of Shokurov and Corti, the most significant new piece of algebraic geometry since Mori's work of the 1980s, and a main focus of HDG02 and WAG07-08. Although we highlight these success stories concerning internal areas of algebraic geometry, each of our activities also had numerous fruitful interactions with neighbouring areas of geometry, algebra and theoretical physics; this is abundantly clear, for example, from a brief glance at our Newton Institute report [HDG]. Our past programs were instrumental in raising the level of UK algebraic geometry and its profile in the world, and attracting new participants into UK mathematics from overseas, e.g.: Corti, Gavril Farkas, Mark Gross, Caucher Birkar, Andrew Kresch, Nikulin, Pukhlikov, Cheltsov, Wendland (and on-going job applicants). [K] J. Koll{\'a}r and others, Flips and abundance for threefolds (Salt Lake City 1991), Ast{\'e}risque 211 [CPR] A. Corti, A. Pukhlikov and M. Reid, Birationally rigid Fano hypersurfaces, in Explicit birational geometry of 3-folds, A. Corti and M. Reid (eds.), CUP 2000, 175--258 [C] A. Corti and others, Flips for 3-fold and 4-folds, book in press at OUP, 170 pp., online at http://www.ma.ic.ac.uk/~acorti/flips.html [BKR] T. Bridgeland, A. King and M. Reid, Mukai implies McKay: the McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc., 14 (2001), 535--554 [HDG] A. Corti, M. Gross and M. Reid, Report on the Newton Institute program "Higher Dimensional Complex Geometry", Feb--Jul 2002, online at http://www.newton.cam.ac.uk/reports/0102/hdg.html 1.2. Short blurb on each of the co-organisers Reid, Brown, Corti, Farkas, Schreyer, Siksek, Wendland Miles Reid (Warwick) was a founder of Mori theory in the 1980, and has continued to pioneer major new research developments, including the higher dimensional McKay correspondence of the 1990s and the application of the algebra of graded rings to explicit constructions in algebraic geometry. He was elected to the Royal Society in 2002, and won the LMS Senior Berwick prize in 2006 for his part in the joint paper [CPR]. He has been extremely successful at organising this type of activities, having led 2 previous Warwick Symposia and a Newton Institute program. Gavin Brown (Kent) studied with Reid in the 1990s, and has collaborated with him ever since on graded rings and applications to Mori flips and constructions of algebraic surfaces and 3-folds via commutative and computer algebra. His project `Algebraic geometry, graded rings and computer algebra' received the top grade `Internationally leading' from the EPSRC assessors, and the Graded Ring Database project that grew out of it extends 20 years of calculations of the graded rings over K3 surfaces and Fano 3-folds, and includes several other explicit lists. Brown developed much of the algebraic geometry in the computational algebra system Magma. He has organised several conferences in algebraic geometry and computational algebra in recent years. Alessio Corti (Imperial) studied with Catanese and Koll\'ar, becoming one of the major players in higher dimensional algebraic geometry. He established the Sarkisov program as a practical tool in explicit birational geometry, and was the first to explore seriously the boundaries of birational rigidity. He won the 2002 LMS Whitehead prize for this work. He led the seminar on higher dimensional flips at the 2002 Newton Institute program, and is the main author of the major book [C]. He is currently breaking new ground with Tom Coates in the study of quantum orbifold cohomology of Fano stacks. Gavril Farkas (Austin Texas and Warwick) In the 6 years since his PhD, Farkas has established himself as a world expert on M_g, the moduli space of curves of genus g; he has introduced locuses on M_g corresponding to Koszul cohomology, contradicting old conjectures and replacing them by a substantially new body of knowledge that links the birational geometry of M_g to topics of current advance in homological algebra; at 35, he is the youngest professor at Warwick since George Lusztig. Frank-Olaf Schreyer (Saarbruecken Faculty of Mathematics and Computer Science) is an expert on algebraic geometry and computer algebra. He studied at Brandeis with David Eisenbud. His syzygy algorithm and his work on Green's conjecture have played a considerable role in the development of computer algebra systems such as Macaulay2 and Singular. Alongside an important body of theoretical work in algebraic geometry, he has pioneered Monte Carlo methods involving varieties over finite fields for determining the existence and moduli of special classes of algebraic varieties. He has been an organiser of several Oberwolfach workshops. Samir Siksek (Warwick) is the first number theorist ever appointed at Warwick. He is a leading expert on the arithmetic of curves, Diophantine equations, and number theoretic algorithms; his algorithms for elliptic curves are an indispensible part of many computer algebra packages. Working with Bugeaud and Mignotte, he has recently solved several famous Diophantine problems that have vexed generations of number theorists, combining the modular methods behind Wiles' proof of Fermat's last theorem with Baker's bounds for linear forms in logarithms. His current work is shedding light on the Hasse principle for curves of higher genus. He has been invited to give a course on his own work at the Institut Henri Poincar\'e (Paris, November 2004), and at the Lorenz Centre (Leiden, May 2007). Katrin Wendland (Augsburg) did her Diploma in global analysis under Werner M\"uller in Bonn. She did her PhD in theoretical physics under Werner Nahm, also in Bonn, but her work is rigorous mathematics. She took part in UK algebraic geometry activities since her appointment at Warwick in 2002, and has recently been appointed to the Chair for Analysis and Geometry at Augsburg. Her expertise is in conformal field theory, string theory, and relations with algebraic and differential geometry. She has extensive experience of organising conferences, for example the 2005 EPSRC LMS Durham symposium on Geometry, Conformal Field Theory and String Theory, and two ABC-KLM workshops.