Warwick Symposium on Algebraic Geometry 2007-08

Algebraic geometry studies the solution sets of systems of polynomial equations. These solution sets are called algebraic varieties, and are viewed as geometric locuses, generalising the circle and hyperbola of analytic geometry. Algebraic geometry is a mature subject, and the geometric points of view and the extensive toolbox it provides for studying varieties apply to a great many problem in mathematics and its applications.Algebraic curves occur as the locus f(x,y)=0 in the plane, where f is a polynomial function of x and y; in low degree (conics and cubics) one gets useful conclusions by explicit manipulations of the equation, but as the degree of f increases, the kind of conclusions one hopes for are more abstract, and necessarily involve more theoretical machinery. One eventually learns to stop worrying that the points of the curve are not parametrised in terms of anything more elementary, and to accept the curve as a primary object of nature, possibly complicated, but to be understood in its own terms and used in subsequent constructions.Rather than the degree, a better invariant of an algebraic curve is its genus, that is, the number of handles (donut-like holes) in its topological model. Especially important is the case division between the three cases g=0 (a sphere) or g=1 (a donut) or g>=2 (a surface with many handles); the case g=1 gives the elliptic curves, that played a key role in Wiles' proof of Fermat's last theorem. For algebraic curves or Riemann surfaces, this trichotomy was clearly perceived already in the 19th century, together with its interpretation in terms of positively curved, flat, or hyperbolic non-Euclidean geometry; the picture of the three cases g=0 or g=1 or g>=2 serves as an icon for the whole subject.The same trichotomy was a distant model for Mori theory or the classification of higher dimensional varieties, one of the most intensively developed area of algebraic geometry from the late 1970s; this work led to Mori's 1990 Fields medal. This classification is at present the subject of a major breakthrough, with the recent announcement of the proof of the minimal model program in all dimensions. The first component of the Warwick symposium will develop and disseminate these new result, and exploit its many applications.Algebraic varieties, the solution sets of simultaneous polynomial equations, provide examples and techniques in number theory and in theoretical physics, in algebra and singularity theory and in other branches of geometry. Even in analysis, which mostly deals in infinite dimensional spaces, the ultimate aim is frequently a reduction to a finite dimensional solution set modelled on algebraic geometry. The Warwick symposium will include components on each of these topics, together with applications of algebraic geometry to other areas of mathematics.