Warwick Symposium on Algebraic Geometry 2007-08

代数几何研究多项式方程组的解集。这些解集被称为代数变异，并被视为几何轨迹，推广了解析几何的圆和双曲线。代数几何是一门成熟的学科，它所提供的研究变量的几何观点和广泛的工具箱适用于数学中的许多问题及其应用。代数曲线出现在平面上的轨迹f(x,y)=0处，其中f是x，y的多项式函数；在低次方程（二次方程和三次方程）中，人们通过对方程的显式操作得到有用的结论，但随着f的程度增加，人们希望得到的结论更加抽象，并且必然涉及更多的理论机制。人们最终学会不再担心曲线上的点没有用任何更基本的东西来参数化，而接受曲线作为自然的主要对象，可能很复杂，但要用它自己的术语来理解，并在随后的构造中使用。而不是程度，代数曲线的一个更好的不变量是它的属，也就是说，它的拓扑模型中的手柄（甜甜圈状孔）的数量。特别重要的是g=0（球面）或g=1（甜甜圈）或g>=2（多柄曲面）这三种情况之间的情况划分；在g=1的情况下给出椭圆曲线，它在怀尔斯证明费马大定理的过程中发挥了关键作用。对于代数曲线或黎曼曲面，这种三分法早在19世纪就已经被清楚地认识到，连同它在正曲线、平面或双曲非欧几里得几何中的解释；三种情况g=0或g=1或g>=2的图片作为整个主题的图标。同样的三分法是Mori理论或高维变量分类的一个遥远的模型，这是1970年代后期代数几何中最集中发展的领域之一；这项工作使森喜朗获得了1990年菲尔兹奖。这种分类目前是一个重大突破的主题，最近宣布了所有维度的最小模型程序的证明。华威研讨会的第一部分将发展和传播这些新成果，并利用其许多应用。代数变量，即联立多项式方程的解集，为数论和理论物理、代数和奇点理论以及其他几何分支提供了例子和技术。即使在分析中，它主要处理无限维空间，最终目标往往是在代数几何模型上简化到有限维解集。华威研讨会将包括这些主题的组成部分，以及代数几何在其他数学领域的应用。

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.