Warwick Symposium on Algebraic Geometry 2007-08

沃里克代数几何研讨会 2007-08

基本信息

批准号：
EP/E060382/1
负责人：
Miles Reid
金额：
$ 20.97万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE060382%2F1
关键词：
Warwick Symposium Algebraic Geometry 2007

项目摘要

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.

代数几何学研究多项式方程系统的解决方案集。这些溶液集称为代数品种，被视为几何源，从而概括了分析几何形状的圆和双曲线。代数几何形状是一个成熟的主题，其几何学观点及其提供的广泛的工具箱用于研究品种适用于数学及其应用中的许多问题。代数曲线发生在座位f（x，y）= 0中的位置曲线。平面，其中f是x和y的多项式函数；在低度（圆锥形和立方体）中，通过明确的方程式操纵获得了有用的结论，但是随着F的程度的增加，人们希望得出的结论更加抽象，并且必然涉及更多的理论机制。人们最终学会了不再担心曲线的点并没有用更多的基本来参数，而要接受曲线作为自然的主要对象，可能很复杂，而是用自己的术语来理解并在后续结构中使用等级是代数曲线的更好不变的是其属的属，即其拓扑模型中的手柄数（甜甜圈孔）。尤其重要的是三种情况G = 0（球形）或g = 1（甜甜圈）或g> = 2（具有多个手柄的表面）之间的情况尤其重要。 Case G = 1给出了椭圆曲线，在Fermat的最后定理证明Wiles的证明中起着关键作用。对于代数曲线或riemann表面，这种三分法已经在19世纪清楚地感知到了它，以及它在正面弯曲，平坦或双曲线非欧盟几何形状方面的解释；三种情况G = 0或g = 1或g> = 2的图片是整个主题的图标。相同的三分法是MORI理论的遥远模型或更高尺寸品种的分类，这是最密集的一个模型之一从1970年代后期开始的代数几何形状的发达地区；这项工作导致了莫里（Mori）1990年的奖牌。目前，此分类是一个重大突破的主题，最近宣布了所有维度最小模型计划的证明。沃里克研讨会的第一个组成部分将开发和传播这些新结果，并利用其许多应用。代数品种，同时多项式方程的解决方案集，在数量理论中提供了示例和技术，在理论物理学中，在代数和奇异理论和奇异理论和奇异理论和奇异理论和奇异理论中以及在其他几何分支中。即使在分析中，主要是在无限尺寸空间中交付的分析中，最终目标通常是减少在代数几何形状上模拟的有限维溶液集。沃里克研讨会将包括这些主题中的每个主题的组成部分，以及代数几何形状到其他数学领域的应用。