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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.

代数几何研究多项式方程组的解集。这些解集被称为代数簇，并被视为几何轨迹，概括了解析几何中的圆和双曲线。代数几何是一门成熟的学科，它所提供的几何观点和广泛的工具箱适用于许多数学问题及其应用.代数曲线在平面上表现为轨迹f（x，y）=0，其中f是x和y的多项式函数;程度低（二次曲线和三次曲线）人们通过方程的显式操作得到有用的结论，但是随着f的次数增加，人们希望得到的那种结论更加抽象，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的点没有用任何更基本的东西来参数化，并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用于随后的构造。代数曲线的一个更好的不变量是它的亏格，而不是次数，也就是说，其拓扑模型中的控制柄（圆环状孔）的数量。特别重要的是在三种情况g=0（一个球面）或g=1（一个圆环）或g>=2（一个有许多柄的曲面）之间的情况划分; g=1的情况给出椭圆曲线，这在怀尔斯证明费马大定理中发挥了关键作用。对于代数曲线或黎曼曲面，这种剖分在世纪就已经被清楚地认识到了，连同它在正弯曲、平坦或双曲非欧几里德几何方面的解释; g=0或g=1或g>=2这三种情况的图像作为整个主题的图标。同样的分裂是Mori理论或高维变量分类的遥远模型，其中一个最密集的发展领域的代数几何从20世纪70年代后期;这项工作导致森的1990年菲尔兹奖。这种分类是目前的一个重大突破的主题，最近宣布在所有维度上证明最小模型程序。沃里克专题讨论会第一部分将发展和传播这些新成果，并探讨其许多应用。代数变种，解集的多项式方程，提供例子和技术，在数论和理论物理，在代数和奇异性理论和其他分支的几何。即使在分析中，主要处理无限维空间，最终的目标往往是减少到有限维的解决方案集代数几何建模。该沃里克专题讨论会将包括组件对每一个这些主题，连同应用代数几何的其他领域的数学。