Algebraic Geometry and Infinite-dimensional Spaces

代数几何和无限维空间

• 批准号: 0500565
• 负责人: Mikhail Kapranov
• 金额: --
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500565&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Infinite; dimensional; Spaces

Kapranov proposes to study infinite-dimensional spaces such as spaces of pathsby using algebro-geometric techniques. The traditional analytic approach to suchspaces leads to many difficulties. The algebro-geometric approach has theadvantage of bypassing these difficulties and at the same time preserving andin fact emphasizing the main conceptual results of the study. He proposes todevelop an approach to Floer cohomology based on the algebro-geometric concept of an ind-scheme. In fact, the very definition of an ind-scheme (known forsome time) is not far removed from Floer's ideas about cycles of ``semi-infinite"dimension. Kapranov proposes to apply this approach to various spaces offormal paths and loops in finite-dimensional varieties. Among other things, heproposes to understand the elliptic cohomology theory by using these spaces,in particular to relate the elliptic cohomology and the derived category ofcoherent sheaves, two objects of recent interest.By studying various "anomalies" on such infinite-dimensional spaces, Kapranovproposes to prove a new Riemann-Roch type theorem involving families ofreal, not complex varieties. The motivation for study of infinite-dimensional spaces comes from physics(string theory) where the fundamental object is not a punctual particle but a string propagating in the space-time. The number of degrees of freedom ofsuch a string is clearly infinite. But working with infinite-dimensional spacesis difficult. The usual problems of convergence familiar from multivariable calculus becomein many cases overwhelming when the number of variables becomes infinite. The algebraic approach proposed by Kapranovcan capture the essense of many problems while maintaining the mathematical rigor andthus preventing one from making mistakes. One example of such a problem is thedeterminantal anomaly: determinants of infinite matrices do not behave as expectedfrom the finite-dimensional case but obey different rules. This leads to a wealth of consequences forthe topology and geometry of infinite-dimensional spaces. The algebraic approachallows one to arrive at and generalize these consequences while minimizing theconsiderable effort needed to even define the infinite determinants.

Kapranov建议使用代数几何技术来研究无限维空间，例如路径空间。传统的分析方法，这样的空间导致了许多困难。代数几何方法的优点是绕过这些困难，并在同一时间保存和事实上强调的主要概念的研究结果。他建议发展一种方法Floer上同调的基础上代数几何概念的ind-scheme。事实上，ind-scheme的定义（已经知道一段时间了）与Floer关于"半无限“维度的循环的思想并没有太大的区别。Kapranov提出将这种方法应用于各种空间的形式路径和循环的有限维品种。除此之外，他提出理解椭圆上同调理论，通过使用这些空间，特别是涉及椭圆上同调和派生类别的相干层，最近的兴趣两个对象。通过研究各种“异常”等无限维空间，Kapranov提出证明一个新的Riemann-Roch型定理涉及家庭的真正的，而不是复杂的品种。研究无限维空间的动机来自物理学（弦理论），其中基本对象不是点状粒子，而是在时空中传播的弦。这样一根弦的自由度显然是无限的。 但在无限维空间工作是困难的。在许多情况下，当变量的数目变成无穷大时，多变量微积分中常见的收敛问题变得难以克服。Kapranov提出的代数方法可以在保持数学严密性的同时抓住许多问题的本质，从而避免犯错误。这样一个问题的一个例子是行列式异常：无限矩阵的行列式不像有限维情况下预期的那样，而是遵循不同的规则。这导致了无穷维空间的拓扑学和几何学的大量结果。代数的方法允许一个到达和推广这些后果，同时尽量减少thesconsiderable努力，甚至需要定义无限的决定因素。