Algebraic Geometry and Infinite-dimensional Spaces

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。这种空间的传统分析方法存在许多困难。代数-几何方法的优点是绕过了这些困难，同时保留并实际上强调了研究的主要概念结果。他提出了一种基于独立方案的代数-几何概念的花上同调方法。事实上，一个独立方案的定义（已经有一段时间了）与Floer关于“半无限”维的循环的想法相差不远。Kapranov建议将这种方法应用于有限维变量的形式路径和循环的各种空间。除此之外，他建议通过使用这些空间来理解椭圆上同调理论，特别是将椭圆上同调和相干束的派生范畴，这两个最近感兴趣的对象联系起来。通过研究无限维空间上的各种“异常”，kapranov提出了一个新的黎曼-罗奇型定理，它涉及实数而非复变元的族。研究无限维空间的动机来自物理学（弦理论），其中基本对象不是准时粒子，而是在时空中传播的弦。这种弦的自由度显然是无限的。但是处理无限维空间是困难的。多变量微积分中常见的收敛问题在很多情况下，当变量的数量变得无限时，就变得难以解决了。kapranov提出的代数方法可以抓住许多问题的本质，同时保持数学的严谨性，从而防止犯错误。这种问题的一个例子是行列式异常：无限矩阵的行列式并不像有限维情况下的预期那样表现，而是遵循不同的规则。这导致了无限维空间的拓扑和几何的大量结果。代数方法允许人们到达并推广这些结果，同时最小化甚至定义无限行列式所需的相当大的努力。

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.