Homological and infinite-dimensional methods in algebraic geometry

Kapranov建议研究几个无限维代数几何对象，如计划和ind计划，以应用于最初由数学物理激发的问题。因此，他建议将复射影簇的椭圆上同调与其派生的凝聚层范畴联系起来。虽然这个问题本质上是“有限维”的，但这里的主要技术工具是Kapranov和Vasserot引入的形式循环的无限维ind-scheme。他进一步提出发展交集理论和理论的投射对偶的ind计划，其中包括交换归纳和投射限制。他建议研究代数几何模型的空间的unparametrized道路使用自由李代数。应用数学在几个领域使有必要研究无限维空间。例如，振动介质（如弦）的状态形成无限维空间。在物理学中的弦理论中，这种图像被提出来描述自然界中的基本相互作用。传统的做法的情况下，使用分析，提出了额外的困难，如问题的收敛系列等Kapranov提出了几何研究的无限维空间使用技术代数。这使人们能够集中精力研究本质现象，在许多情况下，这些现象可以用有限维的直觉来理解，并突出有限维直觉需要修改的情况。

Kapranov proposes to study several infinite-dimensional algebro-geometric objects such as schemes and ind-schemes with the view of applications to problems originally motivated by mathematical physics. Thus, he proposes to relate the elliptic cohomology of a complex projective variety to its derived category of coherent sheaves. While this problem is ``finite-dimensional'' in its nature, the main technical tool here is the infinite-dimensional ind-scheme of formal loops introduced by Kapranov and Vasserot. He further proposes to develop intersection theory and the theory of projective duality for ind-schemes which would, among other things, interchange inductive and projective limits. He proposes to study algebro-geometric models of the spaces of unparametrized paths by using free Lie algebras.Applications of mathematics in several areas make it necessary to study infinite-dimensional spaces. For example, the states of a vibrating medium such as a string, form an infinite-dimensional space. In string theory in physics, such picture was proposed as describing the fundamental interactions in nature.In any approach, study of infinite-dimensional objects is highly nontrivial. The traditional approach to the situation, using analysis, presents additional difficulties such as problems of convergence of series etc. Kapranov proposes a geometric study of infinite-dimensional spaces using techniques from algebra. This allows one to concentrate on essential phenomena which in many cases can be understood using the intuition from finite dimensions and to highlight the cases when the finite-dimensional intuition needs to be modified.