Homological and infinite-dimensional methods in algebraic geometry

卡普拉诺夫建议研究几个无限维代数几何对象，如方案和IND-方案，以期应用于最初由数学物理引起的问题。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提出的形式环的无限维InD格式。他进一步建议发展交集理论和IND-格式的射影对偶理论，其中包括互换归纳极限和射影极限。他建议用自由李代数来研究非参数路径空间的代数几何模型。数学在几个领域的应用使得研究无限维空间成为必要。例如，诸如弦之类的振动介质的状态形成无限维空间。在物理学中的弦理论中，提出了这样的图景，即描述自然界中的基本相互作用。在任何方法中，研究无限维物体都是非常重要的。对于这种情况，传统的方法，使用分析，提出了额外的困难，如级数的收敛问题等。卡普拉诺夫提出了一种使用代数技术来研究无限维空间的几何。这使得人们可以专注于在许多情况下可以使用有限维的直觉来理解的基本现象，并突出有限维的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.