Kapranov proposes to study infinite-dimensional spaces such as spaces of paths by using algebro-geometric techniques. The traditional analytic approach to such spaces leads to many difficulties. The algebro-geometric approach has the advantage of bypassing these difficulties and at the same time preserving and in fact emphasizing the main conceptual results of the study. He proposes to develop 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 for some time) is not far removed from Floer's ideas about cycles of ``semi-infinite" dimension. Kapranov proposes to apply this approach to various spaces of formal paths and loops in finite-dimensional varieties. Among other things, he proposes to understand the elliptic cohomology theory by using these spaces, in particular to relate the elliptic cohomology and the derived category of coherent sheaves, two objects of recent interest. By studying various "anomalies" on such infinite-dimensional spaces, Kapranov proposes to prove a new Riemann-Roch type theorem involving families of real, 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 of such a string is clearly infinite. But working with infinite-dimensional spaces is difficult. The usual problems of convergence familiar from multivariable calculus become in many cases overwhelming when the number of variables becomes infinite. The algebraic approach proposed by Kapranov can capture the essense of many problems while maintaining the mathematical rigor and thus preventing one from making mistakes. One example of such a problem is the determinantal anomaly: determinants of infinite matrices do not behave as expected from the finite-dimensional case but obey different rules. This leads to a wealth of consequences for the topology and geometry of infinite-dimensional spaces. The algebraic approach allows one to arrive at and generalize these consequences while minimizing the considerable effort needed to even define the infinite determinants.
