Index Theorems

The theory of the index of elliptic operators has for a long time been developed in parallel within the framework of two branches of mathematics that, traditionally, are regarded as quite far apart. One of them is the the theory of elliptic equations and boundary value problems—in particular, the theory of singular integral equations. The other is topology and algebraic geometry, where very specific elliptic operators have been considered. A significant role in bringing these two domains together was played by Gel’fand (1960), who posed the problem of topological classification of elliptic operators, in particular, the computation of the index in topological terms. The latter was fully solved by Atiyah and Singer in 1963. The Atiyah-Singer theorem has generated a tremendous amount of interest, which has continued to this day and has exercised an immense influence on the subsequent development and convergence of the theory of differential equations and topology. Thus, for example, the necessity to extend the class of deformations of elliptic operators has led to new algebras of pseudodifferential operators (PDOs). In topology, the Atiyah-Singer theorem has stimulated the further development of K-theory.