He is known for many developments including: the Gelfand representation in Banach algebra; representation theory of the complex classical Lie groups; contributions to distribution theory and measures on infinite-dimensional spaces; the first observation of the connection of automorphic forms with representation (with Fomin); conjectures about the Index theorem; ODEs (Gel'fand-Levitan theory); work on calculus of variations and soliton theory (Gel'fand-Dikii equations); contributions to the *philosophy of cusp forms*; Gel'fand-Fuks cohomology of foliations; Gel'fand-Kirillov dimension; integral geometry; combinatorial definition of the Pontryagin class; Coxeter functors; generalised hypergeometric series; and many other results, particularly in the representation theory for the classical groups.