Gelfand is known for many developments including: • the book
Calculus of Variations (1963), which he co-authored with
Sergei Fomin; •
Gelfand's formula, which expresses the spectral radius as a limit of matrix norms. • the
Gelfand representation in
Banach algebra theory; • the
Gelfand–Mazur theorem in Banach algebra theory; • the
Gelfand–Naimark theorem; • the
Gelfand–Naimark–Segal construction; •
Gelfand–Shilov spaces; • the
Gelfand–Pettis integral; • the
representation theory of the
complex classical Lie groups; • contributions to the theory of
Verma modules in the
representation theory of
semisimple Lie algebras (with I. N. Bernstein and S. I. Gelfand); • contributions to
distribution theory and measures on infinite-dimensional spaces; • the first observation of the connection of
automorphic forms with representations (with
Sergei Fomin); • conjectures about the
Atiyah–Singer index theorem; •
ordinary differential equations (Gelfand–
Levitan theory); • work on
calculus of variations and
soliton theory (Gelfand–Dikii equations); • contributions to the
philosophy of cusp forms; • Gelfand–
Fuchs cohomology of
Lie algebras; •
Gelfand–Kirillov dimension; •
integral geometry; • combinatorial definition of the
Pontryagin class; •
Coxeter functors; •
general hypergeometric functions; • Gelfand–
Tsetlin patterns; • Gelfand–Lokutsievski method; • the
BGG correspondence (with
Joseph Bernstein and
Sergei Gelfand); • and many other results, particularly in the representation theory of
classical groups. Gelfand ran a seminar at
Moscow State University from 1943 until May 1989 (when it continued at
Rutgers University), which covered a wide range of topics and was an important school for many mathematicians. ==Influence outside mathematics==