Israel Gelfand - Work

Work

Gelfand is known for many developments including:

  • the book Calculus of Variations (1963), which he co-authored with Sergei Fomin
  • 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–Fuks cohomology of foliations;
  • Gelfand–Kirillov dimension;
  • integral geometry;
  • combinatorial definition of the Pontryagin class;
  • Coxeter functors;
  • general hypergeometric functions;
  • Gelfand - Tsetlin patterns;
  • and many other results, particularly in the representation theory for the classical groups.

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