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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    With malice toward none, with charity for all, with firmness in the right, as God gives us to see the right, let us strive on to finish the work we are in, to bind up the nation’s wounds, to care for him who shall have borne the battle, and for his widow and his orphan, to do all which may achieve and cherish a just and lasting peace among ourselves, and with all nations.
    Abraham Lincoln (1809–1865)

    My work is the only ground I’ve ever had to stand on. I seem to have a whole superstructure with no foundation—but I’m working on the foundation.
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    What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.
    Boris Pasternak (1890–1960)