Home > Israel Moiseevich Gel'fand

He is known for many developments including:

• the Gelfand representation in Banach algebra theory;
• the Gel'fand-Naimark theorem;
• the Gelfand-Naimark-Segal construction
• the 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 representations (with Fomin);
• conjectures about the index theorem;
• ODEs (Gel'fand-Levitan theory);
• work on calculus of variationsCalculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. Such functionals can for example be formed as integrals involving an unknown function and its and solitonA soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. Solitons are found in many nonlinear physical phenomena, as they are found as solutions of many different nonlinear differential equations. The soliton phenomenon was theory (Gel'fand-Dikii equations);
• contributions to the philosophy of cusp formIn number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion : of the constant coefficient a''. This Fourier expansion exists as a cons;
• Gel'fand-Fuks cohomology of foliationIn mathematics, a foliation structure on a manifold M gives it stripes''. For example if the dimension of M is two, there is a pattern of stripes on the Euclidean plane formed by all lines parallel to the x axis (lines y c , and a foliation on M is a conss;
• Gel'fand-Kirillov dimension;
• integral geometryIn mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. The more traditional usage is that of Santalo and Blaschke. It follows from the classic theorem of Crofton exp;
• combinatorial definition of the Pontryagin classIn mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with index a multiple of four. It applies to real vector bundles. Definition Given a vector bundle over its k th Pontryagin class can;
• Coxeter functors;
• generalised hypergeometric seriesIn mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients a ''a is a rational function of n''. In the case of geometric series the ratio is constant. The series for the exponential;

and many other results, particularly in the representation theory for the classical groups.