Israel Moiseevich Gelfand, Part I

Israel Moiseevich Gelfand Vladimir Retakh, Coordinating Editor

Gelfand with grandaughter. Israel Moiseevich Gelfand, a mathematician compared by Henri Cartan to Poincaré and Hilbert, was born on September 2, 1913, in the small town of Okny (later Red Okny) near Odessa in the Ukraine and died in New Brunswick, New Jersey, USA, on October 5, 2009. Nobody guided Gelfand in his studies. He attended the only school in town, and his mathematics teacher could off...

Gelfand on mathematics and neurophysiology

It is well known that Gelfand’s scientific interests were not limited to mathematics. One of non-mathematical field where Israel Moiseevich Gelfand worked was neurophysiology. In late 1950s, he organized neurophysiological seminar and few years later he spearheaded two neurophysiological research groups: one at the Institute of Biophysics (after 1967, this group moved to the Institute for the P...

HILBERT SERIES OF QUADRATIC ALGEBRAS ASSOCIATED WITH PSEUDO-ROOTS OF NONCOMMUTATIVE POLYNOMIALS Israel Gelfand, Sergei Gelfand,Vladimir Retakh,

The quadratic algebras Qn are associated with pseudo-roots of noncommutative polynomials. We compute the Hilbert series of the algebras Qn and of the dual algebras Q ! n. Introduction Let P (x) = x−a1x n−1 + · · ·+(−1)an be a polynomial over a ring R. Two classical problems concern the polynomial P (x): nvestigation of the solutions of the equation P (x) = 0 and the decomposition of P (x) into ...

Part] I I

Gelfand pairs

Let K ⊂ G be a compact subgroup of a real Lie group G. Denote by D(X) thealgebra of G-invariant differential operators on the homogeneous space X = G/K. ThenX is called commutative or the pair (G,K) is called a Gelfand pair if the algebra D(X)is commutative. Symmetric Riemannian homogeneous spaces introduced by Élie Cartanand weakly symmetric homogeneous spaces introduced by Sel...