Year:    2006

CAM-Net Digest, Vol. 3 (2006), Iss. 3 : p. 4

Abstract

http://www.wolffund.org.il/cat.asp?id=23&cat_title=MATHEMATICS

THE 2005 WOLF FOUNDATION PRIZE IN MATHEMATICS

The Prize Committee for Mathematics has unanimously decided that the 2005 Wolf Prize will be jointly awarded to:

Gregory A. Margulis

Yale University
New Haven, Connecticut, U.S.A.

for his monumental contributions to algebra, in particular to the theory of  lattices in semi-simple Lie groups, and striking applications of this to ergodic  theory, representation theory, number theory, combinatorics, and measure theory,  and Sergei P. Novikov University of Maryland College Park, Maryland, USA; and the L.D. Landau Institute for Theoretical Physics

Moscow, Russia

for his fundamental and pioneering contributions to algebraic and differential topology, and to mathematical physics, notably the introduction of algebraic- geometric methods

At the center of Professor Gregory A. Margulis’s work lies his proof of the Selberg-Piatetskii-Shapiro Conjecture, affirming that lattices in higher rank Lie groups are arithmetic, a question whose origins date back to Poincaré. This was achieved by a remarkable tour de force, in which probabilistic ideas revolving around a non-commutative version of the ergodic theorem were combined with p-adic analysis and with algebraic geometric ideas showing that “rigidity” phenomena, earlier established by Margulis and others, could be formulated  in such a way (“super-rigidity”) as to imply arithmeticity. This work displays stunning technical virtuosity and originality, with both algebraic and analytic methods. The work has subsequently reshaped the ergodic theory of general group actions on manifolds.

In a second tour de force, Margulis solved the 1929 Oppenheim Conjecture, stating that the set of values at integer points of an indefinite irrational  non-degenerate quadratic form in ≥ 3 variables is dense in Rn. This had been  reduced (by Rhagunathan) to a conjecture about unipotent flows on homogeneous spaces, proved by Margulis.

This method transformed to this ergodic setting  a family of questions till then investigated only in analytic number theory.

A third dramatic breakthrough came when Margulis showed that Kazhdan’s “Property  T” (known to hold for rigid lattices) could be used in a single arithmetic lattice construction to solve two apparently unrelated problems. One was the solution to a problem posed by Rusiewicz, about finitely additive measures  on spheres and Euclidean spaces. The other was the first explicit construction  of infinite families of expander graphs of bounded degree, a problem of practical application in the design of efficient communication networks.

Margulis’ work is characterized by extraordinary depth, technical power, creative synthesis of ideas and methods from different areas of mathematics, and a grand architectural unity of its final form. Though his work addresses deep unsolved problems, his solutions are housed in new conceptual and methodological frameworks, of broad and enduring application. He is one of the mathematical giants of  the last half a century.

Professor Sergei P. Novikov is awarded the Wolf Prize for his fundamental  and pioneering contributions to topology and to mathematical physics. His early  work in algebraic and differential topology includes such milestones as the  calculation of cobordism rings and stable homotopy groups, proof of the topological invariance of rational Pontrjagin Classes, formulation of the “Novikov Conjecture” on higher signature invariants, and proof of the existence of closed leaves  in two-dimensional foliations of the 3-sphere.

In the early 1970s, Novikov turned his attention to mathematical physics,  initially contributing to general relativity and conductivity of metals. He  constructed a global version of Morse Theory on manifolds and loop spaces that  had novel applications to quantum field theory (multi-valued action functionals).  His most significant achievements in mathematical physics flow from his introduction  of algebraic-geometric methods to the study of completely integrable systems. These include a systematic study of finite-gap solutions of two-dimensional  integrable systems, formulation of the equivalence of the classification of  algebraic-geometric solutions of the KP equation with the conformal classification  of Riemann surfaces, and work (with Krichever) on “almost commuting” operators, that appear in string theory and matrix models (“Krichever-Novikov algebras,” now widely used in physics).
Novikov made a fundamental and striking contribution to two separate fields  in mathematics, while he is one of those rare mathematicians who brings deep,  key mathematical ideas to bear on difficult pivotal problems of physics, in  ways thatare stunning and compelling for both mathematicians and physicists.