Wolf Prize Laureate in Mathematics 2005
Gregory A. Margulis
Affiliation at the time of the award:
Yale University, USA
“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.”
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.