Grigory Margulis | |
|---|---|
Григорий Маргулис | |
.jpg) Margulis in 2006 | |
| Born | (1946-02-24)February 24, 1946 (age 79) |
| Nationality | Russian,American[1] |
| Education | Moscow State University (BS,MS,PhD) |
| Known for | Diophantine approximation Lie groups Superrigidity theorem Arithmeticity theorem Expander graphs Oppenheim conjecture |
| Awards | Fields Medal (1978) Lobachevsky Prize (1996) Wolf Prize (2005) Abel Prize (2020) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Yale University |
| Thesis | On some aspects of the theory of Anosov flows (1970) |
| Doctoral advisor | Yakov Sinai |
| Doctoral students | Emmanuel Breuillard Hee Oh |
Grigory Aleksandrovich Margulis (Russian:Григо́рий Алекса́ндрович Маргу́лис, first name often given asGregory,Grigori orGregori; born February 24, 1946) is aRussian-American[2]mathematician known for his work onlattices inLie groups, and the introduction of methods fromergodic theory intodiophantine approximation. He was awarded aFields Medal in 1978, aWolf Prize in Mathematics in 2005, and anAbel Prize in 2020 (withHillel Furstenberg), becoming the fifth mathematician to receive the three prizes.[3] In 1991, he joined the faculty ofYale University, where he is currently theErastus L. De Forest Professor of Mathematics.[4]
Margulis was born to aRussian family ofLithuanian Jewish descent inMoscow,Soviet Union. At age 16 in 1962 he won the silver medal at theInternational Mathematical Olympiad. He received his PhD in 1970 from theMoscow State University, starting research inergodic theory under the supervision ofYakov Sinai. Early work withDavid Kazhdan produced theKazhdan–Margulis theorem, a basic result ondiscrete groups. Hissuperrigidity theorem from 1975 clarified an area of classical conjectures about the characterisation ofarithmetic groups amongst lattices inLie groups.
He was awarded theFields Medal in 1978, but was not permitted to travel toHelsinki to accept it in person, allegedly due toantisemitism against Jewish mathematicians in the Soviet Union.[5] His position improved, and in 1979 he visitedBonn, and was later able to travel freely, though he still worked in the Institute of Problems of Information Transmission, a research institute rather than a university. In 1991, Margulis accepted a professorial position atYale University.
Margulis was elected a member of theU.S. National Academy of Sciences in 2001.[6] In 2012 he became a fellow of theAmerican Mathematical Society.[7]
In 2005, Margulis received theWolf Prize for his contributions to theory of lattices and applications to ergodic theory,representation theory,number theory,combinatorics, andmeasure theory.
In 2020, Margulis received theAbel Prize jointly withHillel Furstenberg "For pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics."[8]
Margulis's early work dealt withKazhdan's property (T) and the questions of rigidity and arithmeticity oflattices insemisimple algebraic groups of higher rank over alocal field. It had been known since the 1950s (Borel,Harish-Chandra) that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices, calledarithmetic lattices. It is analogous to considering the subgroupSL(n,Z) of therealspecial linear groupSL(n,R) that consists of matrices withinteger entries. Margulis proved that under suitable assumptions onG (no compact factors andsplit rank greater or equal than two),any (irreducible) latticeΓ in it is arithmetic, i.e. can be obtained in this way. ThusΓ iscommensurable with the subgroupG(Z) ofG, i.e. they agree on subgroups of finiteindex in both. Unlike general lattices, which are defined by their properties, arithmetic lattices are defined by a construction. Therefore, these results of Margulis pave a way for classification of lattices. Arithmeticity turned out to be closely related to another remarkable property of lattices discovered by Margulis.Superrigidity for a latticeΓ inG roughly means that anyhomomorphism ofΓ into the group of real invertiblen ×n matrices extends to the wholeG. The name derives from the following variant:
(The case whenf is anisomorphism is known as thestrong rigidity.) While certain rigidity phenomena had already been known, the approach of Margulis was at the same time novel, powerful, and very elegant.
Margulis solved theBanach–Ruziewicz problem that asks whether theLebesgue measure is the only normalized rotationally invariantfinitely additive measure on then-dimensional sphere. The affirmative solution forn ≥ 4, which was also independently and almost simultaneously obtained byDennis Sullivan, follows from a construction of a certain dense subgroup of theorthogonal group that has property (T).
Margulis gave the first construction ofexpander graphs, which was later generalized in the theory ofRamanujan graphs.
In 1986, Margulis gave a complete resolution of theOppenheim conjecture onquadratic forms and diophantine approximation. This was a question that had been open for half a century, on which considerable progress had been made by theHardy–Littlewood circle method; but to reduce the number of variables to the point of getting the best-possible results, the more structural methods fromgroup theory proved decisive. He has formulated a further program of research in the same direction, that includes theLittlewood conjecture.
