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| {{Infobox scientist
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| |name = Grigory Margulis
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| |image = Grigory Margulis.jpg
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| |image_size = 250px
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| |caption = Grigory Margulis
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| |birth_date = {{birth date and age|1946|02|24|mf=yes}}
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| |birth_place = [[Moscow]], [[Soviet Union]]
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| |death_date =
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| |death_place =
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| |nationality = [[Russia]]n
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| |field = [[Mathematics]]
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| |work_institutions = [[Yale University]]
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| |alma_mater = [[Moscow State University]]
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| |doctoral_advisor = [[Yakov Sinai]]
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| |doctoral_students = [[Emmanuel Breuillard]]<br>[[Bengt Karlsson]]<br>[[Dmitry Kleinbock]]<br>[[Hee Oh]]
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| |known_for = [[Diophantine approximation]]<br>[[Lie group]]s<br>[[Superrigidity theorem]]<br>[[Arithmeticity theorem]]<br>[[Expander graphs]]<br>[[Oppenheim conjecture]]
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| |prizes = [[Fields Medal]] <small>(1978)</small><br>{{nowrap|[[Wolf Prize in Mathematics]] <small>(2005)</small>}}<br>[[Medal of the Collège de France]]<br>[[Alexander von Humboldt Prize]]<br>[[Lobachevsky International Prize]]
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| }}
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| '''Gregori Aleksandrovich Margulis''' ({{lang-ru|Григо́рий Алекса́ндрович Маргу́лис}}, first name often given as '''Gregory''', '''Grigori''' or '''Grigory'''; born February 24, 1946) is a [[Russia]]n [[mathematician]] known for his work on [[lattice (discrete subgroup)|lattices]] in [[Lie group]]s, and the introduction of methods from [[ergodic theory]] into [[diophantine approximation]]. He was awarded a [[Fields Medal]] in 1978 and a [[Wolf Prize]] [[Wolf Prize in Mathematics|in Mathematics]] in 2005, becoming the seventh mathematician to receive both prizes. In 1991, he joined the faculty of [[Yale University]], where he is currently the Erastus L. DeForest Professor of Mathematics.<ref name="YOPA_2005_02_23">{{cite web|url=http://opa.yale.edu/news/article.aspx?id=4312|title=Yale's Margulis Wins 2005 Wolf Prize for Mathematics|date=2005-02-23|publisher=Yale University Office of Public Affairs}}</ref>
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| == Short biography ==
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| Margulis was born in [[Moscow]], [[Soviet Union]]. He received his PhD in 1970 from the [[Moscow State University]], starting research in [[ergodic theory]] under the supervision of [[Yakov Sinai]]. Early work with [[David Kazhdan]] produced the [[Kazhdan's property (T)|Kazhdan–Margulis theorem]], a basic result on [[discrete group]]s. His [[superrigidity theorem]] from 1975 clarified an area of classical conjectures about the characterisation of [[arithmetic group]]s amongst lattices in [[Lie groups]].
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| He was awarded the [[Fields Medal]] in 1978, but was not permitted to travel to [[Helsinki]] to accept in person. His position improved, and in 1979 he visited [[Bonn]], 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 at [[Yale University]].
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| Margulis was elected a member of the [[United States National Academy of Sciences|U.S. National Academy of Sciences]] in 2001.<ref>[http://www.ams.org/notices/200107/people.pdf National Academy of Sciences Elections.] [[Notices of the American Mathematical Society]], vol. 48 (2001), no. 7, p. 722</ref> In 2012 he became a fellow of the [[American Mathematical Society]].<ref>[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-02-02.</ref>
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| In 2005, Margulis received the [[Wolf Prize]] for his contributions to theory of lattices and applications to ergodic theory, [[representation theory]], [[number theory]], [[combinatorics]], and [[measure theory]].
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| == Mathematical contributions ==
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| Early work of Margulis dealt with [[Kazhdan's property (T)]] and the questions of rigidity and arithmeticity of [[lattice (discrete subgroup)|lattices]] in [[semisimple algebraic group]]s of higher rank over a [[local field]]. It had been known since the 1950s ([[Armand Borel|Borel]], [[Harish-Chandra]]) that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices, called ''arithmetic lattices''. It is analogous to considering the subgroup ''SL''(''n'','''Z''') of the [[real numbers|real]] [[special linear group]] ''SL''(''n'','''R''') that consists of matrices with ''integer'' entries. Margulis proved that under suitable assumptions on ''G'' (no compact factors and [[split rank]] greater or equal than two), ''any'' (irreducible) lattice ''Γ'' in it is arithmetic, i.e. can be obtained in this way. Thus ''Γ'' is [[commensurability (mathematics)|commensurable]] with the subgroup ''G''('''Z''') of ''G'', i.e. they agree on subgroups of finite [[Index of a subgroup|index]] 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 lattice ''Γ'' in ''G'' roughly means that any [[group representation|homomorphism]] of ''Γ'' into the group of real invertible ''n'' × ''n'' matrices extends to the whole ''G''. The name derives from the following variant:
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| : If ''G'' and ''G' '', semisimple algebraic groups over a local field without compact factors and whose split rank is at least two and ''Γ'' and ''Γ''<sup><math>'</math></sup> are irreducible lattices in them, then any homomorphism ''f'': ''Γ'' → ''Γ''<sup><math>'</math></sup> between the lattices agrees on a finite index subgroup of ''Γ'' with a homomorphism between the algebraic groups themselves.
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| (The case when ''f'' is an [[isomorphism]] is known as the [[strong rigidity]].) While certain rigidity phenomena had already been known, the approach of Margulis was at the same time novel, powerful, and very elegant.
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| Margulis solved the long-standing [[Banach]]–[[Ruziewicz problem]] that asks whether the [[Lebesgue measure]] is the only normalized rotationally invariant [[Banach measure|finitely additive measure]] on the [[n-sphere|''n''-dimensional sphere]]. The affirmative solution for ''n'' ≥ 4, which was also independently and almost simultaneously obtained by [[Dennis Sullivan]], follows from a construction of a certain dense subgroup of the [[orthogonal group]] that has property (T).
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| Margulis gave the first construction of [[expander graph]]s, which was later generalized in the theory of [[Ramanujan graph]]s.
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| In 1986, Margulis completed the proof of the [[Oppenheim conjecture]] on [[quadratic form]]s and diophantine approximation. This was a question that had been open for half a century, on which considerable progress had been made by the [[Hardy-Littlewood circle method]]; but to reduce the number of variables to the point of getting the best-possible results, the more structural methods from [[group theory]] proved decisive. He has formulated a further program of research in the same direction, that includes the [[Littlewood conjecture]].
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| == Selected publications ==
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| === Books ===
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| * [http://books.google.com/books/about/Discrete_Subgroups_of_Semisimple_Lie_Gro.html?id=V9z2cjHOhVoC ''Discrete subgroups of semisimple Lie groups''], [[Ergebnisse der Mathematik und ihrer Grenzgebiete]] (3) [Results in Mathematics and Related Areas (3)], 17. [[Springer-Verlag]], Berlin, 1991. x+388 pp. ISBN 3-540-12179-X {{MR|1090825}}<ref>{{cite journal|author=Zimmer, Robert J.|authorlink=Robert Zimmer|title=Review: ''Discrete subgroups of semisimple Lie groups'', by G. A. Margulis|journal=Bull. Amer. Math. Soc. (N.S.)|year=1992|volume=27|issue=1|pages=198–202|url=http://www.ams.org/journals/bull/1992-27-01/S0273-0979-1992-00306-3/S0273-0979-1992-00306-3.pdf}}</ref>
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| * ''On some aspects of the theory of Anosov systems''. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian by Valentina Vladimirovna Szulikowska. [[Springer Monographs in Mathematics]]. Springer-Verlag, Berlin, 2004. vi+139 pp. ISBN 3-540-40121-0 {{MR|2035655}}<ref>{{cite journal|author=Parry, William|authorlink=Bill Parry (mathematician)|title=Review: ''On some aspects of the theory of Anosov systems'', by G. A. Margulis, with a survey "Periodic orbits of hyperbolic flows", by Richard Sharp|journal=Bull. Amer. Math. Soc. (N.S.)|year=2005|volume=42|issue=2|pages=257–261|url=http://www.ams.org/journals/bull/2005-42-02/S0273-0979-05-01051-7/S0273-0979-05-01051-7.pdf}}</ref>
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| === Lectures ===
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| * ''Oppenheim conjecture''. Fields Medallists' lectures, 272–327, World Sci. Ser. 20th Century Math., 5, World Sci. Publ., River Edge, NJ, 1997 {{MR|1622909}}
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| * ''Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory''. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 193–215, Math. Soc. Japan, Tokyo, 1991 {{MR|1159213}}
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| === Papers ===
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| * ''Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators''. (Russian) Problemy Peredachi Informatsii 24 (1988), no. 1, 51–60; translation in Problems Inform. Transmission 24 (1988), no. 1, 39–46
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| * ''Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than'' 1, Invent. Math. 76 (1984), no. 1, 93–120 {{MR|0739627}}
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| * ''Some remarks on invariant means'', Monatsh. Math. 90 (1980), no. 3, 233–235 {{MR|0596890}}
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| * ''Arithmeticity of nonuniform lattices in weakly noncompact groups''. (Russian) Funkcional. Anal. i Prilozen. 9 (1975), no. 1, 35–44
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| * ''Arithmetic properties of discrete groups'', Russian Math. Surveys 29 (1974) 107–165 {{MR|0463353}}
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| == References ==
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| {{reflist}}
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| == Further reading ==
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| * {{ cite conference | author=J. Tits | authorlink=Jacques Tits | title=The work of Gregori Aleksandrovitch Margulis | url=http://imu2.zib.de/ICM/ICM1978.1/ | pages=57–63 | editor=Olli Lehto (ed.) | booktitle=Proceedings of the [[International Congress of Mathematicians|ICM]] (Helsinki, 1978) | volume=1 | year=1980 | publisher=[[Finnish Academy of Science and Letters|Academia Scientiarum Fennica]] | location=Helsinki | isbn=951-41-0352-1 | mr=562596 | zbl = 0426.22011 }} ''1978 [[Fields Medal]] citation.''
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| == External links ==
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| * {{MacTutor Biography|id=Margulis}}
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| * {{IMO results |id=9834}}
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| * {{MathGenealogy |id=15062}}
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| {{Fields medalists}}
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| {{Wolf Prize in Mathematics}}
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| {{Authority control |VIAF=24699676 |LCCN=n/90/711576}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME =Margulis, Grigory
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = Russian mathematician
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| | DATE OF BIRTH =February 24, 1946
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| | PLACE OF BIRTH = [[Moscow]], [[Russia]]
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| | DATE OF DEATH =
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Margulis, Grigory}}
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| [[Category:21st-century mathematicians]]
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| [[Category:Members of the United States National Academy of Sciences]]
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| [[Category:Fellows of the American Mathematical Society]]
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| [[Category:Soviet mathematicians]]
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| [[Category:Russian mathematicians]]
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| [[Category:Fields Medalists]]
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| [[Category:Moscow State University alumni]]
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| [[Category:Yale University faculty]]
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| [[Category:Wolf Prize in Mathematics laureates]]
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| [[Category:People from Moscow]]
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| [[Category:1946 births]]
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| [[Category:Living people]]
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| [[Category:Alexander von Humboldt Fellows]]
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| [[Category:International Mathematical Olympiad participants]]
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