Grigory Margulis
Encyclopedia
Gregori Aleksandrovich Margulis is a Russia
n mathematician
known for his far-reaching work on 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 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.
, USSR. 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–Margulis theorem
, a basic result on discrete group
s. His superrigidity theorem from 1975 clarified a whole area of classical conjectures about the characterisation of arithmetic group
s amongst lattices in Lie groups.
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
.
Margulis was elected a member of the U.S. National Academy of Sciences
in 2001.
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.
and the questions of rigidity and arithmeticity of lattices
in semisimple algebraic group
s of higher rank over a local 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, called arithmetic lattices. It is analogous to considering the subgroup SL(n,Z) of the 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 commensurable
with the subgroup G(Z) of G, i.e. they agree on subgroups of finite 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 homomorphism
of Γ into the group of real invertible n × n matrices extends to the whole G. The name derives from the following variant:
(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.
Margulis solved the long-standing Banach–Ruziewicz problem
that asks whether the Lebesgue measure
is the only normalized rotationally invariant finitely additive measure
on the 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).
Margulis gave the first construction of expander graph
s, which was later generalized in the theory of Ramanujan graph
s.
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. This has been widely influential.
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for his far-reaching work on lattices
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, and the introduction of methods from ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
into diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
. He was awarded a Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1978 and a Wolf Prize in Mathematics
Wolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
in 2005, becoming the seventh mathematician to receive both prizes. In 1991, he joined the faculty of Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, where he is currently the Erastus L. DeForest Professor of Mathematics.
Short biography
Margulis was born in MoscowMoscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
, USSR. He received his PhD in 1970 from the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
, starting research in ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
under the supervision of Yakov Sinai. Early work with David Kazhdan
David Kazhdan
David Kazhdan or Každan, Kazhdan, formerly named Dmitry Aleksandrovich Kazhdan , is a Soviet and Israeli mathematician known for work in representation theory.-Life:...
produced the Kazhdan–Margulis theorem
Kazhdan's property (T)
In mathematics, a locally compact topological group G has property if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero...
, a basic result on discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
s. His superrigidity theorem from 1975 clarified a whole area of classical conjectures about the characterisation of arithmetic group
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
s amongst lattices in Lie groups.
He was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1978, but was not permitted to travel to Helsinki
Helsinki
Helsinki is the capital and largest city in Finland. It is in the region of Uusimaa, located in southern Finland, on the shore of the Gulf of Finland, an arm of the Baltic Sea. The population of the city of Helsinki is , making it by far the most populous municipality in Finland. Helsinki is...
to accept in person. His position improved, and in 1979 he visited Bonn
Bonn
Bonn is the 19th largest city in Germany. Located in the Cologne/Bonn Region, about 25 kilometres south of Cologne on the river Rhine in the State of North Rhine-Westphalia, it was the capital of West Germany from 1949 to 1990 and the official seat of government of united Germany from 1990 to 1999....
, 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
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
.
Margulis was elected a member of the U.S. National Academy of Sciences
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
in 2001.
In 2005, Margulis received the Wolf Prize for his contributions to theory of lattices and applications to ergodic theory, representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, and measure theory.
Scientific contributions
Early work of Margulis dealt with Kazhdan's property (T)Kazhdan's property (T)
In mathematics, a locally compact topological group G has property if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero...
and the questions of rigidity and arithmeticity of lattices
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in semisimple algebraic group
Semisimple algebraic group
In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
s of higher rank over a local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
. It had been known since the 1950s (Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
, Harish-Chandra
Harish-Chandra
Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...
) 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 special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
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 commensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...
with the subgroup G(Z) of G, i.e. they agree on subgroups of finite index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
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 homomorphism
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of Γ into the group of real invertible n × n matrices extends to the whole G. The name derives from the following variant:
- 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 ' are irreducible lattices in them, then any homomorphism f: Γ → Γ ' between the lattices agrees on a finite index subgroup of Γ with a homomorphism between the algebraic groups themselves.
(The case when f is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
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.
Margulis solved the long-standing Banach–Ruziewicz problem
Ruziewicz problem
In mathematics, the Ruziewicz problem in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotations, and defined on all Lebesgue measurable sets.This was answered...
that asks whether the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
is the only normalized rotationally invariant finitely additive measure
Banach measure
In mathematics, Banach measure in measure theory may mean a real-valued function on the algebra of all sets , by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round...
on the n-dimensional sphere. The affirmative solution for n ≥ 4, which was also independently and almost simultaneously obtained by Dennis Sullivan
Dennis Sullivan
Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
, follows from a construction of a certain dense subgroup of the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
that has property (T).
Margulis gave the first construction of expander graph
Expander graph
In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion as described below...
s, which was later generalized in the theory of Ramanujan graph
Ramanujan graph
A Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible . Such graphs are excellent spectral expanders....
s.
In 1986, Margulis completed the proof of the Oppenheim conjecture
Oppenheim conjecture
In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened by Davenport and Oppenheim...
on quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
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
Hardy-Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E...
; but to reduce the number of variables to the point of getting the best-possible results, the more structural methods from group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
proved decisive. He has formulated a further program of research in the same direction, that includes the Littlewood conjecture. This has been widely influential.
Books
- Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer GrenzgebieteErgebnisse der Mathematik und ihrer GrenzgebieteErgebnisse der Mathematik und ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics is a series of scholarly monographs published by Springer-Verlag. The title literally means "Results in mathematics and related areas". Most of the books were published in German or English, but there were a...
(3) [Results in Mathematics and Related Areas (3)], 17. Springer-Verlag, Berlin, 1991. x+388 pp. ISBN 3-540-12179-X - 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
Lectures
- Oppenheim conjecture. Fields Medallists' lectures, 272–327, World Sci. Ser. 20th Century Math., 5, World Sci. Publ., River Edge, NJ, 1997
- 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
Papers
- 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
- Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Invent. Math. 76 (1984), no. 1, 93–120
- Some remarks on invariant means, Monatsh. Math. 90 (1980), no. 3, 233–235
- Arithmeticity of nonuniform lattices in weakly noncompact groups. (Russian) Funkcional. Anal. i Prilozen. 9 (1975), no. 1, 35–44
- Arithmetic properties of discrete groups, Russian Math. Surveys 29 (1974) 107–165