George Mackey
Encyclopedia
St. Louis, Missouri
St. Louis is an independent city on the eastern border of Missouri, United States. With a population of 319,294, it was the 58th-largest U.S. city at the 2010 U.S. Census. The Greater St...
, Missouri
Missouri
Missouri is a US state located in the Midwestern United States, bordered by Iowa, Illinois, Kentucky, Tennessee, Arkansas, Oklahoma, Kansas and Nebraska. With a 2010 population of 5,988,927, Missouri is the 18th most populous state in the nation and the fifth most populous in the Midwest. It...
– March 15, 2006 in Belmont
Belmont, Massachusetts
Belmont is a town in Middlesex County, Massachusetts, a suburb of Boston. The population was 24,729 at the 2010 census.- History :Belmont was founded on March 18, 1859 by former citizens of, and land from the bordering towns of Watertown, to the south; Waltham, to the west; and Arlington, then...
, Massachusetts
Massachusetts
The Commonwealth of Massachusetts is a state in the New England region of the northeastern United States of America. It is bordered by Rhode Island and Connecticut to the south, New York to the west, and Vermont and New Hampshire to the north; at its east lies the Atlantic Ocean. As of the 2010...
) was an American 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....
. Mackey earned his bachelor of arts at Rice University
Rice University
William Marsh Rice University, commonly referred to as Rice University or Rice, is a private research university located on a heavily wooded campus in Houston, Texas, United States...
(then the Rice Institute) in 1938 and obtained his Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
in 1942 under the direction of Marshall H. Stone. He joined the Harvard University Mathematics Department in 1943, was appointed Landon T. Clay
Landon T. Clay
Landon T. Clay is an American businessman and founder of the Clay Institute.Clay graduated from Harvard in 1950 in English and was chair of Eaton Vance and a director of ADE Corporation.-References:...
Professor of Mathematics and Theoretical Science in 1969 and remained there until he retired in 1985.
Mackey's main areas of research were in the areas of 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...
, 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....
, and related parts of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. Earlier in his career Mackey did significant work in the duality theory of locally convex spaces, which provided tools for subsequent work in this area, including Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
's work on topological tensor product
Topological tensor product
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products , but for general Banach spaces or locally convex topological vector space...
s.
Mackey was one of the pioneer workers in the intersection of quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
, the theory of infinite-dimensional unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
s of groups, the theory of operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s and noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
. A central role in Mackey's work, both in the theory of group representations and in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, was played by the concepts of system of imprimitivity
System of imprimitivity
The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations...
and induced representation
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
. This idea led naturally to an analysis of the representation theory of semi-direct products in terms of ergodic actions of groups and in some cases a complete classification of such representations. Mackey's results were essential tools in the study of the representation theory of nilpotent Lie group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
s using the method of orbits
Kirillov orbit theory
In mathematics, the orbit method establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra...
developed by Alexandre Kirillov
Alexandre Kirillov
Alexandre Aleksandrovich Kirillov is a Soviet and Russian mathematician, renowned for his works in the fields of representation theory, topological groups and Lie groups. In particular he introduced the orbit method into representation theory....
in the 1960s. His notion of "virtual subgroup", introduced in 1966 using the language of groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
s, had a significant influence 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....
.
Another essential ingredient in Mackey's work was the assignment of a Borel structure
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
to the dual object of a locally compact group
Locally compact group
In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
(specifically a locally compact separable metric group) G. One of Mackey's important conjectures, which was eventually solved by work of James Glimm
James Glimm
James Gilbert Glimm is an American mathematical physicist, and Professor at the State University of New York at Stony Brook.James Glimm was born in Peoria, Illinois, USA on 24 March 1934.- Career :...
on C*-algebras was that G is type I (meaning that all its factor representations are of type I) if and only if the Borel structure of its dual is a standard Borel space.
He has written numerous survey articles connecting his research interests with a large body of mathematics and physics, particularly quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
and statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
.
Mackey was among the first five recipients of William Lowell Putnam fellowships in 1938.
Mackey was an elected member of the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
and the National Academy of Science.
Books
- Mathematical Foundations of Quantum Mechanics (Dover Books on Mathematics) ISBN 0-486-43517-2
- Unitary Group Representations in Physics, Probability, and Number Theory, 402 pages, Benjamin–Cummings Publishing Company (1978), ISBN 0-8053-6703-9
- The theory of Unitary Group Representations (Chicago Lectures in Mathematics) University Of Chicago Press (August 1, 1976) ISBN 0-226-50051-9
- Induced representations of groups and quantum mechanics, Publisher: W. A. Benjamin (1968)
- Mathematical Problems of Relativistic Physics (Lectures in Applied Mathematics Series, Vol 2) by E. E. Segal, George Whitelaw Mackey, Publisher: Amer Mathematical Society (June 1967) ISBN 0-8218-1102-9
- Lectures on the theory of functions of a complex variable Publisher: R. E. Krieger Pub. Co (1977) ISBN 0-88275-531-5
See also
- Mackey topologyMackey topologyIn functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual...
, the finest dual topologyDual topologyIn functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.... - Mackey–Arens theorem characterizes all dual topologies
- Bornological spaceBornological spaceIn mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and functions, in the same way that a topological space possesses the minimum amount of...
s, all of which are Mackey spaces
External links
- George Mackey (1916–2006), Notices of the American Mathematical Society; vol. 54, no. 7 (August 2007).
- George Mackey (1 February 1916–15 March 2006), Proceedings of the American Philosophical Society; vol. 152, no. 4 (December 2008).
- Commemorative website at Harvard Math Department
- Obituary from Harvard Gazette
- Obituary from Boston Globe
- Peter Woit's blog entry on Mackey
- Two letters from George Mackey and the text of his speech "What do Mathematicians Do?, collected by Stephanie Singer