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George Mackey

January 01, 1970

George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.

Career edit

Mackey earned his B.A. at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone.[1] He joined the Harvard University Mathematics Department in 1943, was appointed Landon T. Clay Professor of Mathematics and Theoretical Science in 1969 and remained there until he retired in 1985.

Work edit

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's work on topological tensor products.

Mackey was one of the pioneer workers in the intersection of quantum logic, the theory of infinite-dimensional unitary representations of groups, the theory of operator algebras and noncommutative geometry. A central role in Mackey's work, both in the theory of group representations and in mathematical physics, was played by the concepts of system of imprimitivity and induced representations. 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 groups using the method of orbits developed by Alexandre Kirillov in the 1960s. His notion of "virtual subgroup", introduced in 1966 using the language of groupoids, had a significant influence in ergodic theory.

Another essential ingredient in Mackey's work was the assignment of a Borel structure to the dual object of a locally compact group (specifically a locally compact separable metric group) G. One of Mackey's important conjectures, which was eventually solved by work of James Glimm 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 and statistical mechanics.

Honours and students edit

Mackey was among the first five recipients of William Lowell Putnam fellowships in 1938.[2] He received the Leroy P. Steele Prize in 1975 for his article Ergodic theory and its significance for statistical mechanics and probability theory.[3]

Mackey was an elected member of the American Academy of Arts and Sciences the National Academy of Sciences and the American Philosophical Society.

Lawrence G. Brown, Paul Chernoff, Edward G. Effros, Calvin Moore, Richard Palais, Caroline Series, John Wermer and Robert Zimmer have been doctoral students of Mackey. Andrew Gleason had no PhD, but considered Mackey to be his advisor.

Books edit

  • Mathematical Foundations of Quantum Mechanics[4] (Dover Books on Mathematics, 2004 ISBN 0-486-43517-2 ISBN 978-0-486-43517-6)[5]
  • Unitary Group Representations in Physics, Probability, and Number Theory, 402 pages, Benjamin–Cummings Publishing Company (1978), ISBN 0-8053-6703-9[6]
  • The Theory of Unitary Group Representations (Chicago Lectures in Mathematics) University Of Chicago Press (August 1, 1976) ISBN 0-226-50051-9[7]
  • 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 I. 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 edit

References edit

  1. ^ "George Mackey - The Mathematics Genealogy Project". www.mathgenealogy.org. Retrieved 2024-01-09.
  2. ^ "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 9, 2021.
  3. ^ Mackey, George W. (1974). "Ergodic theory and its significance for statistical mechanics and probability theory". Advances in Mathematics. 12 (2): 178–268. doi:10.1016/S0001-8708(74)80003-4.
  4. ^ Feldman, Jacob (1967). "Review: Foundations of quantum mechanics, by G. Mackey". Bulletin of the American Mathematical Society. 73 (4): 499–500. doi:10.1090/s0002-9904-1967-11717-8.
  5. ^ Berg, Michael (November 19, 2004). "Joint review of Mathematical Foundations of Quantum Mechanics by George W. Mackey; A Taste of Jordan Algebras by Kevin McCrimmon". MAA Review, Mathematical Association of America.
  6. ^ Gustafson, Karl (1980). "Review: Unitary group representations in physics, probability, and number theory, by G. Mackey". Bulletin of the American Mathematical Society. New Series. 2 (1): 225–229. doi:10.1090/s0273-0979-1980-14731-x.
  7. ^ Kunze, Ray A. (1978). "Review: The theory of unitary group representations, by G. Mackey". Bulletin of the American Mathematical Society. 84 (1): 73–75. doi:10.1090/s0002-9904-1978-14410-3.

External links edit

January 01, 1970
george, mackey, george, whitelaw, mackey, february, 1916, march, 2006, american, mathematician, known, contributions, quantum, logic, representation, theory, noncommutative, geometry, bornfebruary, 1916st, louis, missouri, diedmarch, 2006, 2006, aged, belmont,. George Whitelaw Mackey February 1 1916 March 15 2006 was an American mathematician known for his contributions to quantum logic representation theory and noncommutative geometry George MackeyBornFebruary 1 1916St Louis Missouri U S DiedMarch 15 2006 2006 03 16 aged 90 Belmont Massachusetts U S EducationRice University BA Harvard University PhD Known forLocally convex spacesMackey theoryQuantum logicMackey topologyMackey spaceMackey Arens theoremMackey Borel structureAwardsLeroy P Steele Prize 1975 Scientific careerFieldsMathematical analysisInstitutionsHarvard UniversityThesisThe Subspaces of the Conjugate of an Abstract Linear Space 1942 Doctoral advisorMarshall H StoneDoctoral studentsJohn V BreakwellLawrence G BrownPaul ChernoffEdward G EffrosCalvin C MooreRichard PalaisCaroline SeriesJohn WermerRobert ZimmerOther notable studentsAndrew M Gleason Contents 1 Career 2 Work 3 Honours and students 4 Books 5 See also 6 References 7 External linksCareer editMackey earned his B A at Rice University in 1938 and obtained his Ph D at Harvard University in 1942 under the direction of Marshall H Stone 1 He joined the Harvard University Mathematics Department in 1943 was appointed Landon T Clay Professor of Mathematics and Theoretical Science in 1969 and remained there until he retired in 1985 Work editEarlier 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 s work on topological tensor products Mackey was one of the pioneer workers in the intersection of quantum logic the theory of infinite dimensional unitary representations of groups the theory of operator algebras and noncommutative geometry A central role in Mackey s work both in the theory of group representations and in mathematical physics was played by the concepts of system of imprimitivity and induced representations 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 groups using the method of orbits developed by Alexandre Kirillov in the 1960s His notion of virtual subgroup introduced in 1966 using the language of groupoids had a significant influence in ergodic theory Another essential ingredient in Mackey s work was the assignment of a Borel structure to the dual object of a locally compact group specifically a locally compact separable metric group G One of Mackey s important conjectures which was eventually solved by work of James Glimm 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 and statistical mechanics Honours and students editMackey was among the first five recipients of William Lowell Putnam fellowships in 1938 2 He received the Leroy P Steele Prize in 1975 for his article Ergodic theory and its significance for statistical mechanics and probability theory 3 Mackey was an elected member of the American Academy of Arts and Sciences the National Academy of Sciences and the American Philosophical Society Lawrence G Brown Paul Chernoff Edward G Effros Calvin Moore Richard Palais Caroline Series John Wermer and Robert Zimmer have been doctoral students of Mackey Andrew Gleason had no PhD but considered Mackey to be his advisor Books editMathematical Foundations of Quantum Mechanics 4 Dover Books on Mathematics 2004 ISBN 0 486 43517 2 ISBN 978 0 486 43517 6 5 Unitary Group Representations in Physics Probability and Number Theory 402 pages Benjamin Cummings Publishing Company 1978 ISBN 0 8053 6703 9 6 The Theory of Unitary Group Representations Chicago Lectures in Mathematics University Of Chicago Press August 1 1976 ISBN 0 226 50051 9 7 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 I 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 5See also editBornological spaceReferences edit George Mackey The Mathematics Genealogy Project www mathgenealogy org Retrieved 2024 01 09 Putnam Competition Individual and Team Winners Mathematical Association of America Retrieved December 9 2021 Mackey George W 1974 Ergodic theory and its significance for statistical mechanics and probability theory Advances in Mathematics 12 2 178 268 doi 10 1016 S0001 8708 74 80003 4 Feldman Jacob 1967 Review Foundations of quantum mechanics by G Mackey Bulletin of the American Mathematical Society 73 4 499 500 doi 10 1090 s0002 9904 1967 11717 8 Berg Michael November 19 2004 Joint review of Mathematical Foundations of Quantum Mechanics by George W Mackey A Taste of Jordan Algebras by Kevin McCrimmon MAA Review Mathematical Association of America Gustafson Karl 1980 Review Unitary group representations in physics probability and number theory by G Mackey Bulletin of the American Mathematical Society New Series 2 1 225 229 doi 10 1090 s0273 0979 1980 14731 x Kunze Ray A 1978 Review The theory of unitary group representations by G Mackey Bulletin of the American Mathematical Society 84 1 73 75 doi 10 1090 s0002 9904 1978 14410 3 External links editO Connor John J Robertson Edmund F George Mackey MacTutor History of Mathematics Archive University of St Andrews George Mackey at the Mathematics Genealogy Project George Mackey 1916 2006 Notices of the American Mathematical Society vol 54 no 7 August 2007 George Mackey 1 February 1916 15 March 2006 permanent dead link 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 First letter Second letter Speech Retrieved from https en wikipedia org w index php title George Mackey amp oldid 1224054446, wikipedia, wiki, book, books, library,

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