Bertram Kostant

Bertram Kostant
Bertram Kostant at a workshop on “Enveloping Algebras and Geometric Representation Theory” in Oberwolfach, 2009
Born	(1928-05-24)May 24, 1928 Brooklyn, New York
Died	February 2, 2017(2017-02-02) (aged 88) Roslindale, Massachusetts
Nationality	American
Alma mater	Purdue University University of Chicago (PhD)
Known for	Kostant partition function
Awards	Wigner Medal (2016)
Scientific career
Fields	Mathematics
Institutions	Massachusetts Institute of Technology University of California, Berkeley
Thesis	Representations of a Lie algebra and its enveloping algebra on a Hilbert space
Doctoral advisor	Irving Segal
Doctoral students	Marjorie Batchelor James Lepowsky Arlie Petters Stephen Rallis James Harris Simons Birgit Speh Moss Sweedler David Vogan

Bertram Kostant (May 24, 1928 – February 2, 2017)^[1] was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.

Early life and education[]

Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945.^[2] He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D. from the University of Chicago in 1954, under the direction of Irving Segal, where he wrote a dissertation on representations of Lie groups.

Career in mathematics[]

After time at the Institute for Advanced Study, Princeton University, and the University of California, Berkeley, he joined the faculty at the Massachusetts Institute of Technology, where he remained until his retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. He has given several lectures on the Lie group E8.^[3] He has been one of the principal developers of the theory of geometric quantization. His introduction of the theory of prequantization has led to the theory of quantum Toda lattices. The Kostant partition function is named after him. With Gerhard Hochschild and Alex F. T. W. Rosenberg, he is one of the namesakes of the Hochschild–Kostant–Rosenberg theorem which describes the Hochschild homology of some algebras.^[4]

His students include James Harris Simons, James Lepowsky, Moss Sweedler, David Vogan, and Birgit Speh. At present he has more than 100 mathematical descendants.

Awards and honors[]

Kostant's honors include election to the National Academy of Sciences in 1978. In 2012 he became a fellow of the American Mathematical Society.^[5] He was an Invited Speaker of the ICM with talk Orbits and quantization theory in 1970 at Nice.^[6]

Selected publications[]

• Kostant, Bertram (1955). "Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold". Trans. Amer. Math. Soc. 80 (2): 528–542. doi:10.1090/S0002-9947-1955-0084825-8.
• Kostant, Bertram (1959). "A formula for the multiplicity of a weight". Trans. Amer. Math. Soc. 93 (6): 53–73. doi:10.1090/S0002-9947-1959-0109192-6. PMC 528626. PMID 16590246.
• Kostant, Bertram (1961). "Lie algebra cohomology and the generalized Borel-Weil theorem" (PDF). Annals of Mathematics. 74 (2): 329–387. doi:10.2307/1970237. hdl:2027/mdp.39015095258318. JSTOR 1970237.
• Kostant, Bertram (1963). "Lie group representations on polynomial rings". American Journal of Mathematics. 85 (3): 327–404. doi:10.2307/2373130. JSTOR 2373130.
• Kostant, Bertram (1969). "On the existence and irreducibility of certain series of representations". Bulletin of the American Mathematical Society. 75 (4): 627–642. doi:10.1090/S0002-9904-1969-12235-4.
• Kostant, Bertram (1970). "Quantization and unitary representations". In: Lectures in modern analysis and applications III. Lecture Notes in Mathematics 170. 170. pp. 87–208. doi:10.1007/BFb0079068. ISBN 978-3-540-05284-5.
• with Louis Auslander: Auslander, L.; Kostant, B. (1971). "Polarization and unitary representations of solvable Lie groups". Inventiones Mathematicae. 14 (4): 255–354. Bibcode:1971InMat..14..255A. doi:10.1007/BF01389744. S2CID 122009744.
• with Stephen Rallis: Kostant, B.; Rallis, S. (1971). "Orbits and representations associated with symmetric spaces". American Journal of Mathematics. 93 (3): 753–809. doi:10.2307/2373470. JSTOR 2373470.
• Kostant, Bertram (1973). "On convexity, the Weyl group and the Iwasawa decomposition". Annales Scientifiques de l'École Normale Supérieure. 6 (4): 413–455. doi:10.24033/asens.1254.
• Kostant, Bertram (1977). "Graded manifolds, graded Lie theory, and prequantization". In: Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Math 570. 570. pp. 177–306. doi:10.1007/Bfb0087788. ISBN 978-3-540-08068-8.
• Kostant, Bertram (1978). "On Whittaker vectors and representation theory". Inventiones Mathematicae. 48 (2): 101–184. Bibcode:1978InMat..48..101K. doi:10.1007/BF01390249. S2CID 122765132.
• with David Kazhdan and Shlomo Sternberg: Kazhdan, D.; Kostant, B.; Sternberg, S. (1978). "Hamiltonian group actions and dynamical systems of Calogero type". Communications on Pure and Applied Mathematics. 31 (4): 481–507. doi:10.1002/cpa.3160310405.
• Kostant, Bertram (1979). "The solution to a generalized Toda lattice and representation theory". Advances in Mathematics. 34 (3): 195–338. doi:10.1016/0001-8708(79)90057-4.
• with Shrawan Kumar: Kostant, Bertram; Kumar, Shrawan (1986). "The nil Hecke ring and cohomology of GP for a Kac-Moody group G". Advances in Mathematics. 62 (3): 187–237. doi:10.1016/0001-8708(86)90101-5. PMC 323118. PMID 16593661.
• with Shlomo Sternberg: Kostant, Bertram; Sternberg, Shlomo (1987), "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras", Annals of Physics, 176 (1): 49–113, Bibcode:1987AnPhy.176...49K, doi:10.1016/0003-4916(87)90178-3
• "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of ${\displaystyle {\mbox{SL}}(2,\mathbb {R} )}$". Represent. Theory. 4: 181–224. 2000. doi:10.1090/S1088-4165-00-00096-0.
• with Gerhard Hochschild and Alex Rosenberg: Hochschild, G.; Kostant, Bertram; Rosenberg, Alex (2009). "Differential forms on regular affine algebras". In: Collected Papers. New York: Springer. pp. 265–290. doi:10.1007/b94535_14. ISBN 978-0-387-09582-0. (Reprinted from Trans. Amer. Math. Soc., vol. 102, no. 3, March 1962, pp. 383–408)
• Kostant, Bertram (2009). "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group". In: Collected Papers. pp. 130–189. doi:10.1007/b94535_11. ISBN 978-0-387-09582-0. (Reprinted from the Amer. J. Math., vol. 81, no. 4, Oct. 1959)

See also[]

• Kostant's convexity theorem
• Kostant polynomial

Notes[]

References[]

• Kostant's home page at MIT
• Bertram Kostant at the Mathematics Genealogy Project
• Brylinski, Jean-Luc (editor) (1994). Lie Theory and Geometry, In Honor of Bertram Kostant. Boston, Birkhäuser. ISBN 0-8176-3761-3.CS1 maint: extra text: authors list (link)