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Bertram Kostant



Bertram Kostant at a workshop on “Enveloping Algebras and Geometric Representation Theory” in Oberwolfach, 2009


Born  
Died  February 2, 2017 
(aged 88)
Nationality  American 
Alma mater  Purdue University University of Chicago (PhD) 
Known for  Kostant's convexity theorem Kostant partition function Kostant polynomial Geometric quantization Kostant–Parthasarathy–Ranga Rao–Varadarajan determinants HochschildKostantRosenberg theorem 
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 

Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.
Contents
Early life and education
Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945. 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. 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.
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 was a Guggenheim Fellow in 195960 (in Paris), and a Sloan Fellow in 196163. In 1962 he was elected to the American Academy of Arts and Sciences, and in 1978 to the National Academy of Sciences. In 1982 he was a fellow of the Sackler Institute for Advanced Studies at Tel Aviv University. In 1990 he was awarded the Steele Prize of the American Mathematical Society, in recognition of his 1975 paper, “On the existence and irreducibility of certain series of representations.”
In 2001, Kostant was a Chern Lecturer and Chern Visiting Professor at Berkeley. He received honorary degrees from the University of Córdoba in Argentina in 1989, the University of Salamanca in Spain in 1992, and Purdue University in 1997. The latter, from his alma mater, was an honorary Doctor of Science degree, citing his fundamental contributions to mathematics and the inspiration he and his work provided to generations of researchers.
In May 2008, the Pacific Institute for Mathematical Sciences hosted a conference: “Lie Theory and Geometry: the Mathematical Legacy of Bertram Kostant,” at the University of British Columbia, celebrating the life and work of Kostant in his 80th year. In 2012, he was elected to the inaugural class of fellows of the American Mathematical Society. In the last year of his life, Kostant traveled to Rio de Janeiro for the Colloquium on Group Theoretical Methods in Physics, where he received the prestigious Wigner Medal, “for his fundamental contributions to representation theory that led to new branches of mathematics and physics.”
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/S00029947195500848258.
 Kostant, Bertram (1959). "A formula for the multiplicity of a weight". Trans. Amer. Math. Soc. 93 (6): 53–73. doi:10.1090/S00029947195901091926. PMC 528626. PMID 16590246.
 Kostant, Bertram (1961). "Lie algebra cohomology and the generalized BorelWeil theorem". Annals of Mathematics 74 (2): 329–387. doi:10.2307/1970237. http://dml.cz/bitstream/handle/10338.dmlcz/141389/ArchMathRetro_04620105_6.pdf.
 Kostant, Bertram (1963). "Lie group representations on polynomial rings". American Journal of Mathematics 85 (3): 327–404. doi:10.2307/2373130. http://projecteuclid.org/euclid.bams/1183525365.
 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/S000299041969122354.
 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 9783540052845.
 with Louis Auslander: Auslander, L.; Kostant, B. (1971). "Polarization and unitary representations of solvable Lie groups". Inventiones Mathematicae 14 (4): 255–354. doi:10.1007/BF01389744.
 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.
 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. http://eudml.org/doc/81923.
 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 9783540080688.
 Kostant, Bertram (1978). "On Whittaker vectors and representation theory". Inventiones Mathematicae 48 (2): 101–184. doi:10.1007/BF01390249.
 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:10.1016/00018708(79)900574.
 with Shrawan Kumar: Kostant, Bertram; Kumar, Shrawan (1986). "The nil Hecke ring and cohomology of GP for a KacMoody group G". Advances in Mathematics 62 (3): 187–237. doi:10.1016/00018708(86)901015. PMC 323118. PMID 16593661.
 with Shlomo Sternberg:
 "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simplyconnected covering group of ". Represent. Theory 4: 181–224. 2000. doi:10.1090/S1088416500000960.
 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 9780387095820. (Reprinted from Trans. Amer. Math. Soc., vol. 102, no. 3, March 1962, pp. 383–408)
 Kostant, Bertram (2009). "The principal threedimensional subgroup and the Betti numbers of a complex simple Lie group". In: Collected Papers. pp. 130–189. doi:10.1007/b94535_11. ISBN 9780387095820. (Reprinted from the Amer. J. Math., vol. 81, no. 4, Oct. 1959)
See also
 Chern's conjecture (affine geometry)
 Supermanifold
 Symplectic spinor bundle
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