Bertram Kostant

Bertram Kostant, an American mathematician of Jewish heritage, was a towering figure whose intellectual contributions profoundly shaped fields such as representation theory, differential geometry, and mathematical physics. His work, characterized by its depth and originality, continues to resonate through the mathematical landscape.

Early Life and Education

Born on May 24, 1928, in Brooklyn, New York, Kostant’s early life was steeped in the intellectual vibrancy of the city. He distinguished himself early on, graduating from Stuyvesant High School in 1945. This academic prowess propelled him to Purdue University, where he pursued undergraduate studies in mathematics, earning his degree in 1950. The call of deeper mathematical inquiry led him to the hallowed halls of the University of Chicago, where he completed his Ph.D. in 1954. His doctoral work, conducted under the tutelage of the esteemed Irving Segal, delved into the intricate representations of Lie groups, laying the groundwork for much of his later groundbreaking research. His dissertation, titled "Representations of a Lie algebra and its enveloping algebra on a Hilbert space," was a testament to his burgeoning talent and foresight.

Career in Mathematics

Kostant’s post-doctoral journey was a veritable tour of the world’s most prestigious mathematical institutions. He spent time at the Institute for Advanced Study in Princeton, a sanctuary for groundbreaking thought, and also engaged with the vibrant academic community at the University of California, Berkeley. It was at the Massachusetts Institute of Technology (MIT), however, that he found a long-term intellectual home. Joining the faculty, he remained a cornerstone of the MIT mathematics department until his retirement in 1993.

His research interests were vast and interconnected, weaving through representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry, and mathematical physics. He was particularly drawn to the elegance of symplectic geometry. Kostant's lectures on the Lie group E8, a subject of immense complexity and beauty, were legendary, drawing mathematicians and physicists alike.

He stands as one of the principal architects of the theory of geometric quantization, a framework that seeks to bridge the gap between classical and quantum mechanics. His introduction of the concept of prequantization, a crucial step in this process, paved the way for the development of the theory of quantum Toda lattices. The Kostant partition function, a fundamental tool in the study of Lie algebras, bears his name, a testament to its significance.

In collaboration with Gerhard Hochschild and Alex F. T. W. Rosenberg, Kostant contributed to the formulation of the Hochschild–Kostant–Rosenberg theorem. This theorem is pivotal in understanding the Hochschild homology of certain algebras, offering deep insights into their algebraic structure.

Kostant’s influence extended far beyond his own publications; he was a dedicated mentor whose students went on to achieve remarkable success. His doctoral students included such luminaries as James Harris Simons, the founder of Renaissance Technologies, James Lepowsky, Moss Sweedler, David Vogan, and Birgit Speh. The mathematical lineage stemming from Kostant is extensive, with over a hundred mathematicians tracing their academic ancestry back to him.

Awards and Honors

The mathematical community recognized Kostant’s profound contributions with numerous accolades. He was a Guggenheim Fellow from 1959 to 1960, spending time in Paris, and a Sloan Fellow from 1961 to 1963, further solidifying his position as a rising star. In 1962, he was elected to the prestigious American Academy of Arts and Sciences, and in 1978, he received the even greater honor of election to the National Academy of Sciences. The Sackler Institute for Advanced Studies at Tel Aviv University welcomed him as a fellow in 1982.

In 1990, the American Mathematical Society bestowed upon him the Steele Prize, acknowledging the seminal impact of his 1975 paper, “On the existence and irreducibility of certain series of representations.”

His tenure as a Chern Lecturer and Chern Visiting Professor at Berkeley in 2001 further underscored his esteemed status. Kostant received honorary degrees from institutions worldwide: the University of Córdoba in Argentina in 1989, the University of Salamanca in Spain in 1992, and his alma mater, Purdue University, in 1997, which awarded him an honorary Doctor of Science degree. This last honor was a poignant recognition of his foundational contributions and the inspiration he provided to countless researchers.

In May 2008, the Pacific Institute for Mathematical Sciences organized a conference titled “Lie Theory and Geometry: the Mathematical Legacy of Bertram Kostant” at the University of British Columbia. This event, held in celebration of his 80th birthday, brought together scholars to honor his life and work. In 2012, he was inducted into the inaugural class of fellows of the American Mathematical Society.

In the final year of his life, Kostant attended the Colloquium on Group Theoretical Methods in Physics in Rio de Janeiro, where he was awarded the distinguished Wigner Medal. This prestigious award recognized his “fundamental contributions to representation theory that led to new branches of mathematics and physics,” a fitting capstone to a career dedicated to unraveling the universe’s mathematical underpinnings.

Selected Publications

Kostant's prolific output resulted in a body of work that continues to be studied and referenced. His publications span a wide range of topics, reflecting the breadth of his mathematical interests.

1955. "Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold." Transactions of the American Mathematical Society, 80(2), 528–542. This early work explored the intricate relationship between differential geometry and Lie algebra theory.
1959. "A formula for the multiplicity of a weight." Transactions of the American Mathematical Society, 93(6), 53–73. This paper introduced a significant formula for calculating the multiplicities of weights in representation theory.
1961. "Lie algebra cohomology and the generalized Borel-Weil theorem." Annals of Mathematics, 74(2), 329–387. This work provided a crucial link between Lie algebra cohomology and the [Borel-Weil theorem](/Borel–Weil theorem), a cornerstone of representation theory.
1963. "Lie group representations on polynomial rings." American Journal of Mathematics, 85(3), 327–404. Here, Kostant investigated the action of Lie groups on polynomial rings, revealing deep structural properties.
1969. "On the existence and irreducibility of certain series of representations." Bulletin of the American Mathematical Society, 75(4), 627–642. This paper, later recognized with the Steele Prize, dealt with fundamental questions about the existence and irreducibility of representation series.
1970. "Quantization and unitary representations." In Lectures in modern analysis and applications III (pp. 87–208). Lecture Notes in Mathematics, vol. 170. Springer. This work was instrumental in the development of geometric quantization.
1971. With Louis Auslander. "Polarization and unitary representations of solvable Lie groups." Inventiones Mathematicae, 14(4), 255–354. A significant contribution to the understanding of unitary representations of solvable Lie groups.
1971. With Stephen Rallis. "Orbits and representations associated with symmetric spaces." American Journal of Mathematics, 93(3), 753–809. This paper explored the deep connections between orbits of Lie groups and their representations.
1973. "On convexity, the Weyl group and the Iwasawa decomposition." Annales Scientifiques de l'École Normale Supérieure, 6(4), 413–455. This work connected concepts of convexity, the Weyl group, and the Iwasawa decomposition.
1977. "Graded manifolds, graded Lie theory, and prequantization." In Differential Geometrical Methods in Mathematical Physics (pp. 177–306). Lecture Notes in Math, vol. 570. Springer. Kostant further developed the theory of prequantization within the framework of graded manifolds.
1978. "On Whittaker vectors and representation theory." Inventiones Mathematicae, 48(2), 101–184. This paper introduced and studied Whittaker vectors, which play a crucial role in the representation theory of reductive Lie groups.
1978. With David Kazhdan and Shlomo Sternberg. "Hamiltonian group actions and dynamical systems of Calogero type." Communications on Pure and Applied Mathematics, 31(4), 481–507. This work connected Hamiltonian mechanics with group actions and specific types of dynamical systems.
1979. "The solution to a generalized Toda lattice and representation theory." Advances in Mathematics, 34(3), 195–338. Kostant provided a representation-theoretic solution to the generalized Toda lattice, a significant result in mathematical physics.
1986. With Shrawan Kumar. "The nil Hecke ring and cohomology of GP for a Kac-Moody group G." Advances in Mathematics, 62(3), 187–237. This paper explored the structure of Kac-Moody groups through the lens of the nil Hecke ring and cohomology.
1987. With Shlomo Sternberg. "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras." Annals of Physics, 176(1), 49–113. This work delved into the relationship between symplectic reduction, BRST cohomology, and Clifford algebras.
2000. "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of SL(2,R)." Representation Theory, 4, 181–224. This paper connected classical orthogonal polynomials and special functions with the representation theory of SL(2,R).
2009. With Gerhard Hochschild and Alex Rosenberg. "Differential forms on regular affine algebras." In Collected Papers (pp. 265–290). Springer. A reprint of their influential 1962 work on differential forms over affine algebras.
2009. "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group." In Collected Papers (pp. 130–189). Springer. This reprinted work, originally from 1959, explored the relationship between subgroups of simple Lie groups and their Betti numbers.

Selected Publications (Further Detail)

Kostant’s publications were not merely academic exercises; they were landmarks that charted new territories in mathematics. His 1955 paper on "Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold" was an early indication of his ability to connect seemingly disparate areas of mathematics. By examining the holonomy of Riemannian manifolds, he revealed deep connections to the Lie algebra governing infinitesimal motions.

His 1959 paper, "A formula for the multiplicity of a weight," provided a concrete and elegant method for calculating a crucial quantity in representation theory. The multiplicity of a weight in an irreducible representation is a fundamental invariant, and Kostant’s formula offered a powerful tool for its determination.

The 1961 Annals of Mathematics paper, "Lie algebra cohomology and the generalized Borel-Weil theorem," is a seminal work that solidified his reputation. It connected the algebraic structures of Lie algebra cohomology with the geometric constructions of the Borel-Weil theorem, offering a unified perspective on the representation theory of Lie groups.

In "Lie group representations on polynomial rings" (1963), Kostant explored how Lie groups act on the rings of polynomials. This revealed intricate patterns and structures, showing how geometric and algebraic properties intertwine.

The 1969 Bulletin of the American Mathematical Society paper, "On the existence and irreducibility of certain series of representations," was particularly influential. It addressed fundamental questions about whether certain families of representations exist and whether they are irreducible, a critical problem in the classification of representations. This work was later recognized with the prestigious Leroy P. Steele Prize from the American Mathematical Society.

Kostant’s contributions to geometric quantization were further elaborated in his 1970 work, "Quantization and unitary representations." Here, he laid out foundational ideas for constructing quantum theories from classical ones, a program that has had far-reaching implications in mathematical physics.

The collaboration with Louis Auslander in 1971, "Polarization and unitary representations of solvable Lie groups," provided deep insights into the structure of unitary representations for a particularly important class of Lie groups. Similarly, his 1971 work with Stephen Rallis, "Orbits and representations associated with symmetric spaces," uncovered fundamental relationships between the geometric structure of symmetric spaces and the representations of associated Lie groups.

Kostant's 1973 paper, "On convexity, the Weyl group and the Iwasawa decomposition," demonstrated a remarkable ability to connect concepts from different areas of mathematics, linking geometric notions of convexity to the algebraic machinery of the Weyl group and the Iwasawa decomposition.

The development of prequantization continued in his 1977 contribution, "Graded manifolds, graded Lie theory, and prequantization." This work extended the framework of quantization to more sophisticated mathematical structures, particularly graded manifolds.

His 1978 paper, "On Whittaker vectors and representation theory," introduced and analyzed Whittaker vectors, which are essential for understanding the representations of reductive Lie groups and have applications in number theory.

The collaboration with David Kazhdan and Shlomo Sternberg in 1978, "Hamiltonian group actions and dynamical systems of Calogero type," explored the interplay between Hamiltonian mechanics, group actions, and specific integrable systems, such as those of Calogero–Moser type.

Kostant’s 1979 paper in Advances in Mathematics, "The solution to a generalized Toda lattice and representation theory," was a significant achievement, providing a complete representation-theoretic solution to the generalized Toda lattice. This demonstrated the power of representation theory to solve problems in mathematical physics.

His collaborations continued to yield important results. With Shrawan Kumar in 1986, he explored the structure of Kac-Moody groups through the study of the nil Hecke ring and cohomology. In 1987, with Shlomo Sternberg, he published "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras," connecting these advanced topics in mathematical physics and differential geometry.

In 2000, his paper "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of SL(2,R)" showcased his continued interest in the connections between special functions and representation theory, particularly for the group SL(2,R).

The collected works published in 2009, including reprints of earlier seminal papers like "Differential forms on regular affine algebras" (with Hochschild and Rosenberg) and "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group," ensured that his foundational contributions remained readily accessible to a new generation of mathematicians.

Notes

Bertram Kostant, professor emeritus of mathematics, dies at 88. MIT News. February 16, 2017.
Professor Kostant's Homepage. MIT Math Department. Retrieved 2007-10-31.
Bertram Kostant (2008-02-12). "On Some Mathematics in Garrett Lisi's 'E8 Theory of Everything'". UC Riverside mathematics colloquium. Retrieved 2008-06-15.
Porter, Tim (April 8, 2014), "Hochschild-Kostant-Rosenberg theorem", nLab.