David Kazhdan

David Kazhdan

David Kazhdan ( Russian Дми́трий Александрович Каждан , Dmitri Alexandrowitsch Kaschdan ; Hebrew דוד קשדן; *  1946 in Moscow ) is an Israeli mathematician who used to work in the Soviet Union and who deals with representation theory.

Life

David Kazhdan is the son of the well-known Byzantinist Alexander Kazhdan and received his doctorate in 1969 in Moscow under Alexander Kirillow . He was then a member of the Israel Gelfand School in Moscow before emigrating to the USA as a Jew in 1975, where he changed his name. He went to Harvard University , where he became a professor. In 2005 he emigrated to Israel and is currently a professor at the Hebrew University and at Harvard.

Kazhdan worked on representation theory of Lie groups , Lie algebras and algebraic groups . There the Kazhdan-Lusztig conjectures about identities between characters of Verma modules (special representations of Lie algebras) are named after him and George Lusztig , which were independently proven by Alexander Beilinson and Joseph Bernstein or Masaki Kashiwara and Jean-Luc Brylinski in 1981. The polynomials defined by him and Lusztig are also used in the conjectures. They are defined with the help of the Weyl groups for the Lie algebras and were originally investigated by Lusztig and Kazhdan in connection with the geometry of Schubert varieties , which are often investigated as examples of singular algebraic varieties. The Kazhdan-Lusztig polynomials provide a kind of measure for the singularity of these varieties and were associated by Kazhdan and Lusztig with the intersectional cohomology according to MacPherson and Goresky . An important concept in group theory is Kazhdans property T .

Since 2006 he has been a member of the Israel Academy of Sciences. His PhD students include Vladimir Voivodski and Alexander Polishchuk. In 1990 he was a MacArthur Fellow . In 1986 he was invited speaker at the International Congress of Mathematicians in Berkeley ( Algebraic Geometry and Representation Theory ). In 1990 he was elected to the National Academy of Sciences and in 2008 to the American Academy of Arts and Sciences . In 2012 he gave a plenary lecture at the European Congress of Mathematicians (ECM) in Krakow ( Representations of affine Kac-Moody groups over local and global fields ). For 2020, Kazhdan was awarded the Shaw Prize in Mathematics.

He has three sons and one daughter. His son Eli is a politician in Israel, and his son Michael is a professor of computer science at Johns Hopkins University .

Web links

• Homepage in Harvard
• Homepage at Hebrew University

Works (selection)

• with Israel Gelfand : Representations of the group GL (n, K) where K is a local field. Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pp. 95-118. Halsted, New York, 1975.
• with George Lusztig : Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979) no. 2, 165-184.
• with Victor Guillemin : Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980) no. 3, 301-312.
• with George Lusztig : Schubert varieties and Poincaré duality. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 185-203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980.
• with Samuel Patterson : Metaplectic forms. Inst. Hautes Études Sci. Publ. Math. No. 59: 35-142 (1984).
• with Pierre Deligne , Marie-France Vignéras : Représentations des algèbres centrales simples p -adiques. Representations of reductive groups over a local field, 33-117, Travaux en Cours, Hermann, Paris, 1984.
• with George Lusztig : Equivariant K -theory and representations of Hecke algebras. II. Invent. Math. 80 (1985) no. 2, 209-231.
• with Yuval Flicker : Metaplectic correspondence. Inst. Hautes Études Sci. Publ. Math. No. 64: 53-110 (1986).
• with George Lusztig : Proof of the Deligne-Langlands conjecture for Hecke algebras. Invent. Math. 87 (1987) no. 1, 153-215.
• with Yuval Flicker : On the symmetric square: unstable local transfer. Invent. Math. 91 (1988) no. 3, 493-504.
• with George Lusztig : Fixed point varieties on affine flag manifolds. Israel J. Math. 62 (1988) no. 2, 129-168.
• with George Lusztig : Tensor structures arising from affine Lie algebras. I, II. J. Amer. Math. Soc. 6: no. 4, 905-947, 949-1011 (1993). III, IV. Ibid. 7 (1994) no. 2, 335-381, 383-453.
• with Pavel Etingof : Quantization of Lie bialgebras. I. Selecta Math. (NS) 2 (1996) no. 1, 1-41. II, III. ibid. 4 (1998) no. 2, 213-231, 233-269. IV. The coinvariant construction and the quantum KZ equations. ibid. 6 (2000), no. 1, 79-104. V. Quantum vertex operator algebras. ibid. 6 (2000), no. 1, 105-130.
• with Alexander Braverman : The spherical Hecke algebra for affine Kac-Moody groups I. Ann. of Math. (2) 174 (2011), no. 3, 1603-1642.

Remarks

1. ↑ In his work earlier cited in English translation as Každan or Kajdan
2. ↑ Representations of Coxeter groups and Hecke algebras. Inventiones Mathematicae Vol. 53, 1979, p. 191