Joseph Bernstein is an Israeli mathematician, born and educated in Russia (his Russian name is Иосиф Наумович Бернштейн, transliterated as Josif/Iosif Naumovič Bernštejn); his teacher there was Israel Gel'fand. His numerous works (especically with Alexander Beilinson) brought to proliferation of sheaf-theoretic and algebro-geometric methods in representation theory (D-modules, perverse sheaves, Beilinson-Bernstein localization theory), and are now cornerstones of geometric representation theory. With V. Lunts he introduced the appropriate definition of equivariant derived category (in general not equivalent to the (naive) derived category of equivariant sheaves).
Introducing what came to be called BGG resolutions of Lie group representations by Verma modules built using regular differential operators:
On integral differential forms on supermanifolds and the Stokes theorem for supermanifolds:
Introducing the notion of perverse sheaves (and of t-structures on triangulated categories):
See also:
I. N. Bernšteĭn, I. M. Gelʹfand, V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem. (Russian) Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33.
J. N. Bernstein, I. M. Gel’fand, S. I. Gel’fand, Алгебраические расслоения на и задачи линейной алгебры, Функц. анализ и его прил., 12:3 (1978), 66–67, pdf (Russian); Engl. transl. Algebraic bundles over and problems of linear algebra, Funct. Anal. and its Appl. 1978, 12:3, 212–214
A. Beilinson, J. Bernstein, Localisations de –modules, C. R. Acad. Sci. Paris 292 (1981), 15–18.
J. Bernstein, V. Lunts, On nonholonomic irreducible D-modules, Invent. Math. 94, (1988), no. 2, 223–243.
J. Bernstein, V. Lunts, Equivariant sheaves and functors, Springer Lecture Notes in Math. 1578 (1994). MR95k:55012
A. Beĭlinson, J. Bernstein, A proof of Jantzen conjectures, I. M. Gelʹfand Seminar, 1–50, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc. 1993, pdf
On the Langlands program:
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