Born18 February 1938
Died20 September 2019 (aged 81)
Alma materMoscow State University
Known forDiffiety, Vinogradov sequence, Secondary calculus
Scientific career

Alexandre Mikhailovich Vinogradov (Russian: Александр Михайлович Виноградов; 18 February 1938 – 20 September 2019) was a Russian and Italian mathematician. He made important contributions to the areas of differential calculus over commutative algebras, the algebraic theory of differential operators, homological algebra, differential geometry and algebraic topology, mechanics and mathematical physics, the geometrical theory of nonlinear partial differential equations and secondary calculus.

Biography

A.M. Vinogradov was born on 18 February 1938 in Novorossiysk. His father, Mikhail Ivanovich Vinogradov, was a hydraulics scientist, his mother, Ilza Alexandrovna Firer, was a medical doctor. Among his more distant ancestors, his great-grandfather Anton Smagin, a self-taught peasant and a deputy of the State Duma of the second convocation stand out.

In 1955 A.M. Vinogradov entered the Mechanics and Mathematics Department of Moscow State University (Mech-mat), began his Ph.D. in 1960 and completed it in 1964. In 1965, he received a position at the Department of Higher Geometry and Topology of Moscow State University, where he worked until he left the Soviet Union for Italy in 1990. He obtained the next degree (doktorskaya dissertatsiya) in 1984 at the Institute of Mathematics of the Siberian Branch of the USSR Academy of Science in Novosibirsk in Russia. From 1993 to 2010, he held the position of professor in geometry at the University of Salerno in Italy.

Work

Vinogradov published his first works in number theory, together with B.N. Delaunay and D.B. Fuchs when he was a second year undergraduate student. By the end of his undergraduate years, he was contributing to the A.S. Schwartz seminar, and started working on algebraic topology. His PhD thesis (under the formal supervision of V.G. Boltyansky) was devoted to homotopic properties of the embedding spaces of circles into the 2-sphere or the 3-disk. Vinogradov continued working in algebraic and differential topology – in particular, on the Adams spectral sequence – until the early seventies, and he started his own research seminar in 1967. Between the sixties and the seventies, inspired by the ideas of Sophus Lie, he began to investigate the foundations of the geometric theory of partial differential equations. Having become familiar with the work of Spencer, Goldschmidt and Quillen on formal integrability, he turned his attention to the algebraic (in particular, cohomological) component of that theory. In 1972, the short note in Soviet Math Doklady (publishing long texts in the Soviet Union at the time was very difficult) entitled “The logic algebra of the theory of linear differential operators” [1], contained what Vinogradov himself called the main functors of the differential calculus over commutative algebras.

Vinogradov’s approach to nonlinear differential equations as geometric objects, with their general theory and applications, is developed in detail in the monographs [2], [3] and [4], as well as in some articles [5], [6], [22]. He united infinitely prolonged differential equations into a category [7] whose objects, called diffieties (= differential varieties), are studied in the framework of what he called secondary calculus (by analogy with secondary quantization) [8], [9]. One of the central parts of this theory is based on the ${\displaystyle {\cal {C}}}$-spectral sequence (now known as the Vinogradov spectral sequence) [10], [11]. The first term of this spectral sequence gives a unified cohomological approach to various notions and statements, including the Lagrangian formalism with constraints, conservation laws, cosymmetries, the Noether theorem, and the Helmholtz criterion in the inverse problem of the calculus of variations (for arbitrary nonlinear differential operators). A particular case of the ${\displaystyle {\cal {C}}}$-spectral sequence (for an “empty” equation, i.e., for the space of infinite jets) is the so-called variational bicomplex (see also the n-lab article).

Furthermore, Vinogradov introduced the construction of a new bracket on the graded algebra of linear transformations of a cochain complex [12]. The Vinogradov bracket is skew-symmetric and satisfies the Jacobi identity modulo a coboundary. Vinogradov’s construction precursed the general concept of a derived bracket on a differential Loday (or Leibniz) algebra introduced by Y. Kosmann-Schwarzbach in 1996 [13]. These results were also applied to Poisson geometry [14], [15].

Furthermore, together with coauthors, Vinogradov was concerned with the analysis and comparison of various generalizations of Lie (super) algebras, including ${\displaystyle L_{\infty }}$ algebras and Filippov algebras [16].

The research interests of Alexandre M. Vinogradov were also motivated by problems of contemporary physics – for example the structure of Hamiltonian mechanics [23], [24], the dynamics of acoustic beams [17], the equations of magnetohydrodynamics (the so-called Kadomtsev-Pogutse equations appearing in the stability theory of high-temperature plasma in tokamaks) [18] and mathematical questions in general relativity [19], [20], [21]. Considerable attention to the mathematical understanding of the fundamental physical notion of observable is given in the book [4], written by Vinogradov jointly with several participants of his seminar under the pen name of Jet Nestruev.

Contribution to the mathematical community

Prof. A. M. Vinogradov during a lecture

From 1967 until 1990, Vinogradov headed a research seminar at Mekhmat MSU.

From 1998 to 2019, Vinogradov organized and directed the so-called Diffiety Schools in Italy, Russia, and Poland in which were taught the ideas about differential calculus over commutative algebras, the algebraic theory of differential operators, the geometrical theory of nonlinear partial differential equations, the concept of a diffiety, the Vinogradov (C-spectral) sequence and secondary calculus.

He also organized a series of small conferences called “Current Geometry” that took place in Italy from 2000 to 2010, as well as the large Moscow conference “Secondary Calculus and Cohomological Physics” (1997) [9]. Vinogradov was one of the initial organizers of the Schrödinger International Institute in Mathematical Physics in Vienna, as well as of the mathematical journal Differential Geometry and its Applications, remaining one of the editors to his last days.

In 1985, he created a laboratory that studied various aspects of the geometry of differential equations at the Institute of Programming Systems in Pereslavl-Zalessky and headed it until he left for Italy. In 1978, he was one of the organizers and first lecturers in the so-called People's University for students who were not accepted to Mekhmat because they were ethnically Jewish (he ironically called this school the “People’s Friendship University”).

References

1. Vinogradov, A. M. (1972), "The logic algebra for the theory of linear differential operators", Dokl. Akad. Nauk SSSR (in Russian), 205 (5): 1025–1028. English translation: "The logic algebra for the theory of linear differential operators", Soviet Math. Dokl., 13: 1058–1062, 1972.
2. Vinogradov, A.M.; I.S. Krasil’shchik, V.V. Lychagin (1986). Introduction to the geometry of nonlinear differential equations (in Russian). Nauka, Moscow. p. 336. English translation: Introduction to the geometry of nonlinear differential equations. Gordon and Breach science publishers. 1986. p. 441. ISBN 2-88124-051-8.
3. Bocharov, A.V.; A.M. Verbovetsky, A.M. Vinogradov, et al. (I.S. Krasilshchik, A.M. Vinogradov, eds.) (2005). Symmetries and conservation laws for differential equations of mathematical physics. Factorial Press - 380 pp.CS1 maint: multiple names: authors list (link). English translation: I. S. Krasil'shchik, A. M. Vinogradov (eds.) (1999), Symmetries and conservation laws for differential equations of mathematical physics, Transl. Math. Monogr., 182, Providence, RI: American Mathematical Society, ISBN 0-8218-0958-XCS1 maint: extra text: authors list (link).
4. Nestruev, Jet. Smooth manifolds and observables (PDF) (in Russian). MCCME, Moscow, 2000. 300 pp.. English translation: J. Nestruev (2003), Smooth manifolds and observables, Grad. Texts in Math., 220, New York: Springer-Verlag, doi:10.1007/b98871, ISBN 0-387-95543-7.
5. Vinogradov, A.M. (1981), "The geometry of nonlinear differential equations" (PDF), Journal of Soviet Mathematics, 17: 1624–1649, doi:10.1007/BF01084594, S2CID 121310561
6. Vinogradov, A.M. (1984), "Local symmetries and conservation laws", Acta Appl. Math., 2: 21–78, doi:10.1007/BF01405491, S2CID 121860845
7. Vinogradov, A.M. (1984), "Category of partial differential equations", Lecture Notes in Mathematics, 1108: 77–102, doi:10.1007/BFb0099553
8. Vinogradov, A.M. (1998), "Introduction to Secondary Calculus" (PDF), Contemporary Mathematics, 219, Providence, Rhode Island: American Mathematical Society, pp. 241–272
9. Vinogradov, A.M.; M. Henneaux and I.S. Krasil’shchik (eds) (1997). Secondary Calculus and Cohomological Physics. Proc. Conf. Secondary Calculus and Cohomological Physics, August 24–31, 1997, Moscow; Contemporary Mathematics, 1998, V. 219. Amer. Math. Soc., Providence, Rhode Island. doi:10.1090/conm/219/03079.CS1 maint: extra text: authors list (link)
10. Vinogradov, A.M. (1978), "A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints" (PDF), Dokl. Akad. Nauk SSSR (in Russian), 238 (5): 1028–1031. English translation: Soviet Math. Dokl., 19 (1978), 144–148.
11. A. M. Vinogradov (1984), "The ${\displaystyle {\cal {C}}}$-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory", J. Math. Anal. Appl., 100:1: 1–40, doi:10.1016/0022-247X(84)90071-4; A. M. Vinogradov (1984), "The ${\displaystyle {\cal {C}}}$-spectral sequence, Lagrangian formalism, and conservation laws.II. The nonlinear theory", J. Math. Anal. Appl., 100 (1): 41–129, doi:10.1016/0022-247X(84)90072-6.
12. Vinogradov, A.M. (1990), "The union of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators" (PDF), Mat. Zametki (in Russian), 47 (6): 138–140
13. Kosmann-Schwarzbach, Y. (1996), "From Poisson algebras to Gerstenhaber algebras" (PDF), Ann. Inst. Fourier, 46 (5): 1241–1272, doi:10.5802/aif.1547
14. Cabras, A.; A.M Vinogradov (1992), "Extensions of the Poisson bracket to differential forms and multi-vector fields", J. Geom. Phys., 9 (1): 75–100, Bibcode:1992JGP.....9...75C, doi:10.1016/0393-0440(92)90026-W
15. Marmo, G.; G. Vilasi, A.M. Vinogradov (1998), "The local structure of n-Poisson and n-Jacobi manifolds", J. Geom. Phys., 25 (1–2): 141–182, arXiv:physics/9709046, Bibcode:1998JGP....25..141M, doi:10.1016/S0393-0440(97)00057-0
16. Michor, P.W.; A.M. Vinogradov (1996), "n-ary Lie and associative algebras", Rend. Sem. Mat. Univ. Pol. Torino, 53 (3): 373–392, arXiv:math/9801087, Bibcode:1998math......1087M, zbMath.
17. Vinogradov, A.M.; Vorobjev, E.M. (1976), "Application of symmetries to finding exact solutions of the Zabolotskaya-Khokhlov equation" (PDF), Acoust. J., 22 (1): 23–27
18. Gusyatnikova, V.N.; A.V. Samokhin, V.S. Titov, A.M. Vinogradov, V.A. Yumaguzhin (1989), "Symmetries and conservation laws of Kadomtsev-Pogutse equations", Acta Appl. Math., 15 (1–2): 23–64, doi:10.1007/BF00131929, S2CID 124794448CS1 maint: multiple names: authors list (link)
19. Sparano, G.; G. Vilasi, A.M. Vinogradov (2002), "Vacuum Einstein metrics with bidimensional Killing leaves. I. Local aspects", Differential Geometry and Its Applications, 16: 95–120, arXiv:gr-qc/0301020, doi:10.1016/S0926-2245(01)00062-6, S2CID 7992539
20. Sparano, G.; G. Vilasi, A.M. Vinogradov (2002), "Vacuum Einstein metrics with bidimensional Killing leaves. II. Global aspects", Differential Geometry and Its Applications, 17: 15–35, doi:10.1016/S0926-2245(02)00078-5
21. Sparano, G.; G. Vilasi, A.M. Vinogradov (2001), "Gravitational fields with a non-Abelian, bidimensional Lie algebra of symmetries", Physics Letters B, 513 (1–2): 142–146, arXiv:gr-qc/0102112, Bibcode:2001PhLB..513..142S, doi:10.1016/S0370-2693(01)00722-5, S2CID 15766049
22. Vinogradov, A.M. (2016), "Logic of differential calculus and the Zoo of geometric structures", Banach Center Publications, 110: 257–285, doi:10.4064/bc110-0-17, S2CID 119632868
23. Vinogradov, A.M.; I.S. Krasil’shchik (1975), "What is the Hamiltonian formalism?" (PDF), Russian Mathematical Surveys, 30 (1): 177–202, Bibcode:1975RuMaS..30..177V, doi:10.1070/RM1975v030n01ABEH001403
24. Vinogradov, A.M.; B.A. Kupershmidt (1977), "The structures of Hamiltonian mechanics" (PDF), Russian Mathematical Surveys, 32 (4): 177–243, Bibcode:1977RuMaS..32..177V, doi:10.1070/RM1977v032n04ABEH001642