Alexandre Mikhailovich Vinogradov

Not to be confused with the Russian mathematicians Ivan Vinogradov (of Vinogradov's theorem) or Askold Vinogradov (of Bombieri–Vinogradov theorem).

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.^[1]

Alexandre Mikhailovich Vinogradov
Александр Михайлович Виноградов
Born	(1938-02-18)18 February 1938 Novorossiysk, Russia
Died	20 September 2019(2019-09-20) (aged 81) Lizzano in Belvedere, Italy
Alma mater	Moscow State University
Known for	Diffiety, Vinogradov sequence, Secondary calculus
Scientific career
Fields	Mathematics
Institutions	Moscow State University University of Salerno
Doctoral advisor	Vladimir Boltyansky and Boris Delaunay
Website	https://diffiety.mccme.ru/curvita/amv.htm https://gdeq.org/Alexandre_Vinogradov

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, was a self-taught peasant and a deputy of the State Duma of the second convocation.^[1]

Between 1955 and 1960 Vinogradov studied at the Mechanics and Mathematics Department of Moscow State University (Mech-mat). He pursued a PhD at the same institution, defending his thesis in 1964, under the supervision of V.G. Boltyansky.^[2]

After teaching for one year at the Moscow Mining Institute, in 1965 he received a position at the Department of Higher Geometry and Topology of Moscow State University. He obtained his habilitation degree (doktorskaya dissertatsiya) in 1984 at the Institute of Mathematics of the Siberian Branch of the USSR Academy of Science in Novosibirsk in Russia. In 1990 he left the Soviet Union for Italy, and from 1993 to 2010 was professor in geometry at the University of Salerno.^[1]

Research

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 changed research interests and started working on algebraic topology. His PhD thesis was devoted to homotopic properties of the embedding spaces of circles into the 2-sphere or the 3-disk. He continued working in algebraic and differential topology – in particular, on the Adams spectral sequence – until the early seventies.^[3]

Between the sixties and the seventies, inspired by the ideas of Sophus Lie, Vinogradov changed once more research interests and 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, he published a short note containing what he called the main functors of the differential calculus over commutative algebras.^[4]

Vinogradov’s approach to nonlinear differential equations as geometric objects, with their general theory and applications, is developed in details in some monographs^[5][6][7] as well as in some articles.^[8][9][10] He recast infinitely prolonged differential equations into a category^[11] whose objects, called diffieties, are studied in the framework of what he called secondary calculus (by analogy with secondary quantization).^[12][13][14] One of the central parts of this theory is based on the ${\displaystyle {\cal {C}}}$ -spectral sequence (now known as the Vinogradov spectral sequence).^[15][16][17] 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.^[18]

Furthermore, Vinogradov introduced a new bracket on the graded algebra of linear transformations of a cochain complex.^[19] The Vinogradov bracket is skew-symmetric and satisfies the Jacobi identity modulo a coboundary. Vinogradov’s construction is a precursor of the general concept of a derived bracket on a differential Leibniz algebra introduced by Kosmann-Schwarzbach in 1996.^[20] These results were also applied to Poisson geometry.^[21][22]

Together with Peter Michor [de], Vinogradov was concerned with the analysis and comparison of various generalizations of Lie (super) algebras, including ${\displaystyle L_{\infty }}$  algebras and Filippov algebras.^[23] He also developed a theory of compatibility of Lie algebra structures and proved that any finite-dimensional Lie algebra over an algebraically closed field or over ${\displaystyle \mathbb {R} }$  can be assembled in a few steps from two elementary constituents, that he called dyons and triadons.^[24][25] Furthermore, he speculated that this particle-like structures could be related to the ultimate structure of elementary particles.

Vinogradov's research interests were also motivated by problems of contemporary physics – for example the structure of Hamiltonian mechanics,^[26][27] the dynamics of acoustic beams,^[28] the equations of magnetohydrodynamics (the so-called Kadomtsev-Pogutse equations appearing in the stability theory of high-temperature plasma in tokamaks)^[29] and mathematical questions in general relativity.^[30][31][10] Considerable attention to the mathematical understanding of the fundamental physical notion of observable is given in a book written by Vinogradov jointly with several participants of his seminar, under the pen name of Jet Nestruev.^[7]

Contribution to the mathematical community

 

Prof. A. M. Vinogradov during a lecture

From 1967 until 1990, Vinogradov headed a research seminar at Mekhmat, which became a prominent feature in the mathematical life of Moscow. In 1978, he was one of the organisers 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”). 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 was its scientific supervisor until his departure for Italy.^[1]

Vinogradov was one of the initial founder of the mathematical journal Differential Geometry and its Applications, remaining one of the editors from 1991 to his last days.^[32] A special issue of the journal, devoted to the geometry of PDEs, was published in his memory.^[33]

In 1993 he was one of the promoters of the Schrödinger International Institute in Mathematical Physics in Vienna.^[34] In 1997 he organised the large conference Secondary Calculus and Cohomological Physics in Moscow,^[13] which was followed by a series of small conferences called Current Geometry that took place in Italy from 2000 to 2010.^[35]

From 1998 to 2019, Vinogradov organised and directed the so-called Diffiety Schools in Italy, Russia, and Poland,^[36] in which a wide range of courses were taught, in order to prepare students and young researchers to work on the theory of diffieties and secondary calculus.^[37][38]

He supervised 19 PhD students.^[2]

References

1. Astashov, A. M.; Astashova, I. V.; Bocharov, A. V.; Buchstaber, V. M.; Vassiliev, V. A.; Verbovetsky, A. M.; Vershik, A. M.; Veselov, A. P.; Vinogradov, M. M.; Vitagliano, L.; Vitolo, R. F. (2020). "Alexandre Mikhailovich Vinogradov (obituary)" (PDF). Russian Mathematical Surveys. 75 (2): 369–375. doi:10.1070/rm9931. ISSN 0036-0279. S2CID 219049017.
2. "Alexandre Vinogradov - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2021-12-11.
3. Vinogradov, A. M. (1960). "О спектральной последовательности Адамса" [On Adams' spectral sequence]. Dokl. Akad. Nauk SSSR (in Russian). 133 (5): 999–1002 – via All-Russian Mathematical Portal.
   English translation: Vinogradov, A. M. (1960). "On Adam's spectral sequence". Soviet Mathematics. Doklady. 1: 910–913. Zbl 0097.16101.
4. Vinogradov, A. M. (1972). "Алгебра логики теории линейных дифференциальных операторов" [The logic algebra for the theory of linear differential operators]. Dokl. Akad. Nauk SSSR (in Russian). 205 (5): 1025–1028 – via All-Russian Mathematical Portal.
   English translation: Vinogradov, A. M. (1972). "The logic algebra for the theory of linear differential operators". Soviet Mathematics. Doklady. 13: 1058–1062. ISSN 0197-6788.
5. Vinogradov, A.M.; Krasil’shchik, I.S.; Lychagin, V.V. (1986). Introduction to the geometry of nonlinear differential equations (in Russian). Moscow: Nauka. p. 336.
   English translation: Vinogradov, A. M.; Krasilʹshchik, I. S.; Lychagin, V. V. (1986). Geometry of jet spaces and nonlinear partial differential equations. New York, N.Y.: Gordon and Breach Science Publishers. ISBN 2-88124-051-8. OCLC 12551635.
6. Bocharov, A. V.; Krasilʹshchik, I. S.; Vinogradov, A. M. (1999). Symmetries and conservation laws for differential equations of mathematical physics. Providence, R.I.: American Mathematical Society. ISBN 978-1-4704-4596-6. OCLC 1031947580.
7. Nestruev, Jet (2000). Smooth manifolds and observables (PDF) (in Russian). Moscow: MCCME. p. 300.
   English translation: Nestruev, Jet (2003). Smooth Manifolds and Observables. Graduate Texts in Mathematics. Vol. 220. New York: Springer-Verlag. doi:10.1007/b98871. ISBN 978-0-387-95543-8. S2CID 117029379.
   Second revised and expanded edition: Nestruev, Jet (2020). Smooth Manifolds and Observables. Graduate Texts in Mathematics. Vol. 220. Cham: Springer International Publishing. doi:10.1007/978-3-030-45650-4. ISBN 978-3-030-45649-8. S2CID 242759997.
8. Vinogradov, A. M. (1980). "Геометрия нелинейных дифференциальных уравнений" [The geometry of nonlinear differential equations]. Itogi Nauki I Tekhniki. Ser. Probl. Geom. 11. Moscow: 89–134 – via All-Russian Mathematical Portal.
   English translation: Vinogradov, A. M. (1981). "Geometry of nonlinear differential equations". Journal of Soviet Mathematics. 17 (1): 1624–1649. doi:10.1007/BF01084594. ISSN 0090-4104. S2CID 121310561.
9. Vinogradov, A. M. (1984). "Local symmetries and conservation laws". Acta Applicandae Mathematicae. 2 (1): 21–78. doi:10.1007/BF01405491. ISSN 0167-8019. S2CID 121860845.
10. Sparano, G.; Vilasi, G.; Vinogradov, A.M. (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.
11. Vinogradov, A. M. (1984), Borisovich, Yurii G.; Gliklikh, Yurii E.; Vershik, A. M. (eds.), "Category of nonlinear differential equations", Global Analysis — Studies and Applications I, Lecture Notes in Mathematics, vol. 1108, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 77–102, doi:10.1007/bfb0099553, ISBN 978-3-540-13910-2, retrieved 2021-12-11
12. Vinogradov, A.M. (1998). "Introduction to Secondary Calculus" (PDF). Contemporary Mathematics. 219. Providence, Rhode Island: American Mathematical Society: 241–272. doi:10.1090/conm/219/03079. ISBN 9780821808283.
13. Vinogradov, Alexandre (1998), Henneaux, Marc; Krasil′shchik, Joseph; Vinogradov, Alexandre (eds.), "Secondary Calculus and Cohomological Physics", Contemporary Mathematics, vol. 219, Providence, Rhode Island: American Mathematical Society, doi:10.1090/conm/219/03079, ISBN 978-0-8218-0828-3, retrieved 2021-12-11
14. Vinogradov, A. M. (2001). Cohomological analysis of partial differential equations and secondary calculus. Providence, R.I.: American Mathematical Society. ISBN 0-8218-2922-X. OCLC 47296188.
15. Vinogradov, A.M. (1978). "Одна спектральная последовательность, связанная с нелинейным дифференциальным уравнением и алгебро-геометрические основания лагранжевой теории поля со связями" [A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints]. Dokl. Akad. Nauk SSSR (in Russian). 238 (5): 1028–1031 – via All-Russian Mathematical Portal.
    English translation: Vinogradov, A.M. (1978). "A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundation of Lagrangian field theory with constraints". Soviet Math. Dokl. 19 (1): 144–148.
16. Vinogradov, A.M. (1984). "The b-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory". Journal of Mathematical Analysis and Applications. 100 (1): 1–40. doi:10.1016/0022-247X(84)90071-4.
17. Vinogradov, A.M. (1984). "The b-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory". Journal of Mathematical Analysis and Applications. 100 (1): 41–129. doi:10.1016/0022-247X(84)90072-6.
18. "variational bicomplex in nLab". ncatlab.org. Retrieved 2021-12-18.
19. Vinogradov, A.M. (1990). "Объединение скобок Схоутена и Нийенхейса, когомологии и супердифференциальные операторы" [The union of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators]. Mat. Zametki (in Russian). 47 (6): 138–140 – via All-Russian Mathematical Portal.
20. Kosmann-Schwarzbach, Yvette (1996). "From Poisson algebras to Gerstenhaber algebras". Annales de l'Institut Fourier. 46 (5): 1243–1274. doi:10.5802/aif.1547. ISSN 0373-0956.
21. Cabras, A.; Vinogradov, A.M. (1992). "Extensions of the poisson bracket to differential forms and multi-vector fields". Journal of Geometry and Physics. 9 (1): 75–100. Bibcode:1992JGP.....9...75C. doi:10.1016/0393-0440(92)90026-W.
22. Marmo, G.; Vilasi, G.; Vinogradov, A.M. (1998). "The local structure of n-Poisson and n-Jacobi manifolds". Journal of Geometry and Physics. 25 (1–2): 141–182. arXiv:physics/9709046. Bibcode:1998JGP....25..141M. doi:10.1016/S0393-0440(97)00057-0. S2CID 119118335.
23. Michor, Peter W.; Vinogradov, Alexandre M. (1998-01-19). "n-ary Lie and Associative Algebras". Rend. Sem. Mat. Univ. Pol. Torino. 53 (3): 373–392. arXiv:math/9801087. Bibcode:1998math......1087M. Zbl 0928.17029.
24. Vinogradov, A. M. (2017). "Particle-like structure of Lie algebras". Journal of Mathematical Physics. 58 (7): 071703. arXiv:1707.05717. Bibcode:2017JMP....58g1703V. doi:10.1063/1.4991657. ISSN 0022-2488. S2CID 119316544.
25. Vinogradov, A. M. (2018). "Particle-like structure of coaxial Lie algebras". Journal of Mathematical Physics. 59 (1): 011703. Bibcode:2018JMP....59a1703V. doi:10.1063/1.5001787. ISSN 0022-2488.
26. Vinogradov, A M; Krasil'shchik, I S (1975-02-28). "Что такое гамильтонов формализм?" [What is the Hamiltonian formalism?]. Russian Mathematical Surveys (in Russian). 30 (1): 177–202. doi:10.1070/RM1975v030n01ABEH001403. ISSN 0036-0279. S2CID 250915291 – via All-Russian Mathematical Portal.
27. Vinogradov, A M; Kupershmidt, B A (1977-08-31). "Структура гамильтоновой механики" [The structures of hamiltonian mechanics]. Russian Mathematical Surveys (in Russian). 32 (4): 177–243. doi:10.1070/RM1977v032n04ABEH001642. ISSN 0036-0279. S2CID 250805957 – via All-Russian Mathematical Portal.
28. Vinogradov, A. M.; Vorobjev, E. M. (1976). "Applications of symmetries to finding exact solutions of the Zabolotskaya-Khokhlov equation" (PDF). Akust. Zhurnal. (in Russian). 22 (1): 23–27.
29. Gusyatnikova, V. N.; Samokhin, A. V.; Titov, V. S.; Vinogradov, A. M.; Yumaguzhin, V. A. (1989). "Symmetries and conservation laws of Kadomtsev-Pogutse equations (Their computation and first applications)". Acta Applicandae Mathematicae. 15 (1–2): 23–64. doi:10.1007/BF00131929. ISSN 0167-8019. S2CID 124794448.
30. Sparano, G.; Vilasi, G.; Vinogradov, A.M. (2002). "Vacuum Einstein metrics with bidimensional Killing leaves. I. Local aspects". Differential Geometry and Its Applications. 16 (2): 95–120. arXiv:gr-qc/0301020. doi:10.1016/S0926-2245(01)00062-6. S2CID 7992539.
31. Sparano, G.; Vilasi, G.; Vinogradov, A.M. (2002). "Vacuum Einstein metrics with bidimensional Killing leaves. II. Global aspects". Differential Geometry and Its Applications. 17 (1): 15–35. arXiv:gr-qc/0301020. doi:10.1016/S0926-2245(02)00078-5.
32. "Editorial Board - Differential Geometry and its Applications - Journal - Elsevier". www.journals.elsevier.com. Retrieved 2021-12-18.
33. "Differential Geometry and its Applications | Geometry of PDEs' with subtitle 'In memory of Alexandre Mikhailovich Vinogradov | ScienceDirect.com by Elsevier". www.sciencedirect.com. Retrieved 2021-12-18.
34. "ESI Advisory Board".
35. "Conferences - Levi-Civita Institute". www.levi-civita.org. Retrieved 2021-12-18.
36. "Diffiety Schools - Levi-Civita Institute". www.levi-civita.org. Retrieved 2021-12-18.
37. "Diffiety Education Program - Levi-Civita Institute". www.levi-civita.org. Retrieved 2021-12-18.
38. "Statute - Levi-Civita Institute". www.levi-civita.org. Retrieved 2021-12-18.