Alexandre Mikhailovich Vinogradov

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.{{Cite journal|last1=Astashov|first1=A. M.|last2=Astashova|first2=I. V.|last3=Bocharov|first3=A. V.|last4=Buchstaber|first4=V. M.|last5=Vassiliev|first5=V. A.|last6=Verbovetsky|first6=A. M.|last7=Vershik|first7=A. M.|last8=Veselov|first8=A. P.|last9=Vinogradov|first9=M. M.|last10=Vitagliano|first10=L.|last11=Vitolo|first11=R. F.|date=2020|title=Alexandre Mikhailovich Vinogradov (obituary)|url=https://gdeq.org/files/amv-obituary-en.pdf|journal=Russian Mathematical Surveys|volume=75|issue=2|pages=369–375|doi=10.1070/rm9931|s2cid=219049017|issn=0036-0279}}

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.{{Cite web|title=Alexandre Vinogradov - The Mathematics Genealogy Project|url=https://www.genealogy.math.ndsu.nodak.edu/id.php?id=48562|access-date=2021-12-11|website=www.genealogy.math.ndsu.nodak.edu}}

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.{{Cite journal|last1=Astashov|first1=A. M.|last2=Astashova|first2=I. V.|last3=Bocharov|first3=A. V.|last4=Buchstaber|first4=V. M.|last5=Vassiliev|first5=V. A.|last6=Verbovetsky|first6=A. M.|last7=Vershik|first7=A. M.|last8=Veselov|first8=A. P.|last9=Vinogradov|first9=M. M.|last10=Vitagliano|first10=L.|last11=Vitolo|first11=R. F.|date=2020|title=Alexandre Mikhailovich Vinogradov (obituary)|url=https://gdeq.org/files/amv-obituary-en.pdf|journal=Russian Mathematical Surveys|volume=75|issue=2|pages=369–375|doi=10.1070/rm9931|s2cid=219049017|issn=0036-0279}}

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.{{Cite journal|last=Vinogradov|first=A. M.|date=1960|title=О спектральной последовательности Адамса|trans-title=On Adams' spectral sequence|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=23889&option_lang=eng|journal=Dokl. Akad. Nauk SSSR|language=ru|volume=133|issue=5|pages=999–1002|via=All-Russian Mathematical Portal}}
English translation: {{Cite journal|last=Vinogradov|first=A. M.|date=1960|title=On Adam's spectral sequence|url=https://zbmath.org/?q=an:0097.16101|journal=Soviet Mathematics. Doklady|language=English|volume=1|pages=910–913|zbl=0097.16101 }}

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.{{Cite journal|last=Vinogradov|first=A. M.|date=1972|title=Алгебра логики теории линейных дифференциальных операторов|trans-title=The logic algebra for the theory of linear differential operators|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=37058&option_lang=rus&paperid=37058&option_lang=eng|journal=Dokl. Akad. Nauk SSSR|language=ru|volume=205|issue=5|pages=1025–1028|via=All-Russian Mathematical Portal}}
English translation: {{Cite journal|last=Vinogradov|first=A. M.|date=1972|title=The logic algebra for the theory of linear differential operators|url=https://zbmath.org/?q=an%3A0267.58013|journal=Soviet Mathematics. Doklady|language=English|volume=13|pages=1058–1062|issn=0197-6788}}

Vinogradov’s approach to nonlinear differential equations as geometric objects, with their general theory and applications, is developed in details in some monographs{{Cite book|last1=Vinogradov|first1=A.M.|url=https://diffiety.mccme.ru/djvu/vinogradov-krasilshchik-lychagin.djvu|title=Introduction to the geometry of nonlinear differential equations|last2=Krasil’shchik|first2=I.S.|last3=Lychagin|first3=V.V.|publisher=Nauka|year=1986|location=Moscow|pages=336|language=ru}}
English translation: {{Cite book|last1=Vinogradov|first1=A. M.|url=https://www.worldcat.org/oclc/12551635|title=Geometry of jet spaces and nonlinear partial differential equations|last2=Krasilʹshchik|first2=I. S.|last3=Lychagin|first3=V. V.|publisher=Gordon and Breach Science Publishers|year=1986|isbn=2-88124-051-8|location=New York, N.Y.|oclc=12551635}}{{Cite book|last1=Bocharov|first1=A. V.|url=https://bookstore.ams.org/mmono-182|title=Symmetries and conservation laws for differential equations of mathematical physics|last2=Krasilʹshchik|first2=I. S.|last3=Vinogradov|first3=A. M.|date=1999|publisher=American Mathematical Society|isbn=978-1-4704-4596-6|location=Providence, R.I.|oclc=1031947580}}{{Cite book|last=Nestruev|first=Jet|url=https://diffiety.mccme.ru/books/texts/Nestruev.pdf|title=Smooth manifolds and observables|publisher=MCCME|year=2000|location=Moscow|pages=300|language=ru}}
English translation: {{Cite book|last=Nestruev|first=Jet|url=http://link.springer.com/10.1007/b98871|title=Smooth Manifolds and Observables|date=2003|publisher=Springer-Verlag|isbn=978-0-387-95543-8|series=Graduate Texts in Mathematics|volume=220|location=New York|language=en|doi=10.1007/b98871|s2cid=117029379 }}
Second revised and expanded edition: {{Cite book|last=Nestruev|first=Jet|url=http://link.springer.com/10.1007/978-3-030-45650-4|title=Smooth Manifolds and Observables|date=2020|publisher=Springer International Publishing|isbn=978-3-030-45649-8|series=Graduate Texts in Mathematics|volume=220|location=Cham|language=en|doi=10.1007/978-3-030-45650-4|s2cid=242759997}} as well as in some articles.{{Cite journal|last=Vinogradov|first=A. M.|date=1980|title=Геометрия нелинейных дифференциальных уравнений|trans-title=The geometry of nonlinear differential equations|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intg&paperid=121&option_lang=eng|journal=Itogi Nauki I Tekhniki. Ser. Probl. Geom.|publication-place=Moscow|volume=11|pages=89–134|via=All-Russian Mathematical Portal}}
English translation: {{Cite journal|last=Vinogradov|first=A. M.|date=1981|title=Geometry of nonlinear differential equations|url=http://link.springer.com/10.1007/BF01084594|journal=Journal of Soviet Mathematics|language=en|volume=17|issue=1|pages=1624–1649|doi=10.1007/BF01084594|s2cid=121310561|issn=0090-4104|via=}}{{Cite journal|last=Vinogradov|first=A. M.|date=1984|title=Local symmetries and conservation laws|url=http://link.springer.com/10.1007/BF01405491|journal=Acta Applicandae Mathematicae|language=en|volume=2|issue=1|pages=21–78|doi=10.1007/BF01405491|s2cid=121860845|issn=0167-8019}}{{Cite journal|last1=Sparano|first1=G.|last2=Vilasi|first2=G.|last3=Vinogradov|first3=A.M.|date=2001|title=Gravitational fields with a non-Abelian, bidimensional Lie algebra of symmetries|url=https://linkinghub.elsevier.com/retrieve/pii/S0370269301007225|journal=Physics Letters B|language=en|volume=513|issue=1–2|pages=142–146|doi=10.1016/S0370-2693(01)00722-5|arxiv=gr-qc/0102112|bibcode=2001PhLB..513..142S|s2cid=15766049}} He recast infinitely prolonged differential equations into a category{{Citation|last=Vinogradov|first=A. M.|title=Category of nonlinear differential equations|date=1984|url=http://link.springer.com/10.1007/BFb0099553|work=Global Analysis — Studies and Applications I|series=Lecture Notes in Mathematics|volume=1108|pages=77–102|editor-last=Borisovich|editor-first=Yurii G.|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|doi=10.1007/bfb0099553|isbn=978-3-540-13910-2|access-date=2021-12-11|editor2-last=Gliklikh|editor2-first=Yurii E.|editor3-last=Vershik|editor3-first=A. M.}} whose objects, called diffieties, are studied in the framework of what he called secondary calculus (by analogy with secondary quantization).{{Cite journal|last=Vinogradov|first=A.M.|date=1998|title=Introduction to Secondary Calculus|url=https://diffiety.mccme.ru/preprint/98/05_98.pdf|journal=Contemporary Mathematics|location=Providence, Rhode Island|publisher=American Mathematical Society|volume=219|pages=241–272|doi=10.1090/conm/219/03079|isbn=9780821808283}}{{Citation|last=Vinogradov|first=Alexandre|title=Secondary Calculus and Cohomological Physics|date=1998|url=http://www.ams.org/conm/219/|work=Contemporary Mathematics|volume=219|pages=|editor-last=Henneaux|editor-first=Marc|place=Providence, Rhode Island|publisher=American Mathematical Society|language=en|doi=10.1090/conm/219/03079|isbn=978-0-8218-0828-3|access-date=2021-12-11|editor2-last=Krasil′shchik|editor2-first=Joseph|editor3-last=Vinogradov|editor3-first=Alexandre}}{{Cite book|last=Vinogradov|first=A. M.|url=https://www.worldcat.org/oclc/47296188|title=Cohomological analysis of partial differential equations and secondary calculus|date=2001|publisher=American Mathematical Society|isbn=0-8218-2922-X|location=Providence, R.I.|oclc=47296188}} One of the central parts of this theory is based on the $\backslash cal\; C$-spectral sequence (now known as the Vinogradov spectral sequence).{{Cite journal|last=Vinogradov|first=A.M.|date=1978|title=Одна спектральная последовательность, связанная с нелинейным дифференциальным уравнением и алгебро-геометрические основания лагранжевой теории поля со связями|trans-title=A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=41521&option_lang=eng|journal=Dokl. Akad. Nauk SSSR|language=ru|volume=238|issue=5|pages=1028–1031|via=All-Russian Mathematical Portal}}
English translation: {{Cite journal|last=Vinogradov|first=A.M.|date=1978|title=A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundation of Lagrangian field theory with constraints|url=https://www.scopus.com/record/display.uri?eid=2-s2.0-0000950795&origin=inward&txGid=53745bb00536375ba76bd954359ba052|journal=Soviet Math. Dokl.|volume=19|issue=1|pages=144–148}}{{Cite journal|last=Vinogradov|first=A.M.|date=1984|title=The b-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory|journal=Journal of Mathematical Analysis and Applications|language=en|volume=100|issue=1|pages=1–40|doi=10.1016/0022-247X(84)90071-4|doi-access=free}}{{Cite journal|last=Vinogradov|first=A.M.|date=1984|title=The b-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory|journal=Journal of Mathematical Analysis and Applications|language=en|volume=100|issue=1|pages=41–129|doi=10.1016/0022-247X(84)90072-6|doi-access=free}} 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 $\backslash cal\; C$-spectral sequence (for an “empty” equation, i.e., for the space of infinite jets) is the so-called variational bicomplex.{{Cite web|title=variational bicomplex in nLab|url=https://ncatlab.org/nlab/show/variational+bicomplex|access-date=2021-12-18|website=ncatlab.org}}

Furthermore, Vinogradov introduced a new bracket on the graded algebra of linear transformations of a cochain complex.{{Cite journal|last=Vinogradov|first=A.M.|date=1990|title=Объединение скобок Схоутена и Нийенхейса, когомологии и супердифференциальные операторы|trans-title=The union of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=3270&option_lang=eng|journal=Mat. Zametki|language=ru|volume=47|issue=6|pages=138–140|via=All-Russian Mathematical Portal}} 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.{{Cite journal|last=Kosmann-Schwarzbach|first=Yvette|date=1996|title=From Poisson algebras to Gerstenhaber algebras|url=https://aif.centre-mersenne.org/item/AIF_1996__46_5_1243_0/|journal=Annales de l'Institut Fourier|volume=46|issue=5|pages=1243–1274|doi=10.5802/aif.1547|issn=0373-0956|doi-access=free}} These results were also applied to Poisson geometry.{{Cite journal|last1=Cabras|first1=A.|last2=Vinogradov|first2=A.M.|date=1992|title=Extensions of the poisson bracket to differential forms and multi-vector fields|url=https://linkinghub.elsevier.com/retrieve/pii/039304409290026W|journal=Journal of Geometry and Physics|language=en|volume=9|issue=1|pages=75–100|doi=10.1016/0393-0440(92)90026-W|bibcode=1992JGP.....9...75C}}{{Cite journal|last1=Marmo|first1=G.|last2=Vilasi|first2=G.|last3=Vinogradov|first3=A.M.|date=1998|title=The local structure of n-Poisson and n-Jacobi manifolds|url=https://linkinghub.elsevier.com/retrieve/pii/S0393044097000570|journal=Journal of Geometry and Physics|language=en|volume=25|issue=1–2|pages=141–182|doi=10.1016/S0393-0440(97)00057-0|arxiv=physics/9709046|bibcode=1998JGP....25..141M|s2cid=119118335}}

Together with {{Interlanguage link|Peter W. Michor|lt=Peter Michor|de}}, Vinogradov was concerned with the analysis and comparison of various generalizations of Lie (super) algebras, including $L\_\backslash infty$ algebras and Filippov algebras.{{Cite journal|last1=Michor|first1=Peter W.|last2=Vinogradov|first2=Alexandre M.|date=1998-01-19|title=n-ary Lie and Associative Algebras|url=https://zbmath.org/?q=an:0928.17029|journal=Rend. Sem. Mat. Univ. Pol. Torino|volume=53|issue=3|pages=373–392|arxiv=math/9801087|bibcode=1998math......1087M|zbl=0928.17029 }} 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 $\backslash mathbb\{R\}$ can be assembled in a few steps from two elementary constituents, that he called dyons and triadons.{{Cite journal|last=Vinogradov|first=A. M.|date=2017|title=Particle-like structure of Lie algebras|url=http://aip.scitation.org/doi/10.1063/1.4991657|journal=Journal of Mathematical Physics|language=en|volume=58|issue=7|pages=071703|doi=10.1063/1.4991657|arxiv=1707.05717|bibcode=2017JMP....58g1703V|s2cid=119316544|issn=0022-2488}}{{Cite journal|last=Vinogradov|first=A. M.|date=2018|title=Particle-like structure of coaxial Lie algebras|url=http://aip.scitation.org/doi/10.1063/1.5001787|journal=Journal of Mathematical Physics|language=en|volume=59|issue=1|pages=011703|doi=10.1063/1.5001787|bibcode=2018JMP....59a1703V|issn=0022-2488}} 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,{{Cite journal|last1=Vinogradov|first1=A M|last2=Krasil'shchik|first2=I S|date=1975-02-28|title=Что такое гамильтонов формализм?|trans-title=What is the Hamiltonian formalism?|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=4140&option_lang=eng|journal=Russian Mathematical Surveys|language=ru|volume=30|issue=1|pages=177–202|doi=10.1070/RM1975v030n01ABEH001403|s2cid=250915291 |issn=0036-0279|via=All-Russian Mathematical Portal}}{{Cite journal|last1=Vinogradov|first1=A M|last2=Kupershmidt|first2=B A|date=1977-08-31|title=Структура гамильтоновой механики|trans-title=The structures of hamiltonian mechanics|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=3221&option_lang=eng|journal=Russian Mathematical Surveys|language=ru|volume=32|issue=4|pages=177–243|doi=10.1070/RM1977v032n04ABEH001642|s2cid=250805957 |issn=0036-0279|via=All-Russian Mathematical Portal}} the dynamics of acoustic beams,{{Cite journal|last1=Vinogradov|first1=A. M.|last2=Vorobjev|first2=E. M.|date=1976|title=Applications of symmetries to finding exact solutions of the Zabolotskaya-Khokhlov equation|url=http://www.akzh.ru/pdf/1976_1_23-27.pdf|journal=Akust. Zhurnal.|language=ru|volume=22|issue=1 |pages=23–27}} the equations of magnetohydrodynamics (the so-called Kadomtsev-Pogutse equations appearing in the stability theory of high-temperature plasma in tokamaks){{Cite journal|last1=Gusyatnikova|first1=V. N.|last2=Samokhin|first2=A. V.|last3=Titov|first3=V. S.|last4=Vinogradov|first4=A. M.|last5=Yumaguzhin|first5=V. A.|date=1989|title=Symmetries and conservation laws of Kadomtsev-Pogutse equations (Their computation and first applications)|url=http://link.springer.com/10.1007/BF00131929|journal=Acta Applicandae Mathematicae|language=en|volume=15|issue=1–2|pages=23–64|doi=10.1007/BF00131929|s2cid=124794448|issn=0167-8019}} and mathematical questions in general relativity.{{Cite journal|last1=Sparano|first1=G.|last2=Vilasi|first2=G.|last3=Vinogradov|first3=A.M.|date=2002|title=Vacuum Einstein metrics with bidimensional Killing leaves. I. Local aspects|journal=Differential Geometry and Its Applications|language=en|volume=16|issue=2|pages=95–120|doi=10.1016/S0926-2245(01)00062-6|s2cid=7992539|doi-access=free|arxiv=gr-qc/0301020}}{{Cite journal|last1=Sparano|first1=G.|last2=Vilasi|first2=G.|last3=Vinogradov|first3=A.M.|date=2002|title=Vacuum Einstein metrics with bidimensional Killing leaves. II. Global aspects|journal=Differential Geometry and Its Applications|language=en|volume=17|issue=1|pages=15–35|doi=10.1016/S0926-2245(02)00078-5|doi-access=free|arxiv=gr-qc/0301020}} 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.

File:Alexandre_Vinogradov-lect-1.jpg 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.

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.{{Cite web|title=Editorial Board - Differential Geometry and its Applications - Journal - Elsevier|url=https://www.journals.elsevier.com/differential-geometry-and-its-applications/editorial-board|access-date=2021-12-18|website=www.journals.elsevier.com}} A special issue of the journal, devoted to the geometry of PDEs, was published in his memory.{{Cite web|title=Differential Geometry and its Applications {{!}} Geometry of PDEs' with subtitle 'In memory of Alexandre Mikhailovich Vinogradov {{!}} ScienceDirect.com by Elsevier|url=https://www.sciencedirect.com/journal/differential-geometry-and-its-applications/special-issue/10D14H8NVGC|access-date=2021-12-18|website=www.sciencedirect.com|language=en-us}}

In 1993 he was one of the promoters of the Schrödinger International Institute in Mathematical Physics in Vienna.{{Cite web|title=ESI Advisory Board|url=https://www.esi.ac.at/about/people}} In 1997 he organised the large conference Secondary Calculus and Cohomological Physics in Moscow, which was followed by a series of small conferences called Current Geometry that took place in Italy from 2000 to 2010.{{Cite web|title=Conferences - Levi-Civita Institute|url=https://sites.google.com/site/levicivitainstitute/Activities/Conferences|access-date=2021-12-18|website=www.levi-civita.org}}

From 1998 to 2019, Vinogradov organised and directed the so-called Diffiety Schools in Italy, Russia, and Poland,{{Cite web|title=Diffiety Schools - Levi-Civita Institute|url=https://sites.google.com/site/levicivitainstitute/Activities/DiffietySchools|access-date=2021-12-18|website=www.levi-civita.org}} 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.{{Cite web|title=Diffiety Education Program - Levi-Civita Institute|url=https://sites.google.com/site/levicivitainstitute/dep|access-date=2021-12-18|website=www.levi-civita.org}}{{Cite web|title=Statute - Levi-Civita Institute|url=https://sites.google.com/site/levicivitainstitute/home/statute|access-date=2021-12-18|website=www.levi-civita.org}}

He supervised 19 PhD students.

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{{DEFAULTSORT:Vinogradov, Alexandre Mikhailovich}}

Category:1938 births

Category:2019 deaths

Category:Russian expatriates in Italy

Category:20th-century Russian mathematicians

Category:21st-century Russian mathematicians

Category:20th-century Italian mathematicians

Category:21st-century Italian mathematicians

Category:Italian mathematical physicists

Category:Topologists

Category:Algebraic geometers