Alexandre Vinogradov

Site: https://diffiety.mccme.ru/curvita/amv.htm Obituary in the Russian Mathematical Surveys (in Russian) Obituary in the Russian Mathematical Surveys (in English) A special issue Geometry of PDEs' in memory of Alexandre Mikhailovich Vinogradov published by the journal Differential Geometry and its Applications		Wikipedia page: Alexandre Mikhailovich Vinogradov Lectures at the Voronezh Diffiety school, January 26 - February 6, 2015 (Russian) "Assembling Lie Algebras from Lieons", Globus seminar, 11 October 2012 (Russian) "Bella Subbotovskaya and People's University", B. Subbotovskaya Memorial Conference, March 2007 "Mathematics of nonlinear word" by A.Vinogradov, a tv record made on 29 September 2003 (in Russian)

Contribution to mathematics and to the mathematical community

Alexandre Vinogradov, a remarkable mathematician and extraordinary man, was born on February 18, 1938 in Novorossiysk, but almost all his life lived in Moscow. In 1955 he became a student of Mekhmat of Moscow State University and, in 1960, a graduate (PhD track) student there. After obtaining his PhD in 1964, he was soon invited to take a teaching position at the Chair of Higher Geometry and Topology, which he held until he left the Soviet Union for Italy in 1990. He obtained the habilitation degree in 1984. From 1993 to 2010, he held the position of professor at the University of Salerno in Italy.

Being a second year undergraduate student, Vinogradov published two works (with B.N. Delaunay and D.B. Fuchs) in number theory, but by the end of undergraduate years his research interests changed: he began working in algebraic topology. His PhD was devoted to the homotopic properties of the embedding spaces of circles into the 2-sphere or the 3-disk. One of Vinogradov's first works was devoted to the Adams spectral sequence. In 1960, Vinogradov announced the solution of J.F. Adams' problem concerning the relationship between the higher cohomological operations and the Adams filtration in the stable homotopy groups of spheres. Adams wrote a favorable review of that note.

Vinogradov radically changed the direction of his research between the sixties and the seventies. Inspired by the ideas of Sophus Lie, he began to think about the foundations of the geometric theory of PDEs; having become familiar with the work of Spencer, Goldschmidt, and Quillen on formal solvability, he turned his attention to the algebraic (in particular, cohomological) component of that theory. In 1972, the short note "The logic algebra of the theory of linear differential operators" introduced what Vinogradov himself called the main functors of the differential calculus in commutative algebras. On four pages, it was elegantly shown that for the definition and the study of such fundamental notions as vector field, differential form, jet, linear differential operator, etc., the category of modules over a commutative algebra with unit provides an appropriate setting, while the geometric prototypes of these notions occur when, for the algebra, one chooses the algebra of smooth functions on a manifold, and for the modules, the spaces of sections of vector bundles over the manifold.

Vinogradov's approach to nonlinear differential equations as geometric objects, with general theory and applications, is developed in several monographs and articles. He combined infinitely prolonged differential equations into a category. Its objects, diffieties (differential varieties), are studied in the framework of what he called the secondary calculus. One of the central parts of this theory is based on the ${\mathcal {C}}$-spectral sequence (Vinogradov spectral sequence). The term $E_{1}$ of this sequence gives a unified cohomological approach to many scattered concepts and statements, including the Lagrangian formalism with constraints, conservation laws, cosymmetries, the Noether theorems, and the Helmholtz criterion in the inverse problem of the calculus of variations (for arbitrary nonlinear differential operators. The ideas underlying the construction of the ${\mathcal {C}}$-spectral sequence and the results following from these ideas were the first decisive steps in the direction of what is now called "cohomological physics".

Vinogradov introduced the construction of a new bracket on the graded algebra of linear transformations of a cochain complex. The construction preceded the general concept of derived bracket on a differential Loday algebra. The Vinogradov bracket is a skew-symmetric version of the derived bracket generated by the coboundary operator. Derived brackets and their generalizations play an exceptionally important role in modern applications of homotopy Lie algebras, Lie algebroids, etc., and Vinogradov's results are pioneering in this direction. In particular, Vinogradov showed that the classical Schouten bracket (on multivector fields) and the Nijenhuis bracket (on vector fields with coefficients in differential forms) are restrictions of his bracket onto the corresponding subalgebras of superdifferential operators on the algebra of differential forms.

In two last papers he developed a theory of compatibility of Lie algebra structures and proved that any finite-dimensional Lie algebra over an algebraically closed field or over $\mathbb {R}$ can be assembled in a few steps from two elementary constituents, that he called dyons and triadons. Furthermore, Vinogradov speculated that this particle-like structures could be related to the ultimate structure of elementary particles.

Generally speaking, a significant part of Vinogradov's work was highly motivated by the complex and important problems of modern physics. In particular, much attention was paid to the mathematical understanding of the fundamental physical concept of the observable in the book "Smooth manifolds and observables", written by A.M. Vinogradov in co-authorship with the participants of his seminar and published under the pseudonym Jet Nestruev.

Vinogradov's published heritage consists of over a hundred articles and ten monographs. Whatever he worked on, be it the geometry of differential equations, the Schouten and Nijenhuis brackets, mathematical questions of gravitation theory, $n$-ary generalizations of Lie algebras or the structural analysis of the latter, he produced work characterized by a very unorthodox approach, depth, and nontriviality of the obtained results.

The scientific activity of Vinogradov was not limited to the writing of books and articles. For many years he headed a research seminar at Mekhmat at Moscow State University; the seminar was in two parts - mathematical and physical - and became a notable phenomenon in Moscow's mathematical life between 1960 and 1980. He had numerous students (in Russia, Italy, Switzerland, and Poland), nineteen of whom obtained their PhD's under his guidance, six obtained the higher habilitation degree, and one became a corresponding member of the Russian Academy of Sciences. Vinogradov organized and headed Diffiety Schools in Italy, Russia, and Poland. He was the soul of a series of small "Current Geometry" conferences that took place in Italy from 2000 to 2010, as well as of the large Moscow conference "Secondary Calculus and Cohomological Physics".

A.M. Vinogradov was one of the initial organizers of the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna, as well as of the Journal of Differential Geometry and Applications, remaining one of the editors to his last days. In 1985 he created a department 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 he left for Italy. He was one of the organizers and first lecturers in the unofficial school for students who were not accepted to Mekhmat because they were ethnically Jewish.

Alexandre Vinogradov was a versatile person — he played the violin, wrote poetry in Russian and Italian, played for the Mekhnat water-polo team, was an enthusiastic football player. But the most important thing for him was, undoubtedly, mathematics. He was full of bright and fruitful ideas and actively worked until his death on September 20, 2019.

List of publications

Monographs

• A. De Paris, A. M. Vinogradov, Fat Manifolds and Linear Connections, World Scientific, 2008, xii+297 pp., DOI: 10.1142/6904.

• Jet Nestruev, Smooth manifolds and observable, Grad. Texts in Math., 220, New York: Springer-Verlag, pp. XIV+222, 2003, DOI: 10.1007/b98871.
  Russian original: Moscow, MCCMO Publ., 317 pp., 2000.
  Second extended and revised English edition: Grad. Texts in Math., 220, New York: Springer-Verlag, pp. XVIII+433, 2020, DOI: https://doi.org/10.1007/978-3-030-45650-4.

• A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, AMS, series: Translations of Mathematical Monograph, 204, 2001, AMS bookstore.

• I. S. Krasil'shchik , A. M. Vinogradov (eds.), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS, Translations of Mathematical Monograph series, 182, xiv+333 pp., 1999, AMS bookstore, zbl 0911.00032.
  Parallel Russian edition: Moscow, Factorial Publ. House, 461 pp., 1997.
  Second extended and revised Russian edition: Moscow, Factorial Publ. House, 380 pp., 2005.

• D. V. Alekseevski, V. V. Lychagin, A. M. Vinogradov, Basic ideas and concepts of differential geometry, Geometry I. Encycl. Math. Sci. 28, 255 pp., 1991, Mi intf108, MR 1315081, Zbl 0675.53001.
  Russian original: «Modern problems of mathematics: fundamental directions», Vol. 28, 1988, 298 pp., Moscow, VINITI.

• I. S. Krasil'shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics, 1, Gordon and Breach, New York, London. xx+441 pp, 1986.

• A. M. Vinogradov, I. S. Krasil'shchik, V. V. Lychagin, Introduction to geometry of nonlinear differential equations (Russian), «Nauka», Moscow, 336 pp, 1986.

• A. M. Vinogradov, I. S. Krasil'shchik, V. V. Lychagin, Geometry of nonlinear differential equations (Russian), Moscow Institute of Electronic Engineering, 86 pp, 1982.

• A. M. Vinogradov, I. S. Krasil'shchik, V. V. Lychagin, Application of nonlinear differential equations in civil aviation (Russian), Moscow Insitute of Civil Aviation Engineering, 123 pp., 1977.

• A. M. Vinogradov, Algebraic Topology (Russian), Moscow Institute of Electronic Engineering, 232 pp., 1970.

Articles

2015 — 2019

• A. M. Vinogradov, Particle-like structure of coaxial Lie algebras, J. Math. Phys., 59:1 (2018), 011703, 42 pp., DOI: 10.1063/1.5001787, MR 3749205, Zbl 1422.17009.

• A. M. Vinogradov, Particle-like structure of Lie algebras, J. Math. Phys., 58:7 (2017), 071703, 49 pp., arXiv:1707.05717 DOI: 10.1063/1.4991657, MR 3670617, Zbl 1422.17008.

• A. M. Vinogradov, Logic of differential calculus and the zoo of geometric structures, «Geometry of Jets and Fields», Banach Center Publications, 2016, 110, 257-285, 2015, arXiv:1511.06861.

2010 — 2014

• A. M. Vinogradov, Some remarks on contact manifolds, Monge-Ampáre equations and solution singularities, International Journal of Geometric Methods in Modern Physics, 14 pp., 2014, arXiv:1403.1742.

• A. M. Vinogradov, What are symmetries of nonlinear PDEs and what are they themselves? In «Lie and Klein: The Erlangen program and its impact on mathematics and physics» (ed: A. Papadopoulos and L. Ji), European Mathematical Society Publishing House, 45 pp., 2014, arXiv:1308.5861.

• A. M. Vinogradov, Assembling Lie algebras from lieons, arXiv:1205.6096v1 [math.DG], 99 pp., 2012.

• D. Catalano Ferraioli, A. M. Vinogradov, Differential invariants of generic parabolic Monge-Ampere equations, J. Phys. A: Math. Theor., 45, 265204, 24 pp., 2012, arXiv:0811.3947.

• A. De Paris, A. M. Vinogradov, Scalar differential invariants of symplectic Monge-Ampere equations, Cent. Eur. J. Math., 9, no.4, 731-751, 2011, arXiv:1102.0426.

2005 — 2009

• A. M. Vinogradov, On geometry of second order parabolic differential equations in two independent variables, Doklady Akademii Nauk, 2008, 423:5, 588-591, Mi dan189, MR 2498570 (Russian).
  English trans. Doklady Mathematics, 2008, 78, no. 3, 887-890, DOI: 10.1134/S1064562408060227, DIPS-01/08.

• C. Di Pietro, A. M. Vinogradov, A spectral sequence associated with a symplectic manifold, Dokl. Acad. Nauk, 2007, 413:5, 591-593 (Russian), Mi dan662, MR 2458550.
  English translation: Doklady Mathematics, 2007, 75:2, 287-289, arXiv:math/0611138, DOI: 10.1134/S1064562407020287.

• G. Moreno, A. M. Vinogradov, Domains in infinite jet spaces: ${\cal {C}}$-spectral sequence, Dokl. Acad. Nauk, 2007, 4132, 154-157, (Russian), Mi dan689, MR 2456137.
  English translation: Doklady Mathematics, 2007, 75:2, 204-207,arXiv:math/0609079, DOI: 10.1134/S1064562407020081.

• M. Marvan, A. M. Vinogradov and V. A. Yumaguzhin, Differential invariants of generic hyperbolic Monge-Ampere equations, Cent. Eur. J. Math., 2007, 5, no. 1, 105-133, arXiv:nlin/0604038, DOI: 10.2478/s11533-006-0043-4.

• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: Λ_k-1${\cal {C}}$-spectral sequence on infinitely prolonged equations, Dokl. Acad. Nauk, 2007, 416:3, 298-301, Mi dan547, 2458866 MR 2458866, (Russian).
  English translation: Doklady Mathematics, 2007, 76:, 692-695, arXiv:math/0703761, DOI: 10.1134/S1064562407050146.

• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: Λ_k-1${\cal {C}}$-spectral sequence on infinite jets, Dokl. Acad. Nauk, 2007, 416:2, 161-165 (Russian), Mi dan556, MR 2450915.
  English translation: Doklady Mathematics, 2007, 76:2, 673-677, arXiv:math/0703661, DOI: 10.1134/S1064562407050092.

• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: the ${\cal {C}}$-spectral sequence, Dokl. Acad. Nauk, 2007, 414:1, 447-450 (Russian), Mi dan629, MR 2451933.
  English translation: Doklady Mathematics, 2007, 75:3, 403-406, arXiv:math/0610917, DOI: 10.1134/S1064562407030192.

• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: integral calculus, Dokl. Acad. Nauk, 2007, 413:1, 7-10 (Russian), Mi dan697, MR 2447059.
  English translation: Doklady Mathematics, 2007, 75:2, 177-180, arXiv:math/0610914, DOI: 10.1134/S1064562407020019.

• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: Riemannian geometry revisited, Dokl. Acad. Nauk, 2006, 407:2, 151-153 (Russian), Mi dan988, MR 2348307.
  English translation: Doklady Mathematics, 2006, 73:2, 182-184, arXiv:math/0609287, DOI: 10.1134/S1064562406020074.

• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: tensors, Dokl. Acad. Nauk, 2006, 407:1, 16-18 (Russian), Mi dan998, MR 2347355.
  English translation: Doklady Mathematics, 2006, vol. 73, no. 2, pp.169-171, arXiv:math/0605113, DOI: 10.1134/S1064562406020037.

• J. Cortes, A. M. Vinogradov, Hamiltonian theory of constraint impulsive motion, J.Math.Phys., 2006, 47, 042905, 30 pp, arXiv:math/0401380, DOI: 10.1063/1.2192974.

• D. Catalano Ferraioli, A. M. Vinogradov, Ricci flat 4-metrics with bidimensional null orbits. Part II: the Abelian case, Acta Applicandae Mathematicae, 2006, 92:3, 223-239, DIPS 8/2004.

• D. Catalano Ferraioli, A. M. Vinogradov, Ricci flat 4-metrics with bidimensional null orbits. Part I: General aspects and nonabelian case, Acta Applicandae Mathematicae, 2006, 92:3, 209-223, DIPS 7/2004.

• A. M. Vinogradov, M. Marvan, V. A. Yumagughin, Differential invariants of generic hyperbolic Monge-Ampere equations, Dokl. Acad. Nauk, 2005, 405:3, 299-301, (Russian), Mi dan1076, MR 2264293.
  English translation: Doklady Mathematics, 2005, 72:3, 883-885, arXiv:nlin/0604038, DOI: 10.2478/s11533-006-0043-4.

2000 — 2004

• R. Alonso Blanco, A. M. Vinogradov, Green formula and Legendre transformation, Acta Applicandae Mathematicae, 2004, 83, 149-166, DIPS-5/2003, DOI: 10.1023/B:ACAP.0000035594.33327.71.

• G. Vezzosi, A. M. Vinogradov, On higher order analogues of de Rham cohomology, Diff.Geom. and Appl., 2003, 19, 29-59, arXiv:math/0002157, DOI: 10.1016/S0926-2245(03)00014-7.

• F. Pugliese, A. M. Vinogradov, Geometry of Inelastic Collisions, Acta Applicandae Mathematicae, 2002, 72(1), 77-85, DIPS-1/2001, DOI: 10.1023/A:1015230809764.

• G. Sparano, G. Vilasi, A. M. Vinogradov, Vacuum Einstein Metrics with Bidimensional Killing Leaves. I. Local Aspects, Diff.Geom. and Appl., 2002, 16, 95-120, arXiv:gr-qc/0301021, DOI: 10.1016/S0926-2245(02)00078-5.

• G. Sparano, G. Vilasi, A. M. Vinogradov, Vacuum Einstein Metrics with Bidimensional Killing Leaves. II. Global Aspects, Diff.Geom. and Appl., 2002, 17, 15-35, arXiv:gr-qc/0301021, DOI: 10.1016/S0926-2245(02)00078-5.

• A. M. Vinogradov, M. M. Vinogradov, Graded multiple analogs of Lie algebras, Acta Applicandae Mathematicae, 2002, 72, 183-197, DIPS-8/2001, DOI: 10.1023/A:1015281004171.

• F. Pugliese, A. M. Vinogradov, Discontinuous trajectories of Lagrangian systems with singular hypersurface, J. Math.Phys., 2001, 42(1), 309-329, DOI: 10.1063/1.1324653.

• G.Sparano, G.Vilasi, A. M. Vinogradov, Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries, Phys. Lett., Sec. B. 2001, 513, 142-146, arXiv:gr-qc/0102112, DOI: 10.1016/S0370-2693(01)00722-5.

• F. Pugliese, A. M. Vinogradov, On the geometry of singular lagrangians, J.Geom.and Phys., 2000, 35(1), 35-55, DIPS-2/99, DOI: 10.1016/S0393-0440(99)00076-5.

1995 — 1999

• F. Pugliese, A. M. Vinogradov, Jumping oscilator, arXiv:math/9902115 [math.DG], 27 pp., 1999.

• A. M. Vinogradov, Introduction to Secondary Calculus, Contemporary Mathematics, 1998, 219, 241-272, Amer. Math. Soc., Providence, Rhode Island, DIPS-05/98, DOI: 10.1090/conm/219/03079, MR 1640456.

• A. M. Vinogradov, M. M. Vinogradov, 'On multiple generalizations of Lie algebras and Piosson manifolds, Contemporary Mathematics,1998, 219, 273-287, Amer.Math.Soc., Providence, Rhode Island,, DIPS-06/98 DOI: 10.1090/conm/219/03080, MR 1640457.

• G. Marmo, G. Vilasi, A. M. Vinogradov, The local structure of n-Poisson and n-Jacobi manifolds, J. Geom. and Phys., 1998, 25(1-2), 141-182, arXiv:physics/9709046, DOI: 10.1016/S0393-0440(97)00057-0.

• G. Vezzosi, A. M. Vinogradov, Infinitesimal Stokes' formula for higher order de Rham complexes, Acta Applicandae Mathematicae, 1997, 49(3), 311-329, DOI: 10.1023/A:1005811010161.

• P. W. Michor, A. M. Vinogradov, n-ary Lie and associative algebras, Rend. Seminario Matematico di Torino, 1996, 53(4), 373-392, arXiv:math/9801087.

1990 — 1994

• F. Lizzi, G. Marmo, G. Sparano, A. M. Vinogradov, Eikonal type equations for geometrical singularities of solutions in field theory, J. Geom. and Phys., 1994, 14, 211-235, preprint ESI 46(1993), DOI: 10.1016/0393-0440(94)90008-6.

• M. Modugno, A. M. Vinogradov, Some variations on the notion of connection, Annali di Matematica pura ed applicata, 1994, 167, 33-71, DOI: 10.1007/BF01760328.

• A. M. Vinogradov, From symmetries of partial differential equations towards secondary («quantized») calculus, J. Geom. and Phys., 1994, 14, 146-194, DOI: 10.1016/0393-0440(94)90005-1.

• A. Cabras, A. M. Vinogradov, Extension of the Poisson bracket to differential forms and multi-vector fields, J. Geom. and Phys., 1992, 9(1), 75-100, DOI: 10.1016/0393-0440(92)90026-W.

• A.M. Verbovetsky, A. M. Vinogradov and D. M. Gessler, Scalar differential invariants and characteristic classes of homogeneous geometric structures, Mat. Zametki, 1992, 51:6, 15-26, Mi mz4625, MR 1187472, Zbl 0814.57019 (Russian).
  English translation in Math. Notes (1992), 51(5-6), 543-549, DOI: 10.1007/BF01263295.

• A. M. Vinogradov, Scalar differential invariants, diffeties, and characteristic classes, In: Francaviglia M. (Ed.), Mechanics, Analysis and Geometry: 200 Years after Lagrange, Elsevier, Amsterdam, 1991, 379-416, DOI: 10.1016/B978-0-444-88958-4.50020-3.

• A. M. Vinogradov, V. A. Yumaguzhin, Differential invariants of webs on two-dimensional manifolds, Mat. Zametki, 1990, 48:1, 26-37, Mi mz3280, MR 1081890, Zbl 0714.53019 (Russian).
  English translation in Math. Notes, 1991, 48(1), 639-647, DOI: 10.1007/BF01164260.

• A. M. Vinogradov, A common generalization of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators, Mat. Zametki, 1990, 47:6, 138-140, Mi mz3270, MR 1074539, Zbl 0712.58059, (Russian).

1985 — 1989

• I. S. Krasil'shchik, A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math., 1989, 15:1, 161-209, DOI: 10.1007/BF00131935.
  Also in: «Symmetries of Partial Differential Equations», ed. by A. M. Vinogradov, Kluwer Acad. Publ., Dordrecht, Boston, London, 1989, 161-209.

• V. N. Gusyatnikova, A. V. Samokhin, V. S. Titov, A. M. Vinogradov, V. A. Yumaguzhin, Symmetries and conservation laws of Kadomtsev-Pogutse equations (their computation and first applications), Acta Appl. Math. 1989, 15(1), 23-64, DOI: 10.1007/BF00131929.

• A. M. Vinogradov, Symmetries and conservation laws of partial differential equations: basic notions and results, Acta Appl. Math., 1989, 15(1), 3-21, DOI: 10.1007/BF00131928.

• A. M. Vinogradov, An informal introduction to the geometry of jet spaces, Conference on Differential Geometry and Topology (Sardinia, 1988). Rend. Sem. Fac. Sci. Univ. Cagliari, 1988, 15, suppl., 301-333.

• S. I. Senashov, A. M. Vinogradov, Symmetries and conservation laws of 2-dimensional ideal plasticity, Proc. Edinburgh Math. Soc., 31 (1988), 415-439, DOI: 10.1017/S0013091500006817.

• A. M. Vinogradov, Integrability and symmetries, in «Nonlinear waves. Structures and bifurcations», Moscow, «Nauka», 1987, 279-290 (Russian).

• A. M. Astashov, A. M. Vinogradov, On the structure of Hamiltonian operator in field theory, J. Geom. and Phys., (1986), 3:2, 263-287, DOI: 10.1016/0393-0440(86)90022-7.

• A. M. Vinogradov, Geometric singularities of solutions of nonlinear partial differential equations, Proc. Conf. «Differential geometry and its applications», Brno, 1986, Math. Appl. (East European Ser.), Reidel, Dordrecht-Boston, MA, 1987, 27, 359-379,.

• A. M. Vinogradov, A. V. Samokhin, The Cartan-Kähler theorem, Transactions of Seminar on Algebra and Geometry of Differential Equations, Moscow, VINITI, 1986,858-B, 112-132 (Russian).

• A. M. Vinogradov, A. V. Samokhin, On quotiening of partial differential equations, Transactions of the Seminar of Algebra and Geometry of Differential Equations, Moscow, VINITI, 1986,858-B, 133-146 (Russian).

• A. M. Vinogradov, Why is the space 3-dimensional and how may groups be seen?, Acta Appl. Math., 1986, 5(2), 169-180, DOI: 10.1007/BF00046586.

• A. M. Vinogradov, Geometry of differential equations, secondary differential calculus and quantum field theory, Soviet Mathematics (Izvestiya VUZ. Matematika), 1986, 1, 13-21, Mi ivm7465, MR 838427, Zbl 0616.58009 (Russian).
  English translation Soviet Math. (Iz. VUZ), 1986, 30:1, 14–25.

• V. N. Gusyatnikova, A. M. Vinogradov, V. A. Yumaguzhin, Secondary differential operators, J. Geom. and Phys., 1985, 2(2), 23-65, DOI: 10.1016/0393-0440(85)90008-7.

• A. M. Vinogradov, V. N. Gusyatnikova, V. A. Yumaguzhin, Secondary differential operators, Dokl. Akad. Nauk SSSR, 1985, 283:4, 801-805, Mi dan9040, MR 802682, Zbl 0598.58009 (Russian).
  English transl. in Soviet Math.Dokl., 1985, 32:1, 198-202.

1980 — 1984

• A. M. Vinogradov, Category of partial differential equations, «Global Analysis - Studies and Applications I», Lecture Notes in Math., 1984, 1108, 77-102, DOI: 10.1007/BFb0099553.

• A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 1984, 3, 21-78, DOI: 10.1007/BF01405491.

• A. M. Vinogradov, The ${\cal {C}}$-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl., 1984, 100, no. 1, pp. 1-40, DOI: 10.1016/0022-247X(84)90071-4.

• A. M. Vinogradov, The ${\cal {C}}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl., 1984, 100, no. 1, pp. 41-129, DOI: 10.1016/0022-247X(84)90071-4.

• I. S. Krasil'shchik, A. M. Vinogradov, Nonlocal symmetries and the theory of coverings: addendum to to A. M. Vinogradov's «Local symmetries and conservation laws», Acta Appl. Math., 1984, 2(1), 79-96, DOI: 10.1007/BF01405492.

• I. S. Krasil'shchik, A. M. Vinogradov, On the theory of nonlocal symmetries of nonlinear partial differential equations, Dokl. Akad. Nauk SSSR, 1984, 275:5, 1044-1049, Mi dan9705, MR 745842, Zbl 0604.58053 (Russian).
  English transl. in Sov. Math. Dokl., 1984, 20:2, 337-341.

• A. M. Vinogradov, Category of differential equations and its significance for physics, In: Krupka D. (Ed.), Proc. Conf. Diff. Geom. Appl. (Brno, 1984), J.E. Purkynue Univ., Brno, Czechoslovakia, 289-301.

• A. M. Vinogradov, Category of nonlinear differential equations (Russian), addendum to the Russian translation of: J.-F. Pommaret, «Systems of partial differential equations and Lie pseudogroups» (translated by A. V. Bocharov, M. M. Vinogradov and I. S. Krasil'shchik), Moscow, Mir, 400 pp., 1983.

• A. M. Vinogradov, Higher symmetries and conservation laws, in «Group-theoretic methods in physics», 1983, 2, 414-420, Moscow, Nauka, (Russian).

• A. M. Vinogradov, Category of nonlinear differential equations, Equations on manifolds, Novoe v Global. Anal., Voronezh. Gos. Univ., Voronezh, 1982, 26-51 (Russian).

• A. M. Vinogradov, Conservation laws, the Spencer cohomology and the ${\cal {C}}$-spectral sequence, in «Leningrad international topology conference», Leningrad, Nauka, 1982, p. 166 (Russian).

• A. M. Vinogradov, Geometry of nonlinear differential equations, Itogi Nauki i Tekhniki. Ser. Probl. Geom., 11, Moscow, VINITI, 1980, 89–134, Mi intg121, MR 579929, Zbl 0475.58025|0461.58012.
  English translation: Journal of Soviet Mathematics, 17(1), 1624-1649, 1981, DOI: 10.1007/BF01084594

• I. S. Krasil'shchik, A. M. Vinogradov, What is the Hamiltonian formalism?, London Math. Soc. Lect. Notes Ser., 1981, 60, 241-266, DOI: 10.1017/CBO9780511600784.008.

• A. M. Vinogradov, Category of nonlinear differential equations, «XV Voronezh winter mathematical school», Voronezh Gos. Univ. Publ., Moscow, VINITI, 5691, 1981, 9-10 (Russian).

• A. M. Vinogradov, I. S. Krasil'shchik, A method of computing higher symmetries of nonlinear evolution equations and nonlocal symmetries, Dokl.Akad.Nauk SSSR (1980) 253, 1089-1093, Mi dan43819, MR 0583788, Zbl 0498.35076 (Russian).
  English transl. in Soviet Math. Dokl., 1980, 22, 235-239.

• A. M. Vinogradov, Geometry of nonlinear differential equations, Problems in geometry, 1980, 11, 89-134, Moscow, VINITI, Mi intg121, MR 579929, Zbl 0475.58025|0461.58012 (Russian).
  English transl. in J.Sov.Math., 1981, 17, 1624-1649, DOI: 10.1007/BF01084594.

1975 — 1979

• A. M. Vinogradov, Some new homological systems associated with differential calculus over commutative algebras, Uspechi Mat. Nauk, 1979, 34:6, 145-150, Mi umn4163, MR 562827, Zbl 0475.58024|0476.58028, (Russian).
  English transl. in Russian Math. Surveys, 1979, 34:6, 250-255, DOI: 10.1070/RM1979v034n06ABEH003355.

• A. M. Vinogradov, Theory of higher infinitesimal symmetries of nonlinear partial differential equations, Dokl.Akad.Nauk SSSR, 1979, 248:2, 274-278, Mi dan42982, MR 0553187, Zbl 0445.58030 (Russian).
  English transl. in Soviet Math. Dokl., 1979, 20, 985-989.

• A. M. Vinogradov, A spectral system associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Dokl.Akad.Nauk SSSR, 1978, 238:5, 1028-1031, Mi dan41521, MR 0483733, Zbl 0406.58015 (Russian).
  English translation in Soviet Math. Dokl., 1978, 19, 144-148.

• A. M. Vinogradov, Hamiltonian structures in field theory, Dokl.Akad.Nauk SSSR, 1978, 241:1, 18-21, Mi dan41816, MR 0510883, Zbl 0421.70026 (Russian).
  English transl. in Soviet Math. Dokl., 1978, 19:4, 790-794.

• A. M. Vinogradov, On the algebro-geometric foundations of Lagrangian field theory, Dokl.Akad.Nauk SSSR,1977, 236:2, 284-287, Mi dan41214, MR 0501142, Zbl 0403.58005 (Russian).
  English transl. in Soviet Math. Dokl., 1977, 18:5 , pp. 1200-1204.

• A. M. Vinogradov, B. A. Kupershmidt, The structure of Hamiltonian mechanics, Uspechi Mat.Nauk, 1977, 32:4, 175-228, Mi umn3221, MR 501143, Zbl 0365.70016|0383.70020 (Russian).
  English transl. in Russian Math. Surveys, 1977, 32:4, 177--232; also in London Math. Soc. Lect. Notes, 1981, 60, 173-228, DOI: 10.1070/RM1977v032n04ABEH001642.

• A. V. Bocharov, A. M. Vinogradov, The Hamiltonian form of mechanics with friction, non-holonomic mechanics, invariant mechanics, the theory of refraction and impact, addendum II in A. M. Vinogradov, B. A. Kupershmidt, The structure of Hamiltonian mechanics, Uspechi Mat.Nauk, 1977, 32:4, 228-236, Mi umn3221 (Russian).
  English transl. in Russian Math. Surveys, 1977, 32:4, 232-243; also in London Math. Soc. Lect. Notes, 1981, 60, 229-239, DOI: 10.1070/RM1977v032n04ABEH001642.

• A. M. Vinogradov, E. M. Vorobjev, Applications of symmetries to finding exact solutions of the Zabolotskaya-Khokhlov equation, Akust. Zhurnal., 1976, 22:1, 23-27 (Russian).

• A. M. Vinogradov, Theory of symmetries of non-linear differential equations, DEP 2855-74, Moscow, VINITI, 1974, 16 pp. (Russian).

• A. M. Vinogradov, I. S. Krasil'shchik, What is the Hamiltonian formalism?, Uspechi Mat.Nauk, 1975, 30:1, 173-198, Mi umn4140, MR 650307, Zbl 0327.70006 (Russian).
  English transl. in Russian Math. Surveys,1975, 30, 177-202, also in London Math. Soc. Lect. Notes,1981, 60, 241-266, DOI: 10.1070/RM1975v030n01ABEH001403.

1970 — 1974

• A. M. Vinogradov, Multivalued solutions and a principle of classification of nonlinear differential equation, Dokl.Akad.Nauk, 1973, 210:1, 11-14, Mi dan37624, MR 0348799, Zbl 0306.35003, (Russian).
  English transl. in Soviet Math. Dokl., 1973, 14:3, 661-665.

• A. M. Vinogradov, The logic algebra for the theory of linear differential operators, Dokl.Akad.Nauk, 1972, 205:5, 1025-1028, Mi dan37058, MR 0304363, Zbl 0267.58013 (Russian).
  English transl. in Soviet Math. Dokl., 1972, 13:4, 1058-1062.

• A. M. Vinogradov, M. Kushel'man, Generalized Smith's conjecture in dimension four, Siberian Math. Journ., 1972, 13:1, 52-62, Mi smj4435, MR 0298648, Zbl 0232.55014 (Russian).

1965 — 1969

• A. M. Vinogradov, Morse theory for simple closed curves, Dokl.Akad. Nauk SSSR, 1969, 187:1, 11-14, Mi dan34724, MR 0263117, Zbl 0195.25101 (Russian).

• A. M. Vinogradov, S. P. Novikov, Geometric and differential topology, in «History of Soviet Mathematics», Naukova Dumka, Kiev, 1968, 3, 511-529, (Russian).

• A. M. Vinogradov, Some properties of knots, addendum to Russian transl. of «Introduction to knot theory» by R.Crowell and R.Fox, Moscow, Mir, 1967, 284-309 (Russian).

1960 — 1964

• A. M. Vinogradov, Some homotopy properties of the space of imbeddings of the circle into a sphere or a ball, Dokl. Akad. Nauk SSSR, 1964, 156:5, 999-1002, Mi dan34724, MR 0263117, Zbl 0195.25101 (Russian).

• A. M. Vinogradov, On the Adams spectral sequence, Dokl. Akad. Nauk SSSR, 1960, 133:5, 999-1002, Mi dan23889, MR 0124901, Zbl 0097.16101 (Russian).

• A. M. Vinogradov, On the second obstruction for cross sections, Dokl. Akad. Nauk SSSR, 1960,130:4, 723-725, Mi dan39582, MR 0138111, Zbl 0118.18701 (Russian).

1958 — 1959

• B. N. Delaunay, A. M. Vinogradov, Uber den Zussamenhang zwishen den Lagrangeschen Klassen der Irrationalitaten mit begretzen Leilennern und Markoffschen Klassen der extremen Formen, in «Ehren 250 Geburtstages L. Eulers», Akad. Verlag, Berlin, 1959, 101-106.

• A. M. Vinogradov, B. N. Delaunay B.N. and D. B. Fuchs On rational approximations to irrational numbers with bounded partial quotients, Dokl. Akad. Nauk SSSR, 1958, '117:5, Mi dan22736, MR 0101227, Zbl 0080.26502, 862-865 (Russian).

• A. M. Vinogradov, B. N. Delaunay B.N. and D. B. Fuchs, Amendments to Article «On rational approximations to irrational numbers with bounded partial quotients», Dokl. Akad. Nauk SSSR, 1958, 119:6, 1062 p., Mi dan22736 (Russian)

Edited collections and proceedings

• M. Henneaux, I. S. Krasil'shchik, A. M. Vinogradov (Eds.), Secondary calculus and cohomological physics, Proc. conf. «Secondary calculus and cohomological physics», August 24-31, 1997, Moscow; Contemporary Mathematics, 1998, vol. 219.

• I. S. Krasil'shchik, A. M. Vinogradov (Eds.), Algebraic aspects of differential calculus, special issue of Acta Applicandae Mathematicae, 1997, 49:3. Also in The Diffety Inst. Preprint Series, DIPS 1/96 -DIPS 8/96.

• A. M. Vinogradov (Ed.), Symmetries of partial differential equations: conservation laws, applications, algorithms, Kluwer Acad. Publ., Dordrecht, Boston, London, 1989, vi+456 pp.

• A. M. Vinogradov (Ed.), Transactions of the seminar «Algebra and geometry of differential equations», VINITI, 1986, Dep. 858-B, Moscow.

Addendum: doctoral dissertation

• A. M. Vinogradov, Geometry of jet spaces and its applications to the theory of symmetries and conservation laws for partial differential equations, doctoral dissertation, Moscow State University, 1984, 288 pp., (Russian).