Alexandre Vinogradov

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.

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 headed it 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.

Alexandre Vinogradov

Navigation menu

Search