From his Obituary

(on the website of Moscow State University)

On June 6, 2024, Sergey Petrovich Novikov (20.03.1938 - 06.06.2024) passed away, an outstanding mathematician of our time, academician of the Russian Academy of Sciences, head of the Department of Higher Geometry and Topology of the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University. He was an Honored Professor of Moscow State University, laureate of the Grand Gold Medal of the Russian Academy of Sciences named after M.V. Lomonosov, laureate of the Fields Medal of the International Mathematical Union, laureate of many other higher international and Russian scientific prizes and awards.

Sergey Petrovich Novikov was a world-famous scientist, author of fundamental works in the fields of geometry, topology, mechanics, quantum field theory, solid state physics, theory of integrable systems and other sections of theoretical and mathematical physics. Already the first student work of S.P. Novikov on the cohomology of Steenrod algebra attracted the attention of specialists. Sergei Petrovich finished his postgraduate studies with the famous topologist M. Postnikov. He developed the theory of cobordisms, obtained the classification of simply connected manifolds of dimension greater than or equal to five, and managed to make significant progress in calculating stable homotopy groups of spheres. He obtained other important results in the field of algebraic and differential topology. Also S.P. Novikov obtained fundamental results in the theory of foliations. In 1965, Novikov proved that Pontryagin's rational classes are topological invariants of a smooth manifold. This was one of the greatest achievements of the topology of that time. Novikov also proved the homotopy invariance of Pontryagin-Hirzebruch's special integrals on cycles arising from the homological algebra of the fundamental group, which led him to the famous hypothesis of higher signatures. This Novikov hypothesis about higher signatures has had and continues to have a great influence on the development of this important area of topology to this day. For his work in the field of topology in 1970, S.P. Novikov was awarded the Fields Medal of the International Mathematical Union, becoming the first Soviet mathematician in history to receive this highest international mathematical award.

Since the 70s of the last century, S.P. Novikov turned to problems of physical origin, in particular, nonlinear equations of mathematical physics. In his pioneering 1974 work, Novikov considered the periodic problem for the Korteweg-de Vries equation and expressed a truly revolutionary idea: the spectrum of the Sturm-Liouville operator in the periodic case is correctly considered as a Riemann surface, the so-called spectral curve, and the proper function of the Sturm-Liouville operator is set as a meromorphic function on the Jacobian of an algebraic curve depending on the parameter. Novikov's ideas for the explicit interaction of algebraic geometry and the spectral theory of differential operators have now given rise to the famous theory of finite-zone integration, which has been successfully applied to other important equations. In the early 80s, Novikov became one of the creators of another great scientific direction - the Morse-Novikov theory for multi-valued functionals. The achievements of S.P. Novikov were greatly recognized in theoretical physics.

The list of scientific achievements of Sergei Petrovich Novikov is truly huge.

Sergey Petrovich Novikov from the very beginning of his career began to pay great attention to pedagogical work. He wrote fundamental monographs and textbooks. In the early 70s, he radically rebuilt the teaching of geometry courses, linear algebra, created a new course of differential geometry and topology.

S.P. Novikov is the creator of the world-famous scientific school, 24 of his students became doctors of science, among them Academician of the Russian Academy of Sciences I.A. Taymanov and Corresponding Member of the Russian Academy of Sciences V.M. Bukhstaber. In total, more than 40 candidate dissertations were defended under his leadership. [Editorial comment: in Russia, there are two levels of doctoral degrees: "candidate" and "doctor".  The former is closer to the American Ph.D.; the latter is closer to the German Habilitation.]  S.P. Novikov's seminar "Geometry and Mathematical Physics" is world famous.

Sergei Petrovich Novikov has received many awards and prizes for his scientific and scientific-pedagogical achievements. He is a laureate of the Lenin Prize, the Fields Medal of the International Mathematical Union, the highest award of the Russian Academy of Sciences - the Grand Gold Medal named after M.V. Lomonosov, Gold Medal named after Leonard Euler of the Russian Academy of Sciences, Gold Medal named after N.N. Bogolyubov RAS, Lobachevsky International Prize of the Russian Academy of Sciences, Wolf Prize (Israel), as well as a laureate of other prizes and awards, he was also elected an honorary member of many foreign academies and scientific societies.

S.P. Novikov was the President of the Moscow Mathematical Society for more than 10 years, for more than 30 years he was the permanent editor-in-chief of the journal "Successes of Mathematical Sciences", and he was a member of the editorial boards of many leading scientific journals.

Novikov's mathematics in a nutshell

Novikov is known for a huge number of things, but here we just focus on some of the most important ones.  Along with William Browder, Dennis Sullivan, and C.T.C. Wall, he was one of the founders of surgery theory, the branch of topology concerned with the classification of manifolds (especially in dimensions 5 and up, since dimension 2 is classical and dimensions 3 and 4 have special peculiarities that require other techniques).  This theory in principle makes it possible to answer questions such as:

• given a compact space X, when is it homotopy-equivalent to a closed manifold?
• how many closed manifolds (up to homeomorphism, PL homeomorphism, or diffeomorphism) are there in a given homotopy type?

Novikov's work on surgery theory led to his proof that the rational Pontryagin classes, which a priori depend on the choice of differentiable structure, and are definitely not homotopy invariants in general, are actually homeomorphism invariants.  This quite surprising result was mentioned in his Fields Medal citation.  They also led to his famous Novikov conjecture on higher signatures, which is still one of the key open problems in topology and which has led to interesting connections between topology, operator algebras, and representation theory.

Another of his major contributions to topology was the development of the Adams-Novikov spectral sequence, which is now the tool of choice for computing stable homotopy groups of spheres, one of the oldest, but hardest, classical problems in algebraic topology.

Another major focus of Novikov's work was the application of algebraic topology to mathematical physics.  For example, his work on a generalized Morse theory for multi-valued functionals, mentioned in the Russian obituary above, grew out of applications of the topology of Fermi surfaces to solid-state physics.

Novikov's career at Maryland

After some time as a visitor in the period 1992-1996, Novikov joined Maryland as a full-time professor in 1996 and was made a Distinguished University Professor in 1997.  He retired in 2017 and became Distinguished University Professor Emeritus.  During his time at Maryland, he divided his time between IPST and the math department, where he taught many graduate topology courses, supervised several students, and was a very active participant in seminars.  Novikov received the Wolf Prize in mathematics, one of the highest international honors, during the time he was at Maryland.