Sergei Petrovich Novikov: from his student article to the Fields Medal

The talk will focus on the first decade, 1960–1970, of Sergei Petrovich's scientific work. It will present his results and the influence they had on the subsequent development of mathematics. Novikov’s scientific activity clearly demonstrated his talent in choosing a research direction that remained relevant for many years. While still a student, he published the paper [1] that contained a theorem that later became well known as the Milnor-Novikov theorem on the cobordism ring of stably complex manifolds. In 1965, Novikov published the paper [2] in which the fundamental problem: the topological invariance of rational Pontryagin classes of smooth manifolds was proven. In the program of the International Mathematical Congress of 1966 in Moscow, Novikov’s sectional talk was announced, see [3], but he read another talk. The expanded presentation of this talk was published in the paper [4], in which the Adams-Novikov spectral sequence and the formal group of geometric cobordisms were introduced. The outstanding role of these new methods of algebraic topology was shown in the problem of stable homotopy groups of spheres and the problem of manifolds with group actions. This paper was discussed in detail at leading seminars on algebraic topology in various countries, and its methods formed the basis of subsequent results of many authors. In 1970, Novikov published a two-part paper, see [5], in which he proposed a Hermitian analogue of K-theory based on ideas from physics. In the second part of this paper, he formulated a conjecture on the homotopy invariance of higher signatures. This famous Novikov’s problem remains one of the most fruitful in algebraic topology, see [6]. At the 1970 Mathematical Congress, Novikov was awarded the Fields Medal with the following statement: Made important advances in topology, the most well-known being his proof of the topological invariance of the Pontrjagin classes of the differentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces.