Mikhael Gromov (mathematician)

Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory.

He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.

Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union.

His father Leonid Gromov was Russian-Slavic and his mother Lea was of Jewish heritage.

Both were pathologists.

His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich.

Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him.

When Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him.

His work initiated a renewed study of the Oka–Grauert theory, which had been introduced in the 1950s.

Gromov and Vitali Milman gave a formulation of the concentration of measure phenomena.

They defined a "Lévy family" as a sequence of normalized metric measure spaces in which any asymptotically nonvanishing sequence of sets can be metrically thickened to include almost every point.

This closely mimics the phenomena of the law of large numbers, and in fact the law of large numbers can be put into the framework of Lévy families.

Gromov and Milman developed the basic theory of Lévy families and identified a number of examples, most importantly coming from sequences of Riemannian manifolds in which the lower bound of the Ricci curvature or the first eigenvalue of the Laplace–Beltrami operator diverge to infinity.

They also highlighted a feature of Lévy families in which any sequence of continuous functions must be asymptotically almost constant.

These considerations have been taken further by other authors, such as Michel Talagrand.

Gromov studied mathematics at Leningrad State University where he obtained a master's degree in 1965, a doctorate in 1969 and defended his postdoctoral thesis in 1973.

His thesis advisor was Vladimir Rokhlin.

Gromov married in 1967.

In 1970, he was invited to give a presentation at the International Congress of Mathematicians in Nice, France.

However, he was not allowed to leave the USSR.

Still, his lecture was published in the conference proceedings.

Disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14.

In the early 1970s he ceased publication, hoping that this would help his application to move to Israel.

He changed his last name to that of his mother.

He received a coded letter saying that, if he could get out of the Soviet Union, he could go to Stony Brook, where a position had been arranged for him.

When the request was granted in 1974, he moved directly to New York and worked at Stony Brook.

In 1981 he left Stony Brook University to join the faculty of University of Paris VI and in 1982 he became a permanent professor at the Institut des Hautes Études Scientifiques where he remains today.

At the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences in New York since 1996.

He adopted French citizenship in 1992.

Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.

He is also interested in mathematical biology, the structure of the brain and the thinking process, and the way scientific ideas evolve.

Motivated by Nash and Kuiper's isometric embedding theorems and the results on immersions by Morris Hirsch and Stephen Smale, Gromov introduced the h-principle in various formulations.

Modeled upon the special case of the Hirsch–Smale theory, he introduced and developed the general theory of microflexible sheaves, proving that they satisfy an h-principle on open manifolds.

As a consequence (among other results) he was able to establish the existence of positively curved and negatively curved Riemannian metrics on any open manifold whatsoever.

His result is in counterpoint to the well-known topological restrictions (such as the Cheeger–Gromoll soul theorem or Cartan–Hadamard theorem) on geodesically complete Riemannian manifolds of positive or negative curvature.

After this initial work, he developed further h-principles partly in collaboration with Yakov Eliashberg, including work building upon Nash and Kuiper's theorem and the Nash–Moser implicit function theorem.

There are many applications of his results, including topological conditions for the existence of exact Lagrangian immersions and similar objects in symplectic and contact geometry.

His well-known book Partial Differential Relations collects most of his work on these problems.

Later, he applied his methods to complex geometry, proving certain instances of the Oka principle on deformation of continuous maps to holomorphic maps.

Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry".