The papers which Gromov published in the late 1960s include the following (all written in Russian): On a geometric hypothesis of Banach (1967); The number of simplexes in subdivisions of finite complexes (1968); Transversal mappings of foliations (1968); Transversal mappings of foliations (1968); Simplexes inscribed on a hypersurface (1968); and Stable mappings of foliations into manifolds (1969).
Gromov's mathematical contributions, beginning in this period, are described by Hung-Hsi Wu:-
Around 1970, the world of differential geometry was astounded by the news that a young Russian by the name of Mikhael Gromov had proved that any noncompact differential manifold admits a Riemannian metric of positive sectional curvature, and also one of negative sectional curvature. We were also told that this was achieved by a "soft" method of topological sheaves. Moreover, in one and the same setting, Gromov also proved generalizations of both the Hirsch-Smale immersion theorem and the A Phillips submersion theorems. Many more results were promised. Slowly, Gromov's papers (some in collaboration with Ya M Eliashberg and V A Rokhlin) filtered to the West in the early seventies.In 1970 the International Congress of Mathematicians was held in Nice, France. Gromov had been invited to address the Congress but was not allowed to leave the USSR by the Soviet authorities. He did, however, contribute the text of his lecture A topological technique for the construction of solutions of differential equations and inequalities which was published in the Conference Proceedings in 1971. This is one of the papers referred to in the above quote by Wu, as are Gromov's papers (with V A Rokhlin) Imbeddings and immersions in Riemannian geometry (1970), and (with Ya M Eliashberg) Elimination of singularities of smooth mappings (1971). For his outstanding work Gromov was presented with the Award of the Moscow Mathematical Society in 1971. He then presented his Post-doctoral Thesis to Leningrad University in 1973.
In 1974 Gromov left Russia for the United States when he was appointed as Professor of Mathematics at the State University of New York at Stony Brook. In 1979 he gave a course of lectures Structures métriques pour les variétés riemanniennes at Paris VII which have been remarkably influential. These notes were published in 1981 and Gromov wrote in the Preface:-
These notes are from a course given at the University of Paris VII during the last trimester of 1979. Our purpose was to present some of the links that have been established between the curvature of a Riemannian manifold V and its global behaviour. Here, the word 'global' applies not only to the topology of V but also to a family of metric invariants of Riemannian manifolds and mappings between these manifolds. The simplest metric invariants of V are, for example, its volume and its diameter; the dilatation is an important invariant for a mapping from V1 into V2. In fact, such metric invariants also appear in a purely topological context, and they provide an important link between infinitesimal data on V (generally expressed by a condition on the curvature) and the topology of V. For example, the now classical theorem of Bonnet gives an upper bound for the diameter of a manifold V with positive curvature, from which one can deduce the finiteness of the fundamental group of V. A deeper topological study of Riemannian manifolds requires finer metric invariants than diameter or volume; we have tried to present a systematic treatment of these invariants, but this study is far from being as comprehensive as we had hoped.An English edition of these notes was published as Metric structures for Riemannian and non-Riemannian spaces in 1999. Igor Belegradek begins a review as follows:-
The first edition of this book, published in French [Structures métriques pour les variétés riemanniennes (1981)], is considered one of the most influential books in geometry in the last twenty years. Since then the boundary of the field has dramatically exploded. Reflecting this growth, the new English edition has almost quadrupled in size. Among the most substantial additions, each taking over a hundred pages, there is a chapter on convergence of metric spaces with measures, and an appendix on analysis on metric spaces written by Semmes. In addition, numerous remarks, examples, proofs, and open problems are inserted throughout the book. The original text is mostly preserved with new items conveniently indicated by a subscript +.In 2005 Gromov received the Janos Bolyai International Mathematical Prize for this 1999 book. We have been following the remarkable developments which have come from Gromov's 1979 course in Paris. We should return to relate how his career developed from that time. In 1981 he moved from the State University of New York at Stony Brook to the Université de Paris VI and the following year he moved to the Institut des Hautes Études Scientifiques where he was made a permanent member. He has continued to hold this position but has, in addition, had posts in the United States. He was Professor of Mathematics at University of Maryland, College Park from 1991 to 1996, and then Jay Gould Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University from 1996.
We have explained above that the Soviet authorities did not allow Gromov to attend the International Congress of Mathematicians in Nice in 1970 to which he had been invited as a speaker. He was able to accept invitations to speak at the International Congresses of Mathematicians in Helsinki (1978) and Warsaw (1982). In 1986 he was an invited plenary speaker at the International Congress of Mathematicians in Berkeley where he spoke on Soft and hard symplectic geometry. Hung-Hsi Wu writes:-
This is a survey of the recent work on symplectic geometry, with emphasis on the author's own contributions .... First, some explanations about the title. In the author's own words, "Intuitively, 'hard' refers to a strong and rigid structure of a given object, while 'soft' suggests some weak general property of a vast class of objects. ... 'Soft' and 'hard' in this talk are limited to the framework of the global nonlinear analysis concerning the geometry of spaces of maps between smooth manifolds".In 1985 Gromov was a plenary speaker at the British Mathematical Colloquium in Cambridge when he lectured on Differential geometry with and without infinitesimal calculus: anatomy of curvature.
Gromov has received a fantastic collection of major mathematical prizes for his wonderful contributions. These include: the Oswald Veblen Prize in Geometry from the American Mathematical Society (1981):-
... for his work relating topological and geometric properties of Riemannian manifolds;the Prix Élie Cartan of the Académie des Sciences of Paris (1984); the Prix de l'Union des Assurances de Paris (1989); and the Wolf Prize in Mathematics (1993). In 1997 he was awarded the Lobachevsky Medal and, in the same year, he received the Leroy P Steele Prize from the American Mathematical Society for Seminal Contribution to Research:-
... for his paper "Pseudo-holomorphic curves in symplectic manifolds", which revolutionized the subject of symplectic geometry and topology and is central to much current research activity, including quantum cohomology and mirror symmetry.This 1985 paper opens a new effective approach to fundamental problems of symplectic topology. In 1999 Gromov was awarded the Balzan Prize for Mathematics. The Laudatio for this Prize reads:-
Mikhael L Gromov is, without any doubt, one of the greatest geometers of this century. His work is unique through the abundance and the force of the concepts he has created, as well as through the new techniques he has devised and applied to solve problems, often simple to state and to understand, and which seem, at first sight, inaccessible. Some of those problems were long standing, and their unexpected solutions caused wonder and surprise due to the originality and elegance of the method conceived by Gromov: famous instances are his proof of the old conjecture according to which a finitely generated group of polynomial growth has a nilpotent subgroup of finite index, or the beautiful construction (together with I Pyatetski-Shapiro) of non-arithmetic discrete groups of hyperbolic transformations in arbitrary dimension. On the other hand, new techniques developed by Gromov for different purposes led to completely new kinds of problems: one can imagine the great variety of questions arising from the introduction of a natural geometric structure on the set of all (isomorphism classes) of Riemannian manifolds, or from the discovery of many new and remarkable invariants of manifolds (e.g., the K-area, the simplicial volume, the minimal volume, etc.), not to forget important new notions, such as that of hyperbolic groups, which is at the origin of major recent developments in differential geometry. To summarise, Gromov has brought about not only solutions to famous and time-old problems, but also the bases of new fields of study for many scholars. It has been emphasised above that he tends to look at all questions from the geometric side: he translates them in ad hoc geometric terms, and uses his extraordinary geometric intuition to investigate them thoroughly; it should be added that he is also able to treat, in the same way, questions coming from the most diverse branches of mathematics: algebra, analysis, differential equations, probability theory, theoretical physics, etc. Due to the large number of his disciples and the wide repercussions aroused by his important discoveries, Mikhael Gromov has had, and will continue to have, a considerable influence on contemporary mathematics.The next major prize to be presented to Gromov was the 2002 Kyoto Prize:-
The 2002 Kyoto Prize laureate in Basic Sciences is Mikhael Leonidovich Gromov of France. His contributions, including the introduction of a metric structure for families of various geometrical objects, have led to dramatic developments in geometry and many other fields of mathematics. Gromov has pioneered entirely new disciplines in a variety of fields, including geometry and analysis, and has had a substantial impact on all the mathematical sciences. Through the application of innovative ideas and radical nontraditional mathematical methods, he has also solved a great many complicated problems in modern geometry.He received the 2004 Frederic Esser Nemmers Prize in Mathematics from Northwestern University:-
... for his work in Riemannian geometry, which revolutionized the subject; his theory of pseudoholomorphic curves in symplectic manifolds; his solution of the problem of groups of polynomial growth; and his construction of the theory of hyperbolic groups.In 2008 the London Mathematical Society elected Gromov to Honorary Membership of the Society:-
... in recognition of his profound and extraordinary insights whose influence extends far beyond the boundaries of his own field of geometry.Perhaps the most prestigious of all the major awards he has received was the Abel Prize for 2009:-
The Russian-French mathematician Mikhail L Gromov is one of the leading mathematicians of our time. He is known for important contributions in many areas of mathematics, especially geometry. Geometry is one of the oldest fields of mathematics; it has engaged the attention of great mathematicians through the centuries, but has undergone a revolutionary change in the last 50 years. Mikhail Gromov has led some of the most important developments, producing profoundly original general ideas which have resulted in new perspectives on geometry and other areas of mathematics. Gromov's name is forever attached to deep results and important concepts within Riemannian geometry, symplectic geometry, string theory and group theory. The Abel committee says: "Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions to old problems. He has produced deep and original work throughout his career and remains remarkably creative. The work of Gromov will continue to be a source of inspiration for many future mathematical discoveries".Let us end this biography by quoting from G Elek :-
[Gromov] has truly revolutionised geometry; laid the foundations of brand new fields, introduced spectacularly new viewpoints and a philosophy which makes his papers and thoughts unmistakable.Ezra Getzler writes:-
His work is both tremendously elegant and immediately relevant to problems in applied mathematics in a way that reflects his tremendous creativity and excellent taste.
Article by: J J O'Connor and E F Robertson