Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; ; born 23 December 1943) is a Russian-French mathematician known for his work in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
and
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
. He is a permanent member of
Institut des Hautes Études Scientifiques
The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics (also with a small theoretical biology g ...
in France and a professor of mathematics at
New York University
New York University (NYU) is a private university, private research university in New York City, New York, United States. Chartered in 1831 by the New York State Legislature, NYU was founded in 1832 by Albert Gallatin as a Nondenominational ...
.
Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry".
Early years, education and career
Mikhail Gromov was born on 23 December 1943 in Boksitogorsk,
Soviet Union
The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...
. His father Leonid Gromov was Russian-Slavic and his mother Lea was of
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
heritage. Both were
pathologist
Pathology is the study of disease. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in the context of modern medical treatme ...
s. His mother was the cousin of World Chess Champion
Mikhail Botvinnik
Mikhail Moiseyevich Botvinnik (; ; – May 5, 1995) was a Soviet and Russian chess grandmaster who held five world titles in three different reigns. The sixth World Chess Champion, he also worked as an electrical engineer and computer sci ...
, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during
World War II
World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
, 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.Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19 /ref> When Gromov was nine years old, his mother gave him the book '' The Enjoyment of Mathematics'' by
Hans Rademacher
Hans Adolph Rademacher (; 3 April 1892 – 7 February 1969) was a German-born American mathematician, known for work in mathematical analysis and number theory.
Biography
Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göt ...
and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him.
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
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
in
Nice
Nice ( ; ) is a city in and the prefecture of the Alpes-Maritimes department in France. The Nice agglomeration extends far beyond the administrative city limits, with a population of nearly one millionmove 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
Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public university, public research university in Stony Brook, New York, United States, on Long Island. Along with the University at Buffalo, it is on ...
to join the faculty of University of Paris VI and in 1982 he became a permanent professor at the
Institut des Hautes Études Scientifiques
The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics (also with a small theoretical biology g ...
where he remains today. At the same time, he has held professorships at the
University of Maryland, College Park
The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public university, public Land-grant university, land-grant research university in College Park, Maryland, United States. Founded in 1856, UMD i ...
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
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development ...
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux
* ''Immersion'' (album), the third album by Australian group Pendulum
* ''Immersion'' (film), a 2021 ...
s 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 metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s 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
In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...
, proving certain instances of the ''Oka principle'' on deformation of
continuous map
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s to holomorphic maps. 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
In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...
, 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 manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s in which the lower bound of the Ricci curvature or the first eigenvalue of the
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...
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.
Since the seminal 1964 publication of James Eells and Joseph Sampson on harmonic maps, various rigidity phenomena had been deduced from the combination of an existence theorem for harmonic mappings together with a vanishing theorem asserting that (certain) harmonic mappings must be totally geodesic or holomorphic. Gromov had the insight that the extension of this program to the setting of mappings into
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s would imply new results on
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...
s, following Margulis superrigidity. Richard Schoen carried out the analytical work to extend the harmonic map theory to the metric space setting; this was subsequently done more systematically by Nicholas Korevaar and Schoen, establishing extensions of most of the standard
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
theory. A sample application of Gromov and Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are
arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
...
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
says that if a complete Riemannian manifold has
sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...
s which are all sufficiently close to a given positive constant, then must be finitely covered by a sphere. In contrast, it can be seen by scaling that every closed Riemannian manifold has Riemannian metrics whose sectional curvatures are arbitrarily close to zero. Gromov showed that if the scaling possibility is broken by only considering Riemannian manifolds of a fixed diameter, then a closed manifold admitting such a Riemannian metric, with sectional curvatures sufficiently close to zero, must be finitely covered by a nilmanifold. The proof works by replaying the proofs of the Bieberbach theorem and Margulis lemma. Gromov's proof was given a careful exposition by Peter Buser and Hermann Karcher.
In 1979, Richard Schoen and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
showed that the class of
smooth manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topology ...
which admit Riemannian metrics of positive
scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
is topologically rich. In particular, they showed that this class is closed under the operation of connected sum and of
surgery
Surgery is a medical specialty that uses manual and instrumental techniques to diagnose or treat pathological conditions (e.g., trauma, disease, injury, malignancy), to alter bodily functions (e.g., malabsorption created by bariatric surgery s ...
in codimension at least three. Their proof used elementary methods of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s, in particular to do with the
Green's function
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is a linear dif ...
. Gromov and Blaine Lawson gave another proof of Schoen and Yau's results, making use of elementary geometric constructions. They also showed how purely topological results such as Stephen Smale's h-cobordism theorem could then be applied to draw conclusions such as the fact that every closed and simply-connected smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature. They further introduced the new class of ''enlargeable manifolds'', distinguished by a condition in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
. They showed that Riemannian metrics of positive scalar curvature ''cannot'' exist on such manifolds. A particular consequence is that the
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
cannot support any Riemannian metric of positive scalar curvature, which had been a major conjecture previously resolved by Schoen and Yau in low dimensions.
In 1981, Gromov identified topological restrictions, based upon
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s, on manifolds which admit Riemannian metrics of nonnegative sectional curvature. The principal idea of his work was to combine Karsten Grove and Katsuhiro Shiohama's Morse theory for the Riemannian distance function, with control of the distance function obtained from the Toponogov comparison theorem, together with the Bishop–Gromov inequality on volume of geodesic balls. This resulted in topologically controlled covers of the manifold by geodesic balls, to which
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they h ...
arguments could be applied to control the topology of the underlying manifold. The topology of lower bounds on sectional curvature is still not fully understood, and Gromov's work remains as a primary result. As an application of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
, Peter Li and Yau were able to apply their gradient estimates to find similar Betti number estimates which are weaker than Gromov's but allow the manifold to have convex boundary.Li, Peter; Yau, Shing-Tung. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 3-4, 153–201.
In Jeff Cheeger's fundamental compactness theory for Riemannian manifolds, a key step in constructing coordinates on the limiting space is an injectivity radius estimate for
closed manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The onl ...
s. Cheeger, Gromov, and Michael Taylor localized Cheeger's estimate, showing how to use Bishop−Gromov volume comparison to control the injectivity radius in absolute terms by curvature bounds and volumes of geodesic balls. Their estimate has been used in a number of places where the construction of coordinates is an important problem. A particularly well-known instance of this is to show that Grigori Perelman's "noncollapsing theorem" for Ricci flow, which controls volume, is sufficient to allow applications of Richard Hamilton's compactness theory. Cheeger, Gromov, and Taylor applied their injectivity radius estimate to prove Gaussian control of the
heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum ...
, although these estimates were later improved by Li and Yau as an application of their gradient estimates.
Gromov made foundational contributions to
systolic geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and ...
. Systolic geometry studies the relationship between size invariants (such as volume or diameter) of a manifold M and its topologically non-trivial submanifolds (such as non-contractible curves). In his 1983 paper "Filling Riemannian manifolds" Gromov proved that every essential manifold with a Riemannian metric contains a closed non-contractible
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
of length at most .
Gromov−Hausdorff convergence and geometric group theory
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s with the structure of a metric space. More generally, one can define the Gromov-Hausdorff distance between two metric spaces, relative to the choice of a point in each space. Although this does not give a metric on the space of all metric spaces, it is sufficient in order to define "Gromov-Hausdorff convergence" of a sequence of pointed metric spaces to a limit. Gromov formulated an important compactness theorem in this setting, giving a condition under which a sequence of pointed and "proper" metric spaces must have a subsequence which converges. This was later reformulated by Gromov and others into the more flexible notion of an ultralimit.
Gromov's compactness theorem had a deep impact on the field of geometric group theory. He applied it to understand the asymptotic geometry of the word metric of a group of polynomial growth, by taking the limit of well-chosen rescalings of the metric. By tracking the limits of isometries of the word metric, he was able to show that the limiting metric space has unexpected continuities, and in particular that its isometry group is a
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
. As a consequence he was able to settle the Milnor-Wolf conjecture as posed in the 1960s, which asserts that any such group is virtually nilpotent. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces. Important developments on this topic were given by Bruce Kleiner, Bernhard Leeb, and Pierre Pansu, among others.
Another consequence is Gromov's compactness theorem, stating that the set of compact
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
≤ ''D'' is relatively compact in the Gromov–Hausdorff metric. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ ''c'', a class of
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
Gromov's theory of pseudoholomorphic curves is one of the foundations of the modern study of symplectic geometry. Although he was not the first to consider pseudo-holomorphic curves, he uncovered a "bubbling" phenomena paralleling Karen Uhlenbeck's earlier work on Yang–Mills connections, and Uhlenbeck and Jonathan Sack's work on harmonic maps. In the time since Sacks, Uhlenbeck, and Gromov's work, such bubbling phenomena has been found in a number of other geometric contexts. The corresponding
compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generall ...
encoding the bubbling allowed Gromov to arrive at a number of analytically deep conclusions on existence of pseudo-holomorphic curves. A particularly famous result of Gromov's, arrived at as a consequence of the existence theory and the monotonicity formula for minimal surfaces, is the " non-squeezing theorem," which provided a striking qualitative feature of symplectic geometry. Following ideas of
Edward Witten
Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
, Gromov's work is also fundamental for Gromov-Witten theory, which is a widely studied topic reaching into
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
,
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, and symplectic geometry. From a different perspective, Gromov's work was also inspirational for much of Andreas Floer's work.
Yakov Eliashberg and Gromov developed some of the basic theory for symplectic notions of convexity. They introduce various specific notions of convexity, all of which are concerned with the existence of one-parameter families of diffeomorphisms which contract the symplectic form. They show that convexity is an appropriate context for an h-principle to hold for the problem of constructing certain
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the ...
s. They also introduced analogous notions in contact geometry; the existence of convex contact structures was later studied by Emmanuel Giroux.
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
: 1970 (Nice), 1978 (Helsinki), 1983 (Warsaw), 1986 (Berkeley)
* Foreign member of the
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
(1989), the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
(1989), the
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters (, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway.
History
The Royal Frederick University in Christiania was establis ...
, the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
French Academy of Sciences
The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...
(1997)
* Delivered the 2007 Pál Turán Memorial Lectures.