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 Boksitogorsk () is a town and the administrative center of Boksitogorsky District in Leningrad Oblast, Russia, located on the banks of the Pyardomlya River in the basin of the Syas River, east of St. Petersburg. Population: History Th ...

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 Russian(s) may refer to: *Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *A citizen of Russia *Russian language, the most widely spoken of the Slavic languages *''The Russians'', a b ...

-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 Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnasiu ...

, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at

Leningrad State University Saint Petersburg State University (SPBGU; ) is a public research university in Saint Petersburg, Russia, and one of the oldest and most prestigious universities in Russia. Founded in 1724 by a decree of Peter the Great, the university from the be ...

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 ...

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 Pierre and Marie Curie University ( , UPMC), also known as Paris VI, was a public research university in Paris, France, from 1971 to 2017. The university was located on the Jussieu Campus in the Latin Quarter of the 5th arrondissement of Paris, ...

and in 1982 he became a permanent professor at the

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 ...

from 1991 to 1996, and at the

Courant Institute of Mathematical Sciences The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU). Founded in 1935, it is named after Richard Courant, one of the founders of the Courant Institute ...

in New York since 1996. He adopted French citizenship in 1992.

Work

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 ...

,. 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

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 Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley. A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of ...

and

Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty ...

, Gromov introduced the

h-principle In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, su ...

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 In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is dif ...

) 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 Yakov Matveevich Eliashberg (also Yasha Eliashberg; ; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR. Education and career Eliashberg received his PhD, entitled ''Surgery of Singularities of Smooth Mappin ...

, 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 In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...

. 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 map In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...

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

Vitali Milman Vitali Davidovich Milman (; ) (born 23 August 1939) is a mathematician specializing in analysis. He is a professor at the Tel Aviv University. In the past he was a President of the Israel Mathematical Union and a member of the “Aliyah” committ ...

gave a formulation of the

concentration of measure In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random var ...

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 In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...

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 Michel Pierre Talagrand (; born 15 February 1952) is a French mathematician working in probability theory, functional analysis and mathematical physics. Doctor of Science since 1977, he has been, since 1985, directeur de recherches at CNRS and a ...

. Since the seminal 1964 publication of

James Eells James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis. Biography Eells was born on 25 October 1926, in Cleveland, Ohio. Eells studied mathematics at Bowdoin College in Ma ...

and

Joseph Sampson Joseph Sampson (October 16, 1794 – May 21, 1872) was a 19th-century American businessman and merchant. He was among the founding shareholders of Chemical Bank in 1823. Early life Sampson was born in Plympton, Massachusetts in 1794. He wa ...

harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a ...

s, 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 Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps. Earl ...

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 In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions ...

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

In 1978, Gromov introduced the notion of almost flat manifolds. The famous

quarter-pinched sphere theorem In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. I ...

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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

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 In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, the qu ...

. The proof works by replaying the proofs of the Bieberbach theorem and

Margulis lemma In differential geometry, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-space). Roughly, it states that within a fixed r ...

. Gromov's proof was given a careful exposition by Peter Buser and Hermann Karcher. In 1979,

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 In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

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 Herbert Blaine Lawson, Jr. is a mathematician best known for his work in minimal surfaces, calibrated geometry, and algebraic cycles. He is currently a Distinguished Professor of Mathematics at Stony Brook University. He received his PhD fr ...

gave another proof of Schoen and Yau's results, making use of elementary geometric constructions. They also showed how purely topological results such as

h-cobordism theorem In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M ...

could then be applied to draw conclusions such as the fact that every

and

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoint ...

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 In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness ...

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 Jeff Cheeger (born December 1, 1943) is an American mathematician and Silver Professor at the Courant Institute of Mathematical Sciences of New York University. His main interest is differential geometry and its connections with topology and an ...

's fundamental compactness theory for Riemannian manifolds, a key step in constructing coordinates on the limiting space is an

injectivity radius This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provid ...

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 Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his ...

's "noncollapsing theorem" for

Ricci flow In differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion o ...

, 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 Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

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 In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov. Definition A closed manifold ''M'' is called essential if its fundamental class 'M''defines a nonzero element ...

M

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

C(n)\operatorname(M)^

Gromov−Hausdorff convergence and geometric group theory

In 1981, Gromov introduced the Gromov–Hausdorff metric, which endows the set of all

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 In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces X_n. The concept captures the limiting behavior of finite configurations in the X_n spaces employing an ultrafilter to bypass ...

. Gromov's compactness theorem had a deep impact on the field of

geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...

. He applied it to understand the asymptotic geometry of the

word metric In group theory, a word metric on a discrete group G is a way to measure distance between any two elements of G . As the name suggests, the word metric is a metric on G , assigning to any two elements g , h of G a distance d(g,h) that me ...

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 In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group ''G'' is said to b ...

. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces. Important developments on this topic were given by

Bruce Kleiner Bruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory. He received his Ph.D. in 1990 from the University of California, Berkeley. His advisor was Wu-Yi Hsiang. Kleiner is a p ...

, Bernhard Leeb, and

Pierre Pansu Pierre Pansu (born 13 July 1959) is a French mathematician and a member of the Arthur Besse group and a close collaborator of Mikhail Gromov. He is a professor at the Université Paris-Sud 11 and the École Normale Supérieure in Paris. His mai ...

, among others. Another consequence is Gromov's compactness theorem, stating that the set of compact

s with

≥ ''c'' and

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 mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...

in the Gromov–Hausdorff metric. The possible limit points of sequences of such manifolds are

Alexandrov space In geometry, Alexandrov spaces with curvature ≥ ''k'' form a generalization of Riemannian manifolds with sectional curvature ≥ ''k'', where ''k'' is some real number. By definition, these spaces are locally compact complete length spaces wh ...

s of curvature ≥ ''c'', a class of

s studied in detail by Burago, Gromov and Perelman in 1992. Along with

Eliyahu Rips Eliyahu Rips (; ; ; 12 December 1948 – 19 July 2024) was an Israeli mathematician of Latvian origin known for his research in geometric group theory. He became known to the general public following his co-authoring a paper on what is popularl ...

, Gromov introduced the notion of

hyperbolic group In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abs ...

Symplectic geometry

Gromov's theory of

pseudoholomorphic curves In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or ''J''-holomorphic curve) is a smooth map, from a Riemann surface into an almost complex manifold, that satisfies the Cauchy–Riemann equations. Introduced in 1985 ...

is one of the foundations of the modern study of

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

. Although he was not the first to consider pseudo-holomorphic curves, he uncovered a "bubbling" phenomena paralleling

Karen Uhlenbeck Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W ...

's earlier work on Yang–Mills connections, and Uhlenbeck and Jonathan Sack's work on

s. 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 surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s, is the "

non-squeezing theorem The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder vi ...

," 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

. From a different perspective, Gromov's work was also inspirational for much of

Andreas Floer Andreas Floer (; 23 August 1956 – 15 May 1991) was a German mathematician who made seminal contributions to symplectic topology, and mathematical physics, in particular the invention of Floer homology. Floer's first pivotal contribution was a s ...

's work.

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

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

; the existence of convex contact structures was later studied by Emmanuel Giroux.

Prizes and honors

Prizes

* Prize of the Mathematical Society of Moscow (1971) *

Oswald Veblen Prize in Geometry __NOTOC__ The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was funded in 1961 in memory of Oswald Veblen and first issued in 1964. The Veblen Prize is n ...

(

AMS AMS or Ams may refer to: Organizations Companies * Alenia Marconi Systems * American Management Systems * AMS (Advanced Music Systems) * ams AG, semiconductor manufacturer * AMS Pictures * Auxiliary Medical Services Educational institutions ...

) (1981) * Prix Elie Cartan de l'Academie des Sciences de Paris (1984) * Prix de l'Union des Assurances de Paris (1989) *

Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...

(1993) *

Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

for Seminal Contribution to Research (

) (1997) *

Lobachevsky Medal The Lobachevsky Prize, awarded by the Russian Academy of Sciences, and the Lobachevsky Medal, awarded by the Kazan State University, are mathematical awards in honor of Nikolai Ivanovich Lobachevsky. History The Lobachevsky Prize was establishe ...

(1997) *

Balzan Prize The International Balzan Prize Foundation awards four annual monetary prizes to people or organizations who have made outstanding achievements in the fields of humanities, natural sciences, culture, as well as for endeavours for peace and the b ...

for Mathematics (1999) *

Kyoto Prize The is Japan's highest private award for lifetime achievement in the arts and sciences. It is given not only to those that are top representatives of their own respective fields, but to "those who have contributed significantly to the scientific, ...

in Mathematical Sciences (2002) *

Nemmers Prize in Mathematics Larry Nemmers (born July 12, 1943) is a retired educator and better known as a former American football official in the National Football League (NFL). Nemmers made his debut as an NFL official in the 1985 season and continued in this role unt ...

(2004) *

Bolyai Prize The International János Bolyai Prize of Mathematics is an international prize founded by the Hungarian Academy of Sciences. The prize is named after János Bolyai and is awarded every five years to mathematicians for monographs with important new ...

in 2005 * Abel Prize in 2009 "for his revolutionary contributions to geometry"

Honors

* Invited speaker to

: 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 ...

(2011), and the

National Academy of Sciences of Ukraine The National Academy of Sciences of Ukraine (NASU; , ; ''NAN Ukrainy'') is a self-governing state-funded organization in Ukraine that is the main center of development of Science and technology in Ukraine, science and technology by coordinatin ...

(2023). * Member of the

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 Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. In 1940, because of his Jewish origins, he was arrested by History of the Jews in Hun ...

Memorial Lectures.

Publications

Books Major articles

Notes

References

Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Biography After studying from 1948 to 19 ...

,
Encounter with a Geometer, Part I
, '' AMS Notices'', Volume 47, Number 2 * Marcel Berger,
Encounter with a Geometer, Part II
", ''AMS Notices'', Volume 47, Number 3

External links

Personal page at Institut des Hautes Études Scientifiques

*
Anatoly Vershik, "Gromov's Geometry"
{{DEFAULTSORT:Gromov, Mikhail 1943 births Living people Jewish French scientists People from Boksitogorsk Russian people of Jewish descent Russian emigrants to France Foreign associates of the National Academy of Sciences Foreign members of the Russian Academy of Sciences Kyoto laureates in Basic Sciences Differential geometers Russian mathematicians 20th-century French mathematicians 21st-century French mathematicians French people of Russian-Jewish descent Group theorists New York University faculty Wolf Prize in Mathematics laureates Geometers Members of the French Academy of Sciences Members of the Norwegian Academy of Science and Letters Abel Prize laureates Foreign members of the Royal Society Soviet mathematicians University of Maryland, College Park faculty Russian scientists