The Mathematics of Mikhail Gromov
Short biography of Mikhail Gromov
Mikhail Leonidovich Gromov, also known as Michael Gromov or Misha Gromov, was born on 23 December, 1943, in the small town Boksitogorsk close to Leningrad, now Saint Petersburg, in Russia. He received all of his university education at the University of Saint Petersburg. For his doctoral studies, he was a student of the eminent topologist V.A. Rokhlin, obtaining his PhD in 1969 and writing his Post-doctoral Thesis in 1973. In 1974, Gromov emigrated from Russia and he became a naturalized French citizen in 1992. Since 1982, he has been a permanent Professor at the Institut des Hautes Études Scientifiques de Bures-sur-Yvette, and Jay Gould Professor of Mathematics at New York University.
The task of mathematics formulated by Gromov
Let me start the presentation of the work of Mikhail Gromov by a quotation from a speech to the Balzan Prize Foundation in 1999, where he clearly formulates his views of mathematics:
“The task of mathematics and mathematicians is to articulate the visible regularities in the physical and mental worlds, and to find new structural patterns unperceivable by direct intuition and common sense.”
The work of Gromov in a metric perspective
If a single fundamental mathematical concept should be mentioned as central in the work of Gromov, it is the notion of distance which he has introduced in completely surprising situations and exploited with elegance.
The notion of distance is used to measure whether two physical objects, two states of a physical or biological system, or two abstract ideas are mutually close or far apart. A mathematician will speak of an abstract space of the objects in question, called points in this connection, and will think of the distance as a particular assignment of a positive number to each pair of points in the space. Familiar situations deal with ordinary 3-dimensional Euclidean space, where the distance between two points is given by the length of the straight line segment between them. For the inhabitants in a mountain country, this distance is only of modest practical value. A more useful distance in this case is given by the length of the shortest available path between two given locations.
A metric perspective
It is not surprising that a geometrical object M such as a surface in 3-dimensional Euclidean space can have a notion of distance associated to it, so that one can speak of the distance - a nonnegative number denoted d(x,y) - between any two points x and y in M. If the distance d(x,y) satisfies a few natural conditions, M is called a metric space with metric d(x,y).
There are many other contexts where a collection of points can be equipped with a metric and thereby can be viewed as a metric space. An abstract example comes from the mathematical model of colours. In the so-called RGB decimal code, every colour is characterized by a triplet of integers in the range 0 to 255 (three bytes); the first integer determines the amount of Red, the second the amount of Green, and the third integer the amount of Blue. Accordingly, the RGB decimal code for red is given by [255, 0, 0], the code for green by [0, 255, 0], and the code for blue by [0, 0, 255]. The code for yellow is
[255, 255, 0], and the code for white is [255,255,255]. Using the RGB decimal code we can organize the set of all colours in a space with 256×256×256=16.777.216 points. As an appropriate definition of a distance between two colours one can now use the sum of the absolute values of the differences between the three integers in the RGB decimal codes for the two colours. With this definition of metric in the colour space, the distance between red and blue is 510.
In an arbitrary metric space M with metric d(x,y), Gromov shows that one can introduce a kind of product (x∙y)o
- now called the Gromov product - of any two points x and y in M with respect to a fixed reference point ο in M by the formula
(x∙y)o = ½[d(o,x) + d(o,y) - d(x,y)].
Using his product, Gromov goes on to define the notion of a hyperbolic space in the class of all metric spaces, by requiring that the Gromov product satisfies the inequality
(x∙y)o ≥ min{(x∙z)o, (y∙z)o} - δ,
for some δ ≥ 0, and any four points o, x, y, z in M. The inequality for hyperbolicity can be phrased in a more geometrical language using suitably defined geodesic triangles in the metric space, generalizing the intuitive concept of a triangle as a closed figure having three curves of shortest length (geodesics) as sides.
This type of metric space is now called a Gromov hyperbolic space. It provides a global approach to such objects as the classical non-Euclidean plane (the hyperbolic plane), simply-connected Riemannian manifolds with pinched negative sectional curvature and other interesting spaces. Gromov has also applied such hyperbolic properties in the context of group theory.
An interlude about manifolds
Before we can proceed to describe some of the many contributions to mathematics by Gromov, we need to explain the notion of a manifold in some detail.
Curves in the Euclidean plane and surfaces in 3-dimensional Euclidean space are familiar examples of manifolds of dimension 1 and 2. In higher dimensions many interesting examples of manifolds occur as configuration spaces (the possible positions) for mechanical systems.
As a simple example, consider a double pendulum in the Euclidean plane. In relation to an oriented axis, the possible configurations of the double pendulum are determined by two angles, or alternatively, by a pair of points on two circles, which in turn can be identified with a single point on a ring-shaped surface in 3-dimensional space called a torus. The configuration space for a planar double pendulum is, in other words, a torus.
If we consider instead a double pendulum in 3-dimensional Euclidean space, the possible configurations of this double pendulum are determined by a pair of points on two spheres of dimension 2. Hence the configuration space for a spatial double pendulum can be identified with a 4-dimensional manifold - the product of two spheres - in a Euclidean space of six dimensions.
In more detail, an n-dimensional manifold M is a geometrical object, which locally can be described by n parameters (coordinates). Loosely speaking, this entails that each point p in the object M has a neighbourhood U in one-to-one correspondence with an n-dimensional cube of ordered n–tuples (x1,x2, ... ,xn) of real numbers. In other words, the points x in the neighbourhood U of p can each be assigned n coordinates. We call such an identification of a neighbourhood with a cube of ordered n–tuples for a local coordinate system on M. In our intuitive conception of the notions of curve, surface, and solid body in 3-dimensional space, these geometrical objects can locally be parameterized by 1, 2, and 3 parameters, respectively, exhibiting these objects as manifolds of dimension 1, 2, and 3.
Suppose we have an n-dimensional manifold M. If we can cover M by a collection of local coordinate systems where exchange of coordinates in overlapping systems is smooth (differentiable), we say that the manifold is a smooth manifold.
If the dimension of the manifold M is an even number 2n, then the real coordinates in a local coordinate system on M can be organized into n complex coordinates. If the local coordinate systems in a covering of M can be chosen such that exchange of coordinates in overlapping systems is holomorphic (differentiable in the complex sense) then the manifold is a complex manifold.
Turning the space of Riemannian manifolds into a metric space
The study of manifolds involves a local study of geometrical invariants and a global study of topological invariants. Geometrical invariants concern properties of manifolds preserved by isometric mappings (rigid motions) and topological invariants concern properties preserved by more flexible mappings called homeomorphisms, or even by continuous deformations called homotopies.
It has been said that a topologist is a mathematician who cannot tell the difference between a doughnut and a cup of coffee.
During the last 50 years, investigations of the connections between the topology and the geometry of manifolds have received great attention in mathematics, with Gromov as an extraordinary contributor of brilliant new ideas. The notion of curvature is central in this context.
A smooth curve C in a Euclidean plane has a natural concept of curvature attached to it. The curvature of C at a point p can be defined as the reciprocal of the radius in the circle approximating C best possible in a neighbourhood of p. If the plane of C has a preferred sense of rotation, and if C is traversed in a specified direction, then we can give the curvature a sign. The sign is + 1 if C follows the preferred rotation in a neighbourhood of p, and -1 if it follows the opposite rotation.
Similarly, a smooth surface S in 3-dimensional Euclidean space has a natural concept of curvature attached to it. At a point p in the surface S consider all curves through p obtained by sectioning the surface with planes containing the normal (perpendicular) line to the surface at p. The planar curves obtained by this sectioning process (called normal sections) all have a curvature. Giving the curvature of all normal sections through the point p on S a sign depending on the turning of the curve in relation to the normal of the surface, we define the notion of normal curvatures of S at p. One of these curvatures is smallest possible and one is largest possible. The product of the maximal and minimal normal curvatures (with sign) defines the Gaussian curvature, or just curvature, of S at p.
An intrinsic way of viewing curvature of the surface S at a point p is to consider small geodesic triangles in S with a corner at p. A geodesic triangle in S is by definition a closed piece of surface with three points as corners and three curves of shortest length (geodesics) as sides. As already known to Gauss, the curvature of the surface S at p can be determined by taking an appropriate limit of the sum of the interior angles in infinitesimal geodesic triangles with a corner at p. The curvature is zero if this limit is exactly 180 degrees (like in ordinary plane geometry), negative if it less than 180 degrees, and positive if it is greater than 180 degrees. A cylinder has curvature zero, a sphere has positive curvature, and a surface named the pseudosphere has negative curvature.
Negative curvature
Pseudosphere
Curvature 0
Cylinder
Positive curvature
Sphere
In higher dimensions, a smooth manifold M often comes equipped with a metric tensor - a device by which you can measure lengths of curves and angles between curves in the manifold. Technically speaking, the metric tensor is a symmetric, non-degenerate 2-tensor. Using the metric tensor one can define surfaces of geodesic triangles in complete generality in such manifolds M and introduce a notion of sectional curvature, corresponding to Gaussian curvature of the geodesic triangles. Since the general ideas originate in work of Bernhard Riemann, smooth manifolds M equipped with a metric tensor are named Riemannian manifolds, and the study of their geometry is called Riemannian geometry.
Gromov has made stunning contributions to Riemannian geometry. A particular contribution showing his daring imagination at its height, is his definition of a natural metric structure on the set of all (isomorphism classes) of Riemannian manifolds. The definition of the metric exploits an idea of Felix Hausdorff and is now known as the Gromov-Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single, connected moduli space (the set of isomorphism classes). In this moduli space, convergence allows the collapse of dimension unfolding a very rich geometry, which has been exploited in the work of Gromov and several other eminent mathematicians.
As a spectacular specific result on Riemannian manifolds obtained by Gromov we mention that he succeeded in giving estimates of the Betti numbers – important integers associated with the topology of a manifold - solely in terms of a lower bound on the sectional curvature.
As a last example of Gromov’s contributions to the theory of Riemannian manifolds, we mention his proof in 1981 of the rigidity of locally symmetric spaces in the class of all Riemannian manifolds of non-positive curvature, generalizing a classical rigidity theorem for locally symmetric spaces due to George Mostow.
Compatible almost complex structures on symplectic manifolds
Symplectic geometry, also called symplectic topology when global questions are studied, has its origins in the Hamiltonian formulation of classical mechanics where the phase space of a mechanical system without friction has the structure of a symplectic manifold.
Research in symplectic geometry has applications ranging from celestial mechanics to accelerator particle physics, solid body and fluid mechanics. For systems with a finite number of degrees of freedom, the relevant space is an even dimensional manifold, often called the phase space of the system, and identified with the cotangent bundle of the configuration space for the system. This manifold can, in a natural way, be given a so-called symplectic structure. The usefulness of this construct in the study of dynamical systems is the insight which it gives to the existence of, and properties of, quantities that are conserved by the time evolution. Releasing the dynamics from the bonds of coordinates, geometrical (coordinate-free) mechanics has provided tremendous insight into the deep workings of the mechanical universe.
A classical example is provided by a system of three celestial bodies: say the Sun, the Earth and the Moon. The system is driven by Newton's law of gravitation, where the future is determined by the positions and velocities of the three bodies at a given moment. In this case, the phase space of the system is a manifold of dimension 18, where each point in the manifold corresponds to a configuration of possible positions and velocities of the three bodies. Newton's laws can be encoded into a particular transformation of the phase space: each point goes to a new point, corresponding to the positions and velocities of the bodies after one second of motion ruled by Newton's laws. This transformation distorts any conceivable distance in the space and is very far from preserving distances (being a rigid motion). But it does preserve the symplectic structure, which is related to the area of surfaces in the 18-dimensional manifold. However, it is not the usual area and it is difficult to give an intuitive picture of the symplectic structure. Still it is useful to define symplectic invariants since these are preserved by the above transformation and therefore eventually may provide useful information about the mechanical systems.
In general a symplectic manifold is a smooth manifold M equipped with a symplectic form - a device by which you can measure areas of surfaces inside the manifold. Technically speaking, the symplectic form is a closed, non-degenerate 2-form. Symplectic geometry is the study of symplectic manifolds.
Symplectic geometry has a number of similarities and differences with Riemannian geometry. Unlike Riemannian manifolds, symplectic manifolds have no local invariants such as curvature. This is a consequence of a classical theorem of Darboux stating that any point of a 2n-dimensional symplectic manifold has a neighborhood which is isomorphic to a standard symplectic structure on an open set of R2n. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are topological restrictions. For example, every symplectic manifold is necessarily of even dimension and orientable. Among the spheres of any dimension, only the 2-dimensional sphere admits a symplectic form.
An even dimensional manifold M which can be given a special type of complex structure turning it into a Kähler manifold is also a symplectic manifold. A symplectic manifold of this type is said to have an integrable complex structure compatible with the symplectic form.
Most symplectic manifolds are not of Kähler type, and hence do not have an integrable complex structure compatible with the symplectic form. In a path-breaking work of 1985, Mikhail Gromov made the important observation that symplectic manifolds do admit plenty of compatible almost complex structures, and that they satisfy enough of the properties of Kähler manifolds to be useful. But exchanges of coordinates in the manifolds are not holomorphic, i.e. complex differentiable, so cleverness is required.
Gromov has used the existence of almost complex structures on symplectic manifolds to develop a theory of pseudoholomorphic curves, which are special maps of Riemann surfaces (1-dimensional complex manifolds) into almost complex manifolds. The highly original concept of pseudoholomorphic curves has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov-Witten invariants. These invariants play a key role in string theory.
Pioneering work in geometric group theory
Around 1872, Felix Klein made the proposal of using group theory as a tool to study geometry. Geometric group theory may be viewed as Klein's program in reverse, namely to use geometrical methods to study groups. Two major sources of inspiration in this field are low dimensional topology and hyperbolic geometry. Important work of Max Dehn back in the 1920s on the fundamental group of a hyperbolic Riemann surface can be viewed in this context, and work of William Thurston in the late 1970s intimately linking the study of 3-dimensional manifolds to hyperbolic geometry gave a major boost to establish the subject as a field of mathematics in its own right.
A great deal of the pioneering work in geometric group theory has been carried out by Mikhail Gromov. In particular, he has introduced and developed the notion of a hyperbolic group in the early 1980s. By definition a hyperbolic group, also known as a Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. Gromov noticed that many of the results of Dehn concerning the fundamental group of a hyperbolic Riemann surface did not rely either on it having dimension two or even on being a manifold and hold in a much more general context. In a seminal paper of 1987, Gromov proposed a far ranging research program for hyperbolic groups, which has inspired an entire generation of geometric group theorists.
A special highlight among the results of Gromov in geometric group theory is his proof of an old conjecture according to which a finitely generated group of polynomial growth has a nilpotent subgroup of finite index. Another highlight is his contributions to the construction of non-arithmetic discrete groups of hyperbolic transformations in arbitrary dimension.
Contributions to a geometric theory of partial differential equations
Klein bottle
Mikhail Gromov has also made important contributions to analysis. In particular he is known for establishing a homotopy method to solve differential relations, known as the h-principle. The h-principle is a cornerstone in the foundations of a geometric theory of partial differential equations.
This work was initiated in his doctoral dissertation of 1969, where Gromov found an astonishing new method, by which he could generalize a number of immersion theorems for manifolds M of dimension n into manifolds N of dimensions higher than n. An immersion is a mapping of M into N which leaves a smooth image of M in N, but possibly with self-intersections. A well-known example is provided by the image of the 2-dimensional manifold known as the Klein bottle in 3-dimensional Euclidean space.
Theorems about immersions and embeddings were pioneered by Hassler Whitney in the late 1930s and followed by important work of Stephen Smale and Morris Hirsch in the 1960s. In particular Smale’s proof in1962 that one can turn a sphere in Euclidean 3-space ‘inside out’ through a smooth homotopy of immersions struck the mathematical world with complete surprise.
It came as a similar surprise to mathematicians when Gromov presented his elegant ideas on immersion theory. Special cases of his result were subsequently unfolded in works by other eminent mathematicians. Not only in this connection but in all of his work, one can say that Mikhail Gromov has turned the mathematical world inside out and shown us deep interrelations between mathematical concepts from many different mathematical fields.
Concluding remarks
At the beginning of the new millennium, Mikhail Gromov stands out as one of the most creative mathematicians of our time. His work is unique and brings a profoundly original approach to the mathematical subjects he works with, and often transcends his own field of geometry. He has created brilliant new concepts and devised elegant new techniques, which have been applied to problems, some of which were long standing and seemed to be inaccessible. Furthermore, several of the new techniques developed by Gromov for different purposes, led to completely new kinds of problems.
The creativity and originality of Mikhail Gromov as a mathematician rank at the level of Abel and he deserves our warmest congratulations on receiving this prize dedicated to the memory of Abel.
