Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
who is known for his contributions to the fields of
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
,
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
, and
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
. He is widely regarded as one of the greatest living mathematicians.
In the 1990s, partly in collaboration with
Yuri Burago
Yuri Dmitrievich Burago (russian: Ю́рий Дми́триевич Бура́го) (born 1936) is a Russian mathematician. He works in differential geometry, differential and convex geometry.
Education and career
Burago studied at Saint Pete ...
,
Mikhael Gromov, and Anton Petrunin, he made contributions to the study of
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 where t ...
s. In 1994, he proved the
soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of
Ricci flow
In the mathematical fields of 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 analo ...
, and proved the
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
and
Thurston's geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensi ...
, the former of which had been a famous
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...
in mathematics for the past century. The full details of Perelman's work were filled in and explained by various authors over the following several years.
In August 2006, Perelman was offered the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the
Ricci flow
In the mathematical fields of 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 analo ...
", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." On 22 December 2006, the scientific journal ''
Science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
'' recognized Perelman's proof of the
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
as the scientific "
Breakthrough of the Year
The Breakthrough of the Year is an annual award for the most significant development in scientific research made by the AAAS journal ''Science,'' an academic journal covering all branches of science. Originating in 1989 as the ''Molecule of the Ye ...
", the first such recognition in the area of mathematics.
On 18 March 2010, it was announced that he had met the criteria to receive the first Clay
Millennium Prize for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of
Richard S. Hamilton, the mathematician who pioneered the
Ricci flow
In the mathematical fields of 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 analo ...
partly with the aim of attacking the conjecture.
He had previously rejected the prestigious prize of the European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...
in 1996.[
]
Early life and education
Grigori Yakovlevich Perelman was born in Leningrad
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
parents, Yakov (who now lives in Israel)[ and Lyubov (who still lives in Saint Petersburg with Grigori).][ Grigori's mother Lyubov gave up graduate work in mathematics to raise him. Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school mathematics training program.]
His mathematical education continued at the Leningrad Secondary School 239, a specialized school
Specialist schools, also known as specialised schools or specialized schools, are schools which specialise in a certain area or field of curriculum. In some countries, for example New Zealand, the term is used exclusively for schools specialis ...
with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education
Physical education, often abbreviated to Phys Ed. or P.E., is a subject taught in schools around the world. It is usually taught during primary and secondary education, and encourages psychomotor learning by using a play and movement explorati ...
. In 1982, as a member of the Soviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national ...
team competing in the International Mathematical Olympiad
The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except i ...
, an international competition for high school students, he won a gold medal, achieving a perfect score. He continued as a student of The School of Mathematics and Mechanics at the Leningrad State University
Saint Petersburg State University (SPBU; russian: Санкт-Петербургский государственный университет) is a public research university in Saint Petersburg, Russia. Founded in 1724 by a decree of Peter the G ...
, without admission examinations and enrolled to the university.
After completing his PhD in 1990, Perelman began work at the Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences
The Academy of Sciences of the Soviet Union was the highest scientific institution of the Soviet Union from 1925 to 1991, uniting the country's leading scientists, subordinated directly to the Council of Ministers of the Soviet Union (until 1946 ...
, where his advisors were Aleksandr Aleksandrov and Yuri Burago
Yuri Dmitrievich Burago (russian: Ю́рий Дми́триевич Бура́го) (born 1936) is a Russian mathematician. He works in differential geometry, differential and convex geometry.
Education and career
Burago studied at Saint Pete ...
. In the late 1980s and early 1990s, with a strong recommendation from the geometer Mikhail Gromov, Perelman obtained research positions at several universities in the United States. In 1991, Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society
The Saint Petersburg Mathematical Society (russian: Санкт-Петербургское математическое общество) is a mathematical society run by Saint Petersburg mathematicians.
Historical notes
The St. Petersburg Mathe ...
for his work on Aleksandrov's spaces of curvature bounded from below. In 1992, he was invited to spend a semester each at the Courant Institute
The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU), and is among the most prestigious mathematics schools and mathematical sciences research cente ...
in New York University
New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then-Secretary of the Treasury Albert Gallatin.
In 1832, the ...
and Stony Brook University
Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public research university in Stony Brook, New York. Along with the University at Buffalo, it is one of the State University of New York system's ...
where he began work on manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s with lower bounds on 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 ...
. From there, he accepted a two-year Miller Research Fellows The Miller Research Fellows program is the central program of the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science on the University of California Berkeley campus. The program constitutes the support of Research Fellows - a ...
hip at the University of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
in 1993. After having proved the soul conjecture in 1994, he was offered jobs at several top universities in the US, including Princeton
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the nine ...
and Stanford
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is considere ...
, but he rejected them all and returned to the Steklov Institute in Saint Petersburg in the summer of 1995 for a research-only position.[
]
Early research
Convex geometry
In his undergraduate studies, Perelman dealt with issues in the field of convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbe ...
. His first published article studied the combinatorial structures arising from intersections of convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
. With I. V. Polikanova, he established a measure-theoretic formulation of Helly's theorem
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's t ...
. In 1987, the year he began graduate studies, he published an article controlling the size of circumscribed cylinders by that of inscribed sphere
In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and i ...
s.
Negatively curved hypersurfaces
Surfaces of negative curvature were the subject of Perelman's graduate studies. His first result was on the possibility of prescribing the structure of negatively-curved polyhedral surfaces in three-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. He proved that any such metric on the plane which is complete can be continuously immersed as a polyhedral surface. Later, he constructed an example of a smooth hypersurface of four-dimensional Euclidean space which is complete and has Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
negative and bounded away from zero. Previous examples of such surfaces were known, but Perelman's was the first to exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes. As such, his construction provided further obstruction to the extension of a well-known theorem of Nikolai Efimov
Nikolai Vladimirovich Yefimov (russian: Никола́й Влади́мирович Ефи́мов; 31 May 1910 in Orenburg – 14 August 1982 in Moscow) was a Soviet mathematician. He is most famous for his work on generalized Hilbert's problem o ...
to higher dimensions.
Alexandrov spaces
Perelman's first works to a have a major impact on the mathematical literature were in the field of 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 where t ...
s, the concept of which dates back to the 1950s. In a very well-known paper coauthored with Yuri Burago
Yuri Dmitrievich Burago (russian: Ю́рий Дми́триевич Бура́го) (born 1936) is a Russian mathematician. He works in differential geometry, differential and convex geometry.
Education and career
Burago studied at Saint Pete ...
and Mikhael Gromov, Perelman established the modern foundations of this field, with the notion of Gromov–Hausdorff convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.
Gromov–Hausdorff distance
The Gromov–Hausdorff ...
as an organizing principle. In a followup unpublished paper, Perelman proved his "stability theorem," asserting that in the collection of all Alexandrov spaces with a fixed curvature bound, all elements of any sufficiently small metric ball around a compact space are mutually homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
. Vitali Kapovitch, who described Perelman's article as being "very hard to read," later wrote a detailed version of Perelman's proof, making use of some further simplifications.
Perelman developed a version of Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...
on Alexandrov spaces. Despite the lack of smoothness in Alexandrov spaces, Perelman and Anton Petrunin were able to consider the gradient flow of certain functions, in unpublished work. They also introduced the notion of an "extremal subset" of Alexandrov spaces, and showed that the interiors of certain extremal subsets define a stratification
Stratification may refer to:
Mathematics
* Stratification (mathematics), any consistent assignment of numbers to predicate symbols
* Data stratification in statistics
Earth sciences
* Stable and unstable stratification
* Stratification, or st ...
of the space by topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
s. In further unpublished work, Perelman studied DC functions (difference of concave functions) on Alexandrov spaces and established that the set of regular points has the structure of a manifold modeled on DC functions.
For his work on Alexandrov spaces, Perelman was recognized with an invited lecture at the 1994 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 Nevanlinna Prize (to be rename ...
.
Comparison geometry
In 1972, Jeff Cheeger
Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and ...
and Detlef Gromoll
Detlef Gromoll (13 May 1938 – 31 May 2008) was a mathematician who worked in Differential geometry.
Biography
Gromoll was born in Berlin in 1938, and was a classically trained violinist.
After living and attending school in Rosdorf and gra ...
established their important soul theorem
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by ...
. It asserts that every complete Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
of nonnegative 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 poi ...
has a compact nonnegatively curved submanifold, called a ''soul'', whose normal bundle is diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to the original space. From the perspective of homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
, this says in particular that every complete Riemannian metric of nonnegative sectional curvature may be taken to be 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, ...
. Cheeger and Gromoll conjectured that if the curvature is strictly positive somewhere, then the soul can be taken to be a single point, and hence that the original space must be diffeomorphic to Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. In 1994, Perelman gave a short proof of Cheeger and Gromoll's conjecture by establishing that, under the condition of nonnegative sectional curvature, Sharafutdinov's retraction is a submersion. Perelman's theorem is significant in establishing a topological obstruction to deforming a nonnegatively curved metric to one which is positively curved, even at a single point.
Some of Perelman's work dealt with the construction of various interesting Riemannian manifolds with positive 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 ...
. He found Riemannian metrics on the 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 ...
of arbitrarily many complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
s with positive Ricci curvature, bounded diameter, and volume bounded away from zero. Also, he found an explicit complete metric on four-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
with positive Ricci curvature and Euclidean volume growth, and such that the asymptotic cone
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and uses ...
is nonuniquely defined.
Geometrization and Poincaré conjectures
The problems
The Poincaré conjecture, proposed by mathematician Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
in 1904, was throughout the 20th century regarded as a key problem in topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. On the 3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
, defined as the set of points at unit length from the origin in four-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, any loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
can be contracted into a point. Poincaré suggested that a converse might be true: if a 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, ...
three-dimensional manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
has the property that any loop can be contracted into a point, then it must be topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fun ...
to a 3-sphere. 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 facult ...
proved a high-dimensional analogue of Poincaré's conjecture in 1961, and Michael Freedman
Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gen ...
proved the four-dimensional version in 1982. Despite their work, the case of three-dimensional spaces remained completely unresolved. Moreover, Smale and Freedman's methods have had no impact on the three-dimensional case, as their topological manipulations, moving "problematic regions" out of the way without interfering with other regions, seem to require high dimensions in order to work.
In 1982, William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurston ...
developed a novel viewpoint, making the Poincaré conjecture into a small special case of a hypothetical systematic structure theory of topology in three dimensions. His proposal, known as the Thurston geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...
, posited that given any closed three-dimensional manifold whatsoever, there is some collection of two-dimensional spheres and tori inside of the manifold which disconnect the space into separate pieces, each of which can be endowed with a uniform geometric structure. Thurston was able to prove his conjecture under some provisional assumptions. In John Morgan's view, it was only with Thurston's systematic viewpoint that most topologists came to believe that the Poincaré conjecture would be true.
At the same time that Thurston published his conjecture, Richard Hamilton introduced his theory of the Ricci flow
In the mathematical fields of 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 analo ...
. Hamilton's Ricci flow is a prescription, defined by a partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
formally analogous to the heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
, for how to deform a Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
on a manifold. The heat equation, such as when applied in the sciences to physical phenomena such as temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, models how concentrations of extreme temperatures will spread out until a uniform temperature is achieved throughout an object. In three seminal articles published in the 1980s, Hamilton proved that his equation achieved analogous phenomena, spreading extreme curvatures and uniformizing a Riemannian metric, in certain geometric settings. As a byproduct, he was able to prove some new and striking theorems in the field of Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
.
Despite formal similarities, Hamilton's equations are significantly more complex and nonlinear than the heat equation, and it is impossible that such uniformization is achieved without contextual assumptions. In completely general settings, it is inevitable that "singularities" occur, meaning that curvature accumulates to infinite levels after a finite amount of "time" has elapsed. Following Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
's suggestion that a detailed understanding of these singularities could be topologically meaningful, and in particular that their locations might identify the spheres and tori in Thurston's conjecture, Hamilton began a systematic analysis. Throughout the 1990s, he found a number of new technical results and methods, culminating in a 1997 publication constructing a "Ricci flow with surgery" for four-dimensional spaces. As an application of his construction, Hamilton was able to settle a four-dimensional curvature-based analogue of the Poincaré conjecture. Yau has identified this article as one of the most important in the field of geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
, saying that with its publication it became clear that Ricci flow could be powerful enough to settle the Thurston conjecture. The key of Hamilton's analysis was a quantitative understanding of how singularities occur in his four-dimensional setting; the most outstanding difficulty was the quantitative understanding of how singularities occur in three-dimensional settings. Although Hamilton was unable to resolve this issue, in 1999 he published work on Ricci flow in three dimensions, showing that if a three-dimensional version of his surgery techniques could be developed, and if a certain conjecture on the long-time behavior of Ricci flow could be established, then Thurston's conjecture would be resolved. This became known as the Hamilton program.
Perelman's work
In November 2002 and March 2003, Perelman posted two preprint
In academic publishing, a preprint is a version of a scholarly or scientific paper that precedes formal peer review and publication in a peer-reviewed scholarly or scientific journal. The preprint may be available, often as a non-typeset versio ...
s to arXiv
arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...
, in which he claimed to have outlined a proof of Thurston's conjecture. In a third paper posted in July 2003, Perelman outlined an additional argument, sufficient for proving the Poincaré conjecture (but not the Thurston conjecture), the point being to avoid the most technical work in his second preprint. Making use of the Almgren-Pitts min-max theory from the field of geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
, Tobias Colding
Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great grandchild of Ludwig August Colding.
Biography
He was born in Copenhagen, Denmark, to Torben Holck Colding ...
and William Minicozzi
William Philip Minicozzi II is an American mathematician. He was born in Bryn Mawr, Pennsylvania, Bryn Mawr, Pennsylvania, in 1967.
Career
Minicozzi graduated from Princeton University in 1990 and received his Ph.D. from Stanford University in 199 ...
provided a completely alternative proof of the results in Perelman's third preprint.
Perelman's first preprint contained two primary results, both to do with Ricci flow. The first, valid in any dimension, was based on a novel adaptation of Peter Li and Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
's differential Harnack inequalities to the setting of Ricci flow. By carrying out the proof of the Bishop-Gromov inequality for the resulting Li−Yau length functional, Perelman established his celebrated "noncollapsing theorem" for Ricci flow, asserting that local control of the size of the curvature implies control of volumes. The significance of the noncollapsing theorem is that volume control is one of the preconditions of Hamilton's ''compactness theorem''. As a consequence, Hamilton's compactness and the corresponding existence of subsequential limits could be applied somewhat freely.
The "canonical neighborhoods theorem" is the second main result of Perelman's first preprint. In this theorem, Perelman achieved the quantitative understanding of singularities of three-dimensional Ricci flow which had eluded Hamilton. Roughly speaking, Perelman showed that on a microscopic level, every singularity looks either like a cylinder collapsing to its axis, or a sphere collapsing to its center. Perelman's proof of his canonical neighborhoods theorem is a highly technical achievement, based upon extensive arguments by contradiction in which Hamilton's compactness theorem (as facilitated by Perelman's noncollapsing theorem) is applied to construct self-contradictory manifolds.
Other results in Perelman's first preprint include the introduction of certain monotonic quantities and a "pseudolocality theorem" which relates curvature control and isoperimetry
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
. However, despite being major results in the theory of Ricci flow, these results were not used in the rest of his work.
The first half of Perelman's second preprint, in addition to fixing some incorrect statements and arguments from the first paper, used his canonical neighborhoods theorem to construct a Ricci flow with surgery in three dimensions, systematically excising singular regions as they develop. As an immediate corollary of his construction, Perelman resolved a major conjecture on the topological classification in three dimensions of closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
s which admit 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 ...
. His third preprint (or alternatively Colding and Minicozzi's work) showed that on any space satisfying the assumptions of the Poincaré conjecture, the Ricci flow with surgery exists only for finite time, so that the infinite-time analysis of Ricci flow is irrelevant. The construction of Ricci flow with surgery has the Poincaré conjecture as a corollary.
In order to settle the Thurston conjecture, the second half of Perelman's second preprint is devoted to an analysis of Ricci flows with surgery, which may exist for infinite time. Perelman was unable to resolve Hamilton's 1999 conjecture on long-time behavior, which would make Thurston's conjecture another corollary of the existence of Ricci flow with surgery. Nonetheless, Perelman was able to adapt Hamilton's arguments to the precise conditions of his new Ricci flow with surgery. The end of Hamilton's argument made use of Jeff Cheeger
Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and ...
and Mikhael Gromov's theorem characterizing collapsing manifold
In Riemannian geometry, a collapsing or collapsed manifold is an ''n''-dimensional manifold ''M'' that admits a sequence of Riemannian metrics ''g'i'', such that as ''i'' goes to infinity the manifold is close to a ''k''-dimensional space, w ...
s. In Perelman's adaptation, he required use of a new theorem characterizing manifolds in which collapsing is only assumed on a local level. In his preprint, he said the proof of his theorem would be established in another paper, but he did not then release any further details. Proofs were later published by Takashi Shioya and Takao Yamaguchi, John Morgan and Gang Tian, Jianguo Cao and Jian Ge, and 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 ...
and John Lott
John Richard Lott Jr. (born May 8, 1958) is an American economist, political commentator, and gun rights advocate. Lott was formerly employed at various academic institutions and at the American Enterprise Institute conservative think tank. He ...
.
Verification
Perelman's preprints quickly gained the attention of the mathematical community, although they were widely seen as hard to understand since they had been written somewhat tersely. Against the usual style in academic mathematical publications, many technical details had been omitted. It was soon apparent that Perelman had made major contributions to the foundations of Ricci flow
In the mathematical fields of 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 analo ...
, although it was not immediately clear to the mathematical community that these contributions were sufficient to prove the geometrization conjecture or the Poincaré conjecture.
In April 2003, Perelman visited the Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
, Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
, Stony Brook University
Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public research university in Stony Brook, New York. Along with the University at Buffalo, it is one of the State University of New York system's ...
, Columbia University
Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...
, and New York University
New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then-Secretary of the Treasury Albert Gallatin.
In 1832, the ...
to give short series of lectures on his work, and to clarify some details for experts in the relevant fields. In the years afterwards, three detailed expositions appeared, discussed below. Since then, various parts of Perelman's work have also appeared in a number of textbooks and expository articles.
* In June 2003, 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 ...
and John Lott
John Richard Lott Jr. (born May 8, 1958) is an American economist, political commentator, and gun rights advocate. Lott was formerly employed at various academic institutions and at the American Enterprise Institute conservative think tank. He ...
, both then of the University of Michigan
, mottoeng = "Arts, Knowledge, Truth"
, former_names = Catholepistemiad, or University of Michigania (1817–1821)
, budget = $10.3 billion (2021)
, endowment = $17 billion (2021)As o ...
, posted notes on Lott's website which, section by section, filled in details of Perelman's first preprint. In September 2004, their notes were updated to include Perelman's second preprint. Following further revisions and corrections, they posted a version to arXiv on 25 May 2006, a modified version of which was published in the academic journal Geometry & Topology
''Geometry & Topology'' is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sci ...
in 2008. At the 2006 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 Nevanlinna Prize (to be rename ...
, Lott said "It has taken us some time to examine Perelman's work. This is partly due to the originality of Perelman's work and partly to the technical sophistication of his arguments. All indications are that his arguments are correct." In the introduction to their article, Kleiner and Lott explained
:Since its 2008 publication, Kleiner and Lott's article has subsequently been revised twice for corrections, such as for an incorrect statement of Hamilton's important "compactness theorem" for Ricci flow. The latest revision to their article was in 2013.
* In June 2006, the Asian Journal of Mathematics
''The Asian Journal of Mathematics'' is a peer-reviewed scientific journal covering all areas of pure and theoretical applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, ...
published an article by Huai-Dong Cao of Lehigh University
Lehigh University (LU) is a private research university in Bethlehem, Pennsylvania in the Lehigh Valley region of eastern Pennsylvania. The university was established in 1865 by businessman Asa Packer and was originally affiliated with the Epis ...
and Zhu Xiping of Sun Yat-sen University
Sun Yat-sen University (, abbreviated SYSU and colloquially known in Chinese as Zhongda), also known as Zhongshan University, is a national key public research university located in Guangzhou, Guangdong, China. It was founded in 1924 by and nam ...
, giving a complete description of Perelman's proof of the Poincaré and the geometrization conjectures. Unlike Kleiner and Lott's article, which was structured as a collection of annotations to Perelman's papers, Cao and Zhu's article was aimed directly towards explaining the proofs of the Poincaré conjecture and geometrization conjecture. In their introduction, they explain
:Based also upon the title "A Complete Proof of the Poincaré and Geometrization Conjectures – Application of the Hamilton-Perelman Theory of Ricci Flow" and the phrase "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow" from the abstract, some people interpreted Cao and Zhu to be taking credit from Perelman for themselves. When asked about the issue, Perelman said that he couldn't see any new contribution by Cao and Zhu and that they "did not quite understand the argument and reworked it." Additionally, one of the pages of Cao and Zhu's article was essentially identical to one from Kleiner and Lott's 2003 posting. In a published erratum, Cao and Zhu attributed this to an oversight, saying that in 2003 they had taken down notes from the initial version of Kleiner and Lott's notes, and in their 2006 writeup had not realized the proper source of the notes. They posted a revised version to arXiv with revisions in their phrasing and in the relevant page of the proof.
* In July 2006, John Morgan of Columbia University and Gang Tian of the Massachusetts Institute of Technology posted a paper on arXiv in which they provided a detailed presentation of Perelman's proof of the Poincaré conjecture. Unlike Kleiner-Lott and Cao-Zhu's expositions, Morgan and Tian's also deals with Perelman's third paper. On 24 August 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture, in which he declared that Perelman's work had been "thoroughly checked." In 2015, Abbas Bahri pointed out a counterexample to one of Morgan and Tian's theorems, which was later fixed by Morgan and Tian and sourced to an incorrectly computed evolution equation. The error, introduced by Morgan and Tian, dealt with details not directly discussed in Perelman's original work. In 2008, Morgan and Tian posted a paper which covered the details of the proof of the geometrization conjecture. Morgan and Tian's two articles have been published in book form by the Clay Mathematics Institute.
Fields Medal and Millennium Prize
In May 2006, a committee of nine mathematicians voted to award Perelman a Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
for his work on the Ricci flow. However, Perelman declined to accept the prize. Sir John Ball, president of the International Mathematical Union
The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...
, approached Perelman in Saint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
in June 2006 to persuade him to accept the prize. After 10 hours of attempted persuasion over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows: "He proposed to me three alternatives: accept and come; accept and don't come, and we will send you the medal later; third, I don't accept the prize. From the very beginning, I told him I have chosen the third one ... he prizewas completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed." "'I'm not interested in money or fame,' he is quoted to have said at the time. 'I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me.'"
Nevertheless, on 22 August 2006, 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 Nevanlinna Prize (to be rename ...
in Madrid
Madrid ( , ) is the capital and most populous city of Spain. The city has almost 3.4 million inhabitants and a metropolitan area population of approximately 6.7 million. It is the second-largest city in the European Union (EU), and ...
, Perelman was offered the Fields Medal "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow". He did not attend the ceremony, and the presenter informed the congress that Perelman declines to accept the medal, which made him the only person to have ever declined the prize.
He had previously rejected a prestigious prize from the European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...
.[
On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem.] On 8 June 2010, he did not attend a ceremony in his honor at the Institut Océanographique, Paris to accept his $1 million prize. According to Interfax
Interfax (russian: Интерфакс) is a Russian news agency. The agency is owned by Interfax News Agency joint-stock company and is headquartered in Moscow.
History
As the first non-governmental channel of political and economic informatio ...
, Perelman refused to accept the Millennium prize in July 2010. He considered the decision of the Clay Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's scie ...
unfair for not sharing the prize with Richard S. Hamilton, and stated that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."
The Clay Institute subsequently used Perelman's prize money to fund the "Poincaré Chair", a temporary position for young promising mathematicians at the Paris Institut Henri Poincaré
The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrond ...
.
Possible withdrawal from mathematics
Perelman quit his job at the Steklov Institute in December 2005. His friends are said to have stated that he currently finds mathematics a painful topic to discuss; by 2010, some even said that he had entirely abandoned mathematics.
Perelman is quoted in a 2006 article in ''The New Yorker
''The New Yorker'' is an American weekly magazine featuring journalism, commentary, criticism, essays, fiction, satire, cartoons, and poetry. Founded as a weekly in 1925, the magazine is published 47 times annually, with five of these issues ...
'' saying that he was disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to alleged efforts of Fields medalist Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
to downplay Perelman's role in the proof and play up the work of Cao Cao or CAO may refer to:
Mythology
*Cao (bull), a legendary bull in Meitei mythology
Companies or organizations
* Air China Cargo, ICAO airline designator CAO
*CA Oradea, Romanian football club
*CA Osasuna, Spanish football club
*Canadian Assoc ...
and Zhu. Perelman added, "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest." He also said, "It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated."
This, combined with the possibility of being awarded a Fields medal, led him to state he had quit professional mathematics by 2006. He said, "As long as I was not conspicuous, I had a choice. Either to make some ugly thing or, if I didn't do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit." (''The New Yorker'' authors explained Perelman's reference to "some ugly thing" as "a fuss" on Perelman's part about the ethical breaches he perceived.)
It is uncertain whether his resignation from Steklov and subsequent seclusion mean that he has ceased to practice mathematics. Fellow countryman and mathematician Yakov Eliashberg
Yakov Matveevich Eliashberg (also Yasha Eliashberg; russian: link=no, Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR.
Education and career
Eliashberg receiv ...
said that, in 2007 Perelman confided to him that he was working on other things but it was too premature to talk about it. He is said to have been interested in the past in the Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
and the problem of their solutions’ existence and smoothness.
In 2014, Russian media reported that Perelman was working in the field of nanotechnology
Nanotechnology, also shortened to nanotech, is the use of matter on an atomic, molecular, and supramolecular scale for industrial purposes. The earliest, widespread description of nanotechnology referred to the particular technological goal o ...
in Sweden.["Komsomolskaya Pravda" found out where Perelman disappears](_blank)
ANNA VELIGZHANINA However, shortly afterwards, he was spotted again in his native hometown, Saint Petersburg.
Perelman and the media
Perelman has avoided journalists and other members of the media. Masha Gessen
Masha Gessen (born 13 January 1967) is a Russian-American journalist, author, translator and activist who has been an outspoken critic of the president of Russia, Vladimir Putin, and the former president of the United States, Donald Trump.
Gess ...
, author of the mathematician's biography ''Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century'', was unable to meet him.
A Russian documentary about Perelman in which his work is discussed by several leading mathematicians including Mikhail Gromov was released in 2011 under the title "Иноходец. Урок Перельмана" ("Maverick: Perelman's Lesson").
In April 2011, Aleksandr Zabrovsky, producer of "President-Film" studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title ''The Formula of the Universe''. Zabrovsky says that in the interview, Perelman explained why he rejected the one million dollar prize.[
A number of journalists believe that Zabrovky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.
The writer Brett Forrest briefly interacted with Perelman in 2012.
A reporter who had called him was told: "You are disturbing me. I am picking mushrooms."]
Complete publication list
Dissertation
*
Research papers
Unpublished work
See also
* Ancient solution
*Asteroid
An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere.
...
50033 Perelman
*Homology sphere
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
*Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chromo ...
*Hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, res ...
*" Manifold Destiny" (On ''The New Yorker
''The New Yorker'' is an American weekly magazine featuring journalism, commentary, criticism, essays, fiction, satire, cartoons, and poetry. Founded as a weekly in 1925, the magazine is published 47 times annually, with five of these issues ...
'' article)
*Spherical space form conjecture
In geometric topology, the spherical space form conjecture (now a theorem) states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.
History
The conjecture was posed by Heinz Hopf in 1926 after de ...
*Thurston elliptization conjecture
William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.
Relation to other conjectures
A 3-manifold with a Riem ...
*Uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization o ...
Notes
References
* Anderson, M.T. 2005
Singularities of the Ricci flow
Encyclopedia of Mathematical Physics, Elsevier.
*The Associated Press,
Erratum
Revised version (December 2006)
Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture
*
*
*
*
* (an account of Perelman's talk on his proof at MIT; pdf file; also see Sugaku Seminar 2003–10 pp 4–7 for an extended version in Japanese)
*
External links
*
*
*
{{DEFAULTSORT:Perelman, Grigori
1966 births
20th-century Russian mathematicians
21st-century Russian mathematicians
Differential geometers
Fields Medalists
International Mathematical Olympiad participants
Jewish Russian scientists
Living people
New York University staff
Mathematicians from Saint Petersburg
Russian Jews
Saint Petersburg State University alumni
Soviet Jews
Soviet mathematicians
Topologists
University of California, Berkeley fellows