The Poincaré Conjecture Clay Research Conference Resolution of the Poincaré Conjecture Institut Henri Poincaré Paris, France, June 8–9, 2010
American Mathematical Society
Clay Mathematics Institute
Clay Mathematics Proceedings
The Poincaré Conjecture
Clay Research Conference
Resolution of the Poincaré Conjecture
Institut Henri Poincaré
Paris, France, June 8–9, 2010
2010 Mathematics Subject Classiﬁcation. Primary 53-02, 53C44, 53C99, 53D45, 57-02,
Front cover photo of Henri Poincar´
e is in the public domain.
Back cover image of the m011 horoball diagram is from D. Gabai, R. Meyerhoﬀ, and
P. Milley, Mom technology and volumes of hyperbolic 3-manifolds. Comment. Math.
Helv. 86 (2011), 145–188. Used with permission by the Swiss Mathematical Society.
Back cover photo of Grigoriy Perelman is from The Oberwolfach Photo Collection,
courtesy of the Archives of the Mathematisches Forschungsinstitut Oberwolfach. Used
with the permission of George Bergman.
Library of Congress Cataloging-in-Publication Data
e conjecture : Clay Mathematics Institute Research Conference, resolution of the
e conjecture, Institute Henri Poincar´
e, Paris, France, June 8-9, 2010 / James Carlson,
pages cm. — (Clay mathematics proceedings : volume 19)
“Clay Mathematics Institute.”
Includes bibliographical references.
ISBN 978-0-8218-9865-9 (alk. paper)
e conjecture–Congresses. 2. Three-manifolds (Topology)–Congresses. 3. Manifolds
I. Carlson, James A., 1946- editor.
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Permissions and Acknowledgments
Geometry in 2, 3 and 4 Dimensions
100 Years of Topology: Work Stimulated by Poincar´
e’s Approach to
John W. Morgan
The Evolution of Geometric Structures on 3-Manifolds
Curtis T. McMullen
Invariants of Manifolds and the Classiﬁcation Problem
Simon K. Donaldson
Volumes of Hyperbolic 3-Manifolds
David Gabai, Robert Meyerhoff, and Peter Milley
Manifolds: Where do we come from? What are we? Where are we going?
Geometric Analysis on 4-Manifolds
In 1904, the eminent French mathematician Henri Poincar´
e formulated the
conjecture that bears his name and that motivated much of the research in ge-
ometry and topology for the next one hundred years. Among these developments
were the theory of knots, homotopy theory, surgery theory, and the formulation
by William Thurston of the geometrization conjecture, a sweeping statement that
e’s conjecture and which gave structure and order to the set of
all 3-dimensional manifolds.
In 2000, at a meeting in Paris, the Clay Mathematics Institute (CMI), founded
by Mr. Landon T. Clay, announced the establishment of the seven Millennium
Prize Problems. For the solution of each one, a prize of $1,000,000 was oﬀered.
e conjecture was one of those seven problems.
In November 2002 came a major development: Grigoriy Perelman posted the
ﬁrst of three papers announcing a proof of the conjecture on arXiv.org. His an-
nouncement set oﬀ a ﬂurry of excitement in the mathematical world. Perelman
gave talks at MIT, SUNY-Stony Brook, Princeton, and the University of Penn-
sylvania. Seminars were organized to understand what Perelman had done, and
several groups of researchers set about the task of carefully verifying and validat-
ing his work. CMI supported several of these eﬀorts (Kleiner and Lott, Morgan
and Tian), and it organized a working seminar in Princeton in 2004 devoted to
Perelman’s second paper. On March 10 of 2010, CMI announced award of the
Millennium Prize for the Poincar´
e conjecture to Grigoriy Perelman. The citation
The Clay Mathematics Institute hereby awards the Millennium
Prize for resolution of the Poincar´
e conjecture to Grigoriy Perel-
man. The Poincar´
e conjecture is one of the seven Millennium
Prize Problems established by CMI in 2000. The Prizes were
conceived to record some of the most diﬃcult problems with
which mathematicians were grappling at the turn of the sec-
ond millennium; to elevate in the consciousness of the general
public the fact that in mathematics, the frontier is still open
and abounds in important unsolved problems; to emphasize the
importance of working towards a solution of the deepest, most
diﬃcult problems; and to recognize achievement in mathematics
of historical magnitude.
The decision to award the prize was made on the basis of deliberations by a Special
Advisory Committee appointed to consider the correctness and attribution of the
solution, the CMI Scientiﬁc Advisory Board, and the CMI Board of Directors. The
committee members were Simon Donaldson, David Gabai, Mikhail Gromov, Ter-
ence Tao, and Andrew Wiles (Special Advisory Committee), James Carlson, Simon
Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles
(Scientiﬁc Advisory Board), and Landon T. Clay, Lavinia D. Clay, and Thomas M.
Clay (Board of Directors).
On June 8 and 9 of 2010, a conference on the conjecture was held at the
Institut Henri Poincar´
e in Paris. Most of the lectures given there are featured in
this volume. They provide an overview of the conjecture—its history, its inﬂuence
on the development of mathematics, and, ﬁnally, its proof. Sadly, there is no article
by William Thurston, who passed away in 2012.
Grigoriy Perelman did not accept the Millennium Prize, just as he did not ac-
cept the Fields Medal in 2006. What is important, however, is what Perelman gave
to mathematics: the solution to a long-standing problem of historical signiﬁcance,
and a set of new ideas and new tools with which better to understand the geometry
of manifolds. One chapter of mathematics ends and another begins.
Press Release of March 10, 2010
The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy
Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for
resolution of the Poincar´
e conjecture. The citation for the award reads:
The Clay Mathematics Institute hereby awards the Millennium
Prize for resolution of the Poincar´
e conjecture to Grigoriy Perel-
e conjecture is one of the seven Millennium Prize Problems established
by CMI in 2000. The Prizes were conceived to record some of the most diﬃcult
problems with which mathematicians were grappling at the turn of the second
millennium; to elevate in the consciousness of the general public the fact that in
mathematics, the frontier is still open and abounds in important unsolved prob-
lems; to emphasize the importance of working towards a solution of the deepest,
most diﬃcult problems; and to recognize achievement in mathematics of historical
The award of the Millennium Prize to Dr Perelman was made in accord with
their governing rules: recommendation ﬁrst by a Special Advisory Committee (Si-
mon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles),
then by the CMI Scientiﬁc Advisory Board (James Carlson, Simon Donaldson,
Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with ﬁnal
decision by the Board of Directors (Landon T. Clay, Lavinia D. Clay, and Thomas
James Carlson, President of CMI, said today that “resolution of the Poincar´
conjecture by Grigoriy Perelman brings to a close the century-long quest for the
solution. It is a major advance in the history of mathematics that will long be
Carlson went on to announce that CMI and the Institut Henri
e (IHP) will hold a conference to celebrate the Poincar´
e conjecture and its
resolution June 8 and 9 in Paris. The program will be posted on www.claymath.org.
In addition, on June 7, there will be a press brieﬁng and public lecture by Etienne
Ghys at the Institut Oceanographique, near the IHP.
Reached at his oﬃce at Imperial College, London for his reaction, Fields Medal-
ist Dr. Simon Donaldson said, “I feel that Poincar´
e would have been very satisﬁed
to know both about the profound inﬂuence his conjecture has had on the develop-
ment of topology over the last century and the surprising way in which the problem
was solved, making essential use of partial diﬀerential equations and diﬀerential ge-
PRESS RELEASE OF MARCH 10, 2010
e’s conjecture and Perelman’s proof
Formulated in 1904 by the French mathematician Henri Poincar´
e, the con-
jecture is fundamental to achieving an understanding of three-dimensional shapes
(compact manifolds). The simplest of these shapes is the three-dimensional sphere.
It is contained in four-dimensional space, and is deﬁned as the set of points at a
ﬁxed distance from a given point, just as the two-dimensional sphere (skin of an
orange or surface of the earth) is deﬁned as the set of points in three-dimensional
space at a ﬁxed distance from a given point (the center).
Since we cannot directly visualize objects in n-dimensional space, Poincar´
asked whether there is a test for recognizing when a shape is the three-sphere by
performing measurements and other operations inside the shape. The goal was to
recognize all three-spheres even though they may be highly distorted. Poincar´
found the right test (simple connectivity, see below). However, no one before Perel-
man was able to show that the test guaranteed that the given shape was in fact a
In the last century, there were many attempts to prove, and also to disprove,
e conjecture using the methods of topology. Around 1982, however,
a new line of attack was opened. This was the Ricci ﬂow method pioneered and
developed by Richard Hamilton. It was based on a diﬀerential equation related
to the one introduced by Joseph Fourier 160 years earlier to study the conduction
of heat. With the Ricci ﬂow equation, Hamilton obtained a series of spectacular
results in geometry. However, progress in applying it to the conjecture eventually
came to a standstill, largely because formation of singularities, akin to formation
of black holes in the evolution of the cosmos, deﬁed mathematical understanding.
Perelman’s breakthrough proof of the Poincar´
e conjecture was made possible
by a number of new elements. He achieved a complete understanding of singularity
formation in Ricci ﬂow, as well as the way parts of the shape collapse onto lower-
dimensional spaces. He introduced a new quantity, the entropy, which instead
of measuring disorder at the atomic level, as in the classical theory of heat ex-
change, measures disorder in the global geometry of the space. This new entropy,
like the thermodynamic quantity, increases as time passes. Perelman also intro-
duced a related local quantity, the L-functional, and he used the theories originated
by Cheeger and Aleksandrov to understand limits of spaces changing under Ricci
ﬂow. He showed that the time between formation of singularities could not become
smaller and smaller, with singularities becoming spaced so closely—inﬁnitesimally
close—that the Ricci ﬂow method would no longer apply. Perelman deployed his
new ideas and methods with great technical mastery and described the results he
obtained with elegant brevity. Mathematics has been deeply enriched.
Some other reactions
Fields medalist Stephen Smale, who solved the analogue of the Poincar´
jecture for spheres of dimension ﬁve or more, commented that: “Fifty years ago I
was working on Poincar´
e’s conjecture and thus hold a long-standing appreciation
for this beautiful and diﬃcult problem. The ﬁnal solution by Grigoriy Perelman is
a great event in the history of mathematics.”
Donal O’Shea, Professor of Mathematics at Mt. Holyoke College and author of
e Conjecture, noted: “Poincar´
e altered twentieth-century mathematics
by teaching us how to think about the idealized shapes that model our cosmos. It
HISTORY AND BACKGROUND
is very satisfying and deeply inspiring that Perelman’s unexpected solution to the
e conjecture, arguably the most basic question about such shapes, oﬀers to
do the same for the coming century.
History and Background
In the latter part of the nineteenth century, the French mathematician Henri
e was studying the problem of whether the solar system is stable. Do the
planets and asteroids in the solar system continue in regular orbits for all time, or
will some of them be ejected into the far reaches of the galaxy or, alternatively,
crash into the sun? In this work he was led to topology, a still new kind of mathe-
matics related to geometry, and to the study of shapes (compact manifolds) of all
The simplest such shape was the circle, or distorted versions of it such as the
ellipse or something much wilder: lay a piece of string on the table, tie one end to
the other to make a loop, and then move it around at random, making sure that
the string does not touch itself. The next simplest shape is the two-sphere, which
we ﬁnd in nature as the idealized skin of an orange, the surface of a baseball, or
the surface of the earth, and which we ﬁnd in Greek geometry and philosophy as
the “perfect shape”. Again, there are distorted versions of the shape, such as the
surface of an egg, as well as still wilder objects. Both the circle and the two-sphere
can be described in words or in equations as the set of points at a ﬁxed distance
from a given point (the center). Thus it makes sense to talk about the three-sphere,
the four-sphere, etc. These shapes are hard to visualize, since they naturally are
contained in four-dimensional space, ﬁve-dimensional space, and so on, whereas we
live in three-dimensional space. Nonetheless, with mathematical training, shapes
in higher-dimensional spaces can be studied just as well as shapes in dimensions
two and three.
In topology, two shapes are considered the same if the points of one correspond
to the points of another in a continuous way. Thus the circle, the ellipse, and the
wild piece of string are considered the same. This is much like what happens in
the geometry of Euclid. Suppose that one shape can be moved, without changing
lengths or angles, onto another shape. Then the two shapes are considered the
same (think of congruent triangles). A round, perfect two-sphere, like the surface
of a ping-pong ball, is topologically the same as the surface of an egg.
In 1904 Poincar´
e asked whether a three-dimensional shape that satisﬁes the
“simple connectivity test” is the same, topologically, as the ordinary round three-
sphere. The round three-sphere is the set of points equidistant from a given point
in four-dimensional space. His test is something that can be performed by an
imaginary being who lives inside the three-dimensional shape and cannot see it
from“outside.” The test is that every loop in the shape can be drawn back to the
point of departure without leaving the shape. This can be done for the two-sphere
and the three-sphere. But it cannot be done for the surface of a doughnut, where
a loop may get stuck around the hole in the doughnut.
The question raised became known as the Poincar´
e conjecture. Over the years,
many outstanding mathematicians tried to solve it—Poincar´
e himself, Whitehead,
Bing, Papakirioukopolos, Stallings, and others. While their eﬀorts frequently led
to the creation of signiﬁcant new mathematics, each time a ﬂaw was found in the
proof. In 1961 came astonishing news. Stephen Smale, then of the University of
PRESS RELEASE OF MARCH 10, 2010
California at Berkeley (now at the City University of Hong Kong) proved that the
analogue of the Poincar´
e conjecture was true for spheres of ﬁve or more dimensions.
The higher-dimensional version of the conjecture required a more stringent version
e’s test; it asks whether a so-called homotopy sphere is a true sphere.
Smale’s theorem was an achievement of extraordinary proportions. It did not,
however, answer Poincar´
e’s original question. The search for an answer became all
the more alluring.
Smale’s theorem suggested that the theory of spheres of dimensions three and
four was unlike the theory of spheres in higher dimension. This notion was con-
ﬁrmed a decade later, when Michael Freedman, then at the University of California,
San Diego, now of Microsoft Research Station Q, announced a proof of the Poincar´
conjecture in dimension four. His work used techniques quite diﬀerent from those
of Smale. Freedman also gave a classiﬁcation, or kind of species list, of all simply
connected four-dimensional manifolds.
Both Smale (in 1966) and Freedman (in 1986) received Fields medals for their
work. There remained the original conjecture of Poincar´
e in dimension three. It
seemed to be the most diﬃcult of all, as the continuing series of failed eﬀorts, both
to prove and to disprove it, showed. In the meantime, however, there came three
developments that would play crucial roles in Perelman’s solution of the conjecture.
The ﬁrst of these developments was William Thurston’s geometrization con-
jecture. It laid out a program for understanding all three-dimensional shapes in a
coherent way, much as had been done for two-dimensional shapes in the latter half
of the nineteenth century. According to Thurston, three-dimensional shapes could
be broken down into pieces governed by one of eight geometries, somewhat as a
molecule can be broken into its constituent, much simpler atoms. This is the origin
of the name, “geometrization conjecture.”
A remarkable feature of the geometrization conjecture was that it implied the
e conjecture as a special case. Such a bold assertion was accordingly thought
to be far, far out of reach—perhaps a subject of research for the twenty-second
century. Nonetheless, in an imaginative tour de force that drew on many ﬁelds of
mathematics, Thurston was able to prove the geometrization conjecture for a wide
class of shapes (Haken manifolds) that have a suﬃcient degree of complexity. While
these methods did not apply to the three-sphere, Thurston’s work shed new light on
the central role of Poincar´
e’s conjecture and placed it in a far broader mathematical
Limits of Spaces
The second current of ideas did not appear to have a connection with the
e conjecture until much later. While technical in nature, the work, in which
the names of Cheeger and Perelman ﬁgure prominently, has to do with how one
can take limits of geometric shapes, just as we learned to take limits in beginning
calculus class. Think of Zeno and his paradox: you walk half the distance from
where you are standing to the wall of your living room. Then you walk half the
remaining distance. And so on. With each step you get closer to the wall. The
wall is your “limiting position,” but you never reach it in a ﬁnite number of steps.
Now imagine a shape changing with time. With each ”step” it changes shape, but
can nonetheless be a “nice” shape at each step—smooth, as the mathematicians
say. For the limiting shape the situation is diﬀerent. It may be nice and smooth,
or it may have special points that are diﬀerent from all the others, that is, singular
points, or “singularities.” Imagine a Y-shaped piece of tubing that is collapsing: as
time increases, the diameter of the tube gets smaller and smaller. Imagine further
that one second after the tube begins its collapse, the diameter has gone to zero.
Now the shape is diﬀerent: it is a Y shape of inﬁnitely thin wire. The point where
the arms of the Y meet is diﬀerent from all the others. It is the singular point of
this shape. The kinds of shapes that can occur as limits are called Aleksandrov
spaces, named after the Russian mathematician A. D. Aleksandrov who initiated
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