The Poincaré Conjecture Clay Research Conference Resolution of the Poincaré Conjecture Institut Henri Poincaré Paris, France, June 8–9, 2010 - bet 2
|and developed their theory.
The third development concerns diﬀerential equations. These equations involve
rates of change in the unknown quantities of the equation, e.g., the rate of change of
the position of an apple as it falls from a tree towards the earth’s center. Diﬀerential
equations are expressed in the language of calculus, which Isaac Newton invented
in the 1680s in order to explain how material bodies (apples, the moon, and so on)
move under the inﬂuence of an external force. Nowadays physicists use diﬀerential
equations to study a great range of phenomena: the motion of galaxies and the
stars within them, the ﬂow of air and water, the propagation of sound and light,
the conduction of heat, and even the creation, interaction, and annihilation of
elementary particles such as electrons, protons, and quarks.
In our story, conduction of heat and change of temperature play a special role.
This kind of physics was ﬁrst treated mathematically by Joseph Fourier in his 1822
eorie Analytique de la Chaleur. The diﬀerential equation that governs
change of temperature is called the heat equation. It has the remarkable property
that as time increases, irregularities in the distribution of temperature decrease.
Diﬀerential equations apply to geometric and topological problems as well as
to physical ones. But one studies not the rate at which temperature changes, but
rather the rate of change in some geometric quantity as it relates to other quantities
such as curvature. A piece of paper lying on the table has curvature zero. A sphere
has positive curvature. The curvature is a large number for a small sphere, but
is a small number for a large sphere such as the surface of the earth. Indeed, the
curvature of the earth is so small that its surface has sometimes mistakenly been
thought to be ﬂat. For an example of negative curvature, think of a point on the
bell of a trumpet. In some directions the metal bends away from your eye; in others
it bends towards it.
An early landmark in the application of diﬀerential equations to geometric
problems was the 1963 paper of J. Eells and J. Sampson. The authors introduced
the “harmonic map equation,” a kind of nonlinear version of Fourier’s heat equa-
tion. It proved to be a powerful tool for the solution of geometric and topological
problems. There are now many important nonlinear heat equations—the equations
for mean curvature ﬂow, scalar curvature ﬂow, and Ricci ﬂow.
Also notable is the Yang-Mills equation, which came into mathematics from the
physics of quantum ﬁelds. In 1983 this equation was used to establish very strong
restrictions on the topology of four-dimensional shapes on which it was possible
to do calculus . These results helped renew hopes of obtaining other strong
PRESS RELEASE OF MARCH 10, 2010
geometric results from analytic arguments—that is, from calculus and diﬀerential
equations. Optimism for such applications had been tempered to some extent by
the examples of Ren´
e Thom (on cycles not representable by smooth submanifolds)
and Milnor (on diﬀeomorphisms of the six-sphere).
The diﬀerential equation that was to play a key role in solving the Poincar´
conjecture is the Ricci ﬂow equation. It was discovered two times, independently.
In physics, the equation originated with the thesis of Friedan , although it was
perhaps implicit in the work of Honerkamp . In mathematics it originated with
the 1982 paper of Richard Hamilton . The physicists were working on the renor-
malization group of quantum ﬁeld theory, while Hamilton was interested in geo-
metric applications of the Ricci ﬂow equation itself. Hamilton, now at Columbia
University, was then at Cornell University.
On the left-hand side of the Ricci ﬂow equation is a quantity that expresses
how the geometry changes with time—the derivative of the metric tensor, as the
mathematicians like to say. On the right-hand side is the Ricci tensor, a measure
of the extent to which the shape is curved. The Ricci tensor, based on Riemann’s
theory of geometry (1854), also appears in Einstein’s equations for general relativity
(1915). Those equations govern the interaction of matter, energy, curvature of
space, and the motion of material bodies.
The Ricci ﬂow equation is the analogue, in the geometric context, of Fourier’s
heat equation. The idea, grosso modo, for its application to geometry is that, just
as Fourier’s heat equation disperses temperature, the Ricci ﬂow equation disperses
curvature. Thus, even if a shape was irregular and distorted, Ricci ﬂow would grad-
ually remove these anomalies, resulting in a very regular shape whose topological
nature was evident. Indeed, in 1982 Hamilton showed that for positively curved,
simply connected shapes of dimension three (compact three-manifolds) the Ricci
ﬂow transforms the shape into one that is ever more like the round three-sphere.
In the long run, it becomes almost indistinguishable from this perfect, ideal shape.
When the curvature is not strictly positive, however, solutions of the Ricci ﬂow
equation behave in a much more complicated way. This is because the equation is
nonlinear. While parts of the shape may evolve towards a smoother, more regular
state, other parts might develop singularities. This richer behavior posed serious
diﬃculties. But it also held promise: it was conceivable that the formation of
singularities could reveal Thurston’s decomposition of a shape into its constituent
Hamilton was the driving force in developing the theory of Ricci ﬂow in math-
ematics, both conceptually and technically. Among his many notable results is his
1999 paper , which showed that in a Ricci ﬂow, the curvature is pushed towards
the positive near a singularity. In that paper Hamilton also made use of the col-
lapsing theory  mentioned earlier. Another result , which played a crucial role
in Perelman’s proof, was the Hamilton Harnack inequality, which generalized to
positive Ricci ﬂows a result of Peter Li and Shing-Tung Yau for positive solutions
of Fourier’s heat equation.
PERELMAN ANNOUNCES A SOLUTION OF THE POINCAR ´
Hamilton had established the Ricci ﬂow equation as a tool with the potential
to resolve both conjectures as well as other geometric problems.
serious obstacles barred the way to a proof of the Poincar´
e conjecture. Notable
among these obstacles was lack of an adequate understanding of the formation of
singularities in Ricci ﬂow, akin to the formation of black holes in the evolution of
the cosmos. Indeed, it was not at all clear how or if formation of singularities could
be understood. Despite the new front opened by Hamilton, and despite continued
work by others using traditional topological tools for either a proof or a disproof,
progress on the conjectures came to a standstill.
Such was the state of aﬀairs in 2000, when John Milnor wrote an article describ-
ing the Poincar´
e conjecture and the many attempts to solve it. At that writing,
it was not clear whether the conjecture was true or false, and it was not clear
which method might decide the issue. Analytic methods (diﬀerential equations)
were mentioned in a later version (2004). See  and .
Perelman announces a solution of the Poincar´
It was thus a huge surprise when Grigoriy Perelman announced, in a series of
preprints posted on ArXiv.org in 2002 and 2003, a solution not only of the Poincar´
conjecture, but also of Thurston’s geometrization conjecture , , .
The core of Perelman’s method of proof is the theory of Ricci ﬂow. To its
applications in topology he brought not only great technical virtuosity, but also
new ideas. One was to combine collapsing theory in Riemannian geometry with
Ricci ﬂow to give an understanding of the parts of the shape that were collapsing
onto a lower-dimensional space. Another was the introduction of a new quantity, the
entropy, which instead of measuring disorder at the atomic level, as in the classical
theory of heat exchange, measures disorder in the global geometry of the space.
Perelman’s entropy, like the thermodynamic entropy, is increasing in time: there is
no turning back. Using his entropy function and a related local version (the L-length
functional), Perelman was able to understand the nature of the singularities that
formed under Ricci ﬂow. There were just a few kinds, and one could write down
simple models of their formation. This was a breakthrough of ﬁrst importance.
Once the simple models of singularities were understood, it was clear how to
cut out the parts of the shape near them as to continue the Ricci ﬂow past the
times at which they would otherwise form. With these results in hand, Perelman
showed that the formation times of the singularities could not run into Zeno’s wall:
imagine a singularity that occurs after one second, then after half a second more,
then after a quarter of a second more, and so on. If this were to occur, the “wall,”
which one would reach two seconds after departure, would correspond to a time
at which the mathematics of Ricci ﬂow would cease to hold. The proof would be
unattainable. But with this new mathematics in hand, attainable it was.
The posting of Perelman’s preprints and his subsequent talks at MIT, SUNY-
Stony Brook, Princeton, and the University of Pennsylvania set oﬀ a worldwide
eﬀort to understand and verify his groundbreaking work. In the US, Bruce Kleiner
and John Lott wrote a set of detailed notes on Perelman’s work. These were posted
online as the veriﬁcation eﬀort proceeded. A ﬁnal version was posted to ArXiv.org
in May 2006, and the refereed article appeared in Geometry and Topology in 2008.
This was the ﬁrst time that work on a problem of such importance was facilitated
via a public website. John Morgan and Gang Tian wrote a book-long exposition
PRESS RELEASE OF MARCH 10, 2010
of Perelman’s proof, posted on ArXiv.org in July of 2006, and published by the
American Mathematical Society in CMI’s monograph series (August 2007). These
expositions, those by other teams, and, importantly, the multi-year scrutiny of the
mathematical community, provided the needed veriﬁcation. Perelman had solved
e conjecture. After a century’s wait, it was settled!
Among other articles that appeared following Perelman’s work is a paper in
the Asian Journal of Mathematics, posted on ArXiv.org in June of 2006 by the
American-Chinese team, Huai-Dong Cao (Lehigh University) and Xi-Ping Zhu
(Zhongshan University). Another is a paper by the European group of Bessieres,
Besson, Boileau, Maillot, and Porti, posted on ArXiv.org in June of 2007. It was
accepted for publication by Inventiones Mathematicae in October of 2009. It gives
an alternative approach to the last step in Perelman’s proof of the geometrization
conjecture. Perelman’s proof of the Poincar´
e and geometrization conjectures is
a major mathematical advance. His ideas and methods have already found new
applications in analysis and geometry; surely the future will bring many more.
JC, March 18, 2010
1. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature
bounded. I and II, J. Diﬀerential Geom. Volume 23, Number 3 (1986); Volume 32, Number 1
(1990), 269–298. MR1064875 (92a:53066)
2. S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Diﬀerential
Geom., 18, (1983), 279–315. MR710056 (85c:57015)
3. D. Friedan, Nonlinear Models in 2 + epsilon Dimensions, Annals of Physics 163, 318-419
(1985) MR811072 (87f:81130)
4. R. Hamilton, Three-manifolds with positive Ricci curvature, Journal of Diﬀerential Geometry,
vol. 17, 255–306 (1982) MR664497 (84a:53050)
5. R. Hamilton, Non-singular solutions of the Ricci ﬂow on three-manifolds, Comm. Anal. Geom.
7(4), 695–729 (1999) MR1714939 (2000g:53034)
6. R. Hamilton, The Harnack estimate for Ricci ﬂow, Journal of Diﬀerential Geometry, vol. 37,
225–243 (1993) MR1198607 (93k:58052)
7. J. Honerkamp, (CERN), Chiral multiloops, Nucl. Phys. B36, 130–140 (1972)
8. J. Milnor, The Poincar´
e Conjecture (2000) www.claymath.org/sites/default/ﬁles/poincare.pdf
9. J. Milnor, The Poincar´
e Conjecture, in The Millennium Prize Problems, J. Carlson, A. Jaﬀe,
A. Wiles, eds, AMS (2004) www.claymath.org/publications/special-editions/millennium-
10. G. Perelman, The entropy formula for the Ricci ﬂow and its geometric applications, arXiv.org,
November 11, 2002
11. G. Perelman, Ricci ﬂow with surgery on three-manifolds, arXiv.org, March 10, 2003
12. G. Perelman, Finite extinction time for the solutions to the Ricci ﬂow on certain three-
manifolds, arXiv.org, July 17, 2003
Permissions and Acknowledgments
The American Mathematical Society gratefully acknowledges the following par-
ties for granting permission to reprint their imagery in this volume:
Image of Henri Poincar´
e, p. 2 and front cover, public domain
Images of Carl Friedrich Gauss, Niels Henrik Abel, and Bernhard Riemann, p. 3,
Photo of Edward Witten, p. 5, courtesy of the Institute for Advanced Study.
Photo of William Hodge, p. 4, courtesy of The Royal Society.
Photo of Henri Cartan/Jean-Pierre Serre, p. 4, Photo 39 from I Have a Photo-
graphic Memory by Paul R. Halmos.
c American Mathematical Society (1987).
Used with permission.
Photo of Simon Donaldson, p. 5, c Peter Badge/Typos1 in cooperation with the
Heidelberg Laureate Forum–all rights reserved. Used with permission.
Photos of William Thurston, p. 3, and Grigoriy Perelman, p. 2 and back cover from
the Oberwolfach Photo Collection, courtesy of the Archives of the Mathematisches
Forschungsinstitut Oberwolfach. Used with the permission of George Bergman.
Photo of Vaughn Jones, p. 5, from the Oberwolfach Photo Collection, courtesy of
the Archives of the Mathematisches Forschungsinstitut Oberwolfach.
Photo of Solomon Lefschetz, p. 4, courtesy of the American Mathematical Society.
m011 horoball diagram, ﬁgure 5, p. 68 and back cover is from D. Gabai, R. Mey-
erhoﬀ, and P. Milley, “Mom technology and volumes of hyperbolic 3-manifolds”,
Comment. Math. Helv. 86 (2011), 145–188. Used with permission by the Swiss
Clay Mathematics Proceedings
Volume 19, 2014
Geometry in 2, 3 and 4 Dimensions
Ten years ago, when the Millennium challenge of the Clay Mathematical Insti-
tute was launched here in Paris, I was one of the two speakers tasked with presenting
the problems. The other was John Tate and 7 problems were divided between us,
broadly based on a geometry/algebra division.
So it is appropriate that I introduce today’s session devoted to the ﬁrst of
the seven problems to be solved. The whole world now knows that the century
old conjecture made by Henri Poincar´
e, the leading French mathematician of his
time, has been conclusively settled by the young Russian mathematician Grigoriy
Perelman. It is without question a great event to be celebrated and the ten years
we have had to wait is a short period for a problem of this importance. Time
will tell how many decades will pass before the remaining six millennium problems
succumb to the skill and eﬀorts of the young mathematicians of the 21st century. I
myself am optimistic that we will not have to wait too long for the next occasion,
though I expect another presenter will be required.
In the subsequent lectures there will be more specialized presentations of the
mathematical aspects of Perelman’s proof, both about its achievement and about
directions that it opens up for the future. My aim is to put things into a his-
torical context by reviewing in broad terms the history of geometry over the past
two centuries. As we know the Poincar´
e conjecture is about characterizing the 3-
dimensional sphere in topological terms and its resolution by Perelman, combined
with the earlier brilliant work of William Thurston, provides an essentially com-
plete understanding of compact 3-dimensional manifolds. As such it sits on the
cusp between the classical geometry of surfaces and the still emerging geometry of
4 dimensions, which may occupy mathematicians (and physicists) for many years
After my historical review I will move on to discuss relations between geometry
and physics which have enjoyed a remarkable renaissance in recent years. I will
conclude with a speculative peep into the future, indicating some of the problems
that lie ahead.
2. Historical context: dimensions 2 and 3
The single most important idea in diﬀerential geometry is that of curvature,
pioneered by Gauss and then by Riemann. It is remarkable how far-reaching this
c 2014 Michael Atiyah
e (1854–1912) and Grigoriy Perelman
has proved to be, and we can trace its evolution through the increase in dimensions.
Roughly we can divide the history of geometry into three eras.
19th century: dealing with 2 dimensions and the scalar cuvature R
20th century: dealing with 3 dimensions and the Ricci curvature R
21st century: dealing with 4 dimensions and the Riemann curvature R
Of course this is a great oversimpliﬁcation and the boundaries between centuries are
ﬂuid. Moreover the Riemann curvature remains a basic object in higher-dimensional
diﬀerential geometry. Nevertheless the work of Simon Donaldson has clearly shown
the unique properties of 4-dimensions, and this poses the current challenge. The
theory of (compact oriented) surfaces bridged the gaps between topology, diﬀeren-
tial geometry and algebraic geometry, with the seminal ideas being those of Niels
Henrik Abel. The outcome was the classiﬁcation of surfaces into 3 types depending
on the genus g.
g = 0
g = 1
≥ 2 general case negative curvature
It was Poincar´
e who laid the foundation of topology with the notion of homology
as the “counting of holes” of diﬀerent dimensions and the introduction of the fun-
damental group. The Poincar´
e conjecture originated when Poincar´
e realized that
there was a 3-manifold with no homology other than the 3-sphere. This was the
famous “fake” 3-sphere, arising from the icosahedron, whose symmetries appear in
the fundamental group. This led Poincar´
e to formulate his famous conjecture:
A compact simply connected 3-manifold is topologically a sphere.
In the 20th century topology became a central topic and in 3 dimensions William
Thurston outlined a comprehensive programme in which all 3-manifolds were in-
cluded. Again, as in 2 dimensions, the classiﬁcation involved the curvature. The
3- sphere typiﬁed positive curvature and hyperbolic 3-manifolds typiﬁed negative
curvature, with a total of 8 diﬀerent types in all as building bricks for general
3-manifolds. Perelman’s proof, for the 3-sphere, extends naturally to the whole
GEOMETRY IN 2, 3 AND 4 DIMENSIONS
Gauss (1777–1855), Abel (1802–1829) and Riemann
William Thurston (1946–2012)
Thurston programme and this brings to a close a century’s work on the geometry
of 3 dimensions. It is deﬁnitely the end of an era.
3. Complex algebraic geometry
While the move from dimension 2 to dimension 3 appears to be the obvious
step there is a sense in which one should move from 2 to 4. This comes from the
consideration of complex algebraic geometry. For complex dimension 1 this theory
was started by Abel and continued by Riemann. For algebraic varieties of complex
dimension n the real dimension is 2n, so the case n = 2 leads to 4-dimensional real
The key ﬁgures in the topology of higher-dimensional algebraic varieties were
Lefschetz, Hodge, Cartan and Serre.
Solomon Lefschetz (1884–1972); William Hodge (1903–
1975); Henri Cartan (1904–2008) and Jean-Pierre Serre (1926–)
While general algebraic geometry was one of the major developments of the
second half of the 20th century, the topology of real 4-manifolds had a great surprise
in store when Simon Donaldson made spectacular discoveries opening up an entirely
This work of Donaldson emerged as a by-product of new ideas in physics, another
example of which were the new knot invariants discovered by Vaughan Jones. This
led to extensive developments linking geometry (particularly in low dimensions)
to quantum physics. Much of this was due to Edward Witten and his colleagues.
So the 21st century has begun with the end of a chapter on 3-dimensions and with
new problems emerging in 3 and 4 dimensions for the future.
GEOMETRY IN 2, 3 AND 4 DIMENSIONS
Simon Donaldson (1957–)
Edward Witten (1951–) and Vaughan Jones (1952–)
Relate Jones quantum invariants to Perelman-Thurston.
Understand the structure of simply-connected 4-manifolds and
the relation to physics.
Let me end with some personal speculations on the relation between geometry
and physics. As we know Einstein extended the 3 dimensions of space to a 4-
dimensional space-time where curvature embodies gravitational force. An idea due
essentially to Hermann Weyl shows how an extra 5th dimension incorporates the
Maxwell electro-magnetic ﬁeld. While the 5-dimensional space has an indeﬁnite
metric of signature (4, 1) we can ignore time and get a 4-dimensional Riemannian
manifold. Here Donaldson’s theory comes naturally into its own and I am attracted
by the idea that the phenomena he unearthed should play a key role in physics. I
am exploring the possible role of such Riemannian 4-manifolds as models of nuclear
matter, in which topology will relate directly to physics. These ideas are related
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