Dennis Sullivan
Encyclopedia
Dennis Parnell Sullivan (born February 12, 1941) is an American
mathematician
. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein
Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.
in 1963 from Rice University
and his doctorate in 1966 from Princeton University
. His Ph.D.
thesis, entitled Triangulating homotopy equivalences, was written under the supervision of William Browder
, and was a major contribution to surgery theory
. He was a permanent member of the Institut des Hautes Études Scientifiques
from 1974 to 1997.
Sullivan is one of the founders of the surgery method of classifying high-dimensional manifold
s, along with Browder
, Sergei Novikov and C. T. C. Wall
. In homotopy theory, Sullivan put forward the radical concept that spaces could directly be localised
, a procedure hitherto applied to the algebraic constructs made from them. He founded (along with Daniel Quillen) rational homotopy theory
.
The Sullivan conjecture
, proved in its original form by Haynes Miller, states that the classifying space
BG of a finite group
G is sufficiently different from any finite CW complex
X, that it maps to such an X only 'with difficulty'; in a more formal statement, the space of all mappings BG to X, as pointed space
s and given the compact-open topology
, is weakly contractible
. This area has generated considerable further research. (Both these matters are discussed in his 1970 MIT notes.)
is named after him and the English
mathematician Bill Parry
.
In 1987, he proved Thurston's
conjecture about the approximation
of the Riemann map by circle packings
together with Burton Rodin
.
, the 1981 Prix Élie Cartan of the French Academy of Sciences
, the King Faisal International Prize for Science in 1994, the 2004 National Medal of Science
, the 2006 AMS Steele Prize for Lifetime Achievement and the Wolf Prize in Mathematics
in 2010 for "his contributions to algebraic topology and conformal dynamics".
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.
Work in topology
He received his B.A.Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
in 1963 from Rice University
Rice University
William Marsh Rice University, commonly referred to as Rice University or Rice, is a private research university located on a heavily wooded campus in Houston, Texas, United States...
and his doctorate in 1966 from Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. His Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
thesis, entitled Triangulating homotopy equivalences, was written under the supervision of William Browder
William Browder (mathematician)
William Browder is an American mathematician, specializing in algebraic topology, differential topology and differential geometry...
, and was a major contribution to surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
. He was a permanent member of the Institut des Hautes Études Scientifiques
Institut des Hautes Études Scientifiques
The Institut des Hautes Études Scientifiques is a French institute supporting advanced research in mathematics and theoretical physics...
from 1974 to 1997.
Sullivan is one of the founders of the surgery method of classifying high-dimensional manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, along with Browder
William Browder (mathematician)
William Browder is an American mathematician, specializing in algebraic topology, differential topology and differential geometry...
, Sergei Novikov and C. T. C. Wall
C. T. C. Wall
Charles Terence Clegg Wall is a leading British mathematician, educated at Marlborough and Trinity College, Cambridge. He is an emeritus professor of the University of Liverpool, where he was first appointed Professor in 1965...
. In homotopy theory, Sullivan put forward the radical concept that spaces could directly be localised
Localization of a topological space
In mathematics, well behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in ....
, a procedure hitherto applied to the algebraic constructs made from them. He founded (along with Daniel Quillen) rational homotopy theory
Rational homotopy theory
In mathematics, rational homotopy theory is the study of the rational homotopy type of a space, which means roughly that one ignores all torsion in the homotopy groups...
.
The Sullivan conjecture
Sullivan conjecture
In mathematics, Sullivan conjecture can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions of a finite group G. The most elementary formulation, however, is in terms of the...
, proved in its original form by Haynes Miller, states that the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
BG of a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
G is sufficiently different from any finite CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
X, that it maps to such an X only 'with difficulty'; in a more formal statement, the space of all mappings BG to X, as pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
s and given the compact-open topology
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
, is weakly contractible
Weakly contractible
In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.-Property:It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible.-Example:...
. This area has generated considerable further research. (Both these matters are discussed in his 1970 MIT notes.)
Work in dynamics
In 1985, he proved the No wandering domain theorem. The Parry-Sullivan invariantParry-Sullivan invariant
In mathematics, the Parry–Sullivan invariant is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical systems...
is named after him and the English
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
mathematician Bill Parry
Bill Parry (mathematician)
Professor William Parry FRS was an English mathematician. During his research career, he was highly active in the study of dynamical systems, and, in particular, ergodic theory, and made significant contributions to these fields. He is considered to have been at the forefront of the introduction...
.
In 1987, he proved Thurston's
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
conjecture about the approximation
of the Riemann map by circle packings
Circle packing theorem
The circle packing theorem describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles whose interiors are disjoint...
together with Burton Rodin
Burton Rodin
Burt Rodin is an American mathematician known for his research in conformal mapping and Riemann surfaces. He was a professor at the University of California, San Diego 1970–1994 where he was Chair of the Mathematics Department 1977–1981. He became Professor Emeritus in June 1994.He...
.
Awards and honors
Awards include the 1971 Oswald Veblen Prize in GeometryOswald Veblen Prize in Geometry
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen...
, the 1981 Prix Élie Cartan of the French Academy of Sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
, the King Faisal International Prize for Science in 1994, the 2004 National Medal of Science
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...
, the 2006 AMS Steele Prize for Lifetime Achievement and the Wolf Prize in Mathematics
Wolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
in 2010 for "his contributions to algebraic topology and conformal dynamics".