Dennis Parnell Sullivan (born February 12, 1941) is an American

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 ...

known for his work in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a

distinguished professor Distinguished Professor is an academic title given to some top tenured professors in a university, school, or department. Some distinguished professors may have endowed chairs. In the United States Often specific to one institution, titles such ...

at Stony Brook University. Sullivan was awarded the Wolf Prize in Mathematics in 2010 and the Abel Prize in 2022.

Early life and education

Sullivan was born in Port Huron, Michigan, on February 12, 1941.. His family moved to Houston soon afterwards. He entered Rice University to study chemical engineering but switched his major to mathematics in his second year after encountering a particularly motivating mathematical theorem. The change was prompted by a special case of the

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 ...

, according to which, in his own words: He received his

Bachelor of Arts Bachelor of arts (BA or AB; from the Latin ', ', or ') is a bachelor's degree awarded for an undergraduate program in the arts, or, in some cases, other disciplines. A Bachelor of Arts degree course is generally completed in three or four year ...

degree from Rice in 1963. He obtained his

Doctor of Philosophy A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

from

Princeton University 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 ...

in 1966 with his thesis, ''Triangulating homotopy equivalences'', under the supervision of William Browder.

Career

Sullivan worked at the University of Warwick on a

NATO The North Atlantic Treaty Organization (NATO, ; french: Organisation du traité de l'Atlantique nord, ), also called the North Atlantic Alliance, is an intergovernmental military alliance between 30 member states – 28 European and two No ...

Fellowship from 1966 to 1967. He was a Miller Research Fellow 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 ...

from 1967 to 1969 and then a Sloan Fellow at

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 ...

from 1969 to 1973. He was a visiting scholar at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...

in 1967–1968, 1968–1970, and again in 1975. Sullivan was an associate professor at Paris-Sud University from 1973 to 1974, and then became a permanent professor at the

Institut des Hautes Études Scientifiques The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics. It is located in Bures-sur-Yvette, jus ...

(IHÉS) in 1974. In 1981, he became the Albert Einstein Chair in Science (Mathematics) at the

Graduate Center, City University of New York The Graduate School and University Center of the City University of New York (CUNY Graduate Center) is a public research institution and post-graduate university in New York City. Serving as the principal doctorate-granting institution of the C ...

and reduced his duties at the IHÉS to a half-time appointment. He joined the mathematics faculty at Stony Brook University in 1996 and left the IHÉS the following year. Sullivan was involved in the founding of the Simons Center for Geometry and Physics and is a member of its board of trustees.

Research

Topology

Geometric topology

Along with Browder and his other students, Sullivan was an early adopter of

surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...

, particularly for classifying high-dimensional manifolds. His thesis work was focused on the ''

Hauptvermutung The ''Hauptvermutung'' of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the s ...

''. In an influential set of notes in 1970, Sullivan put forward the radical concept that, within homotopy theory, spaces could directly "be broken into boxes" (or ''localized''), a procedure hitherto applied to the algebraic constructs made from them. The Sullivan conjecture, proved in its original form by Haynes Miller, states that the

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'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...

''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 In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...

s and given the

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 ...

, is weakly contractible. Sullivan's conjecture was also first presented in his 1970 notes. Sullivan and

Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...

(independently) created rational homotopy theory in the late 1960s and 1970s. It examines "rationalizations" of simply connected topological spaces with homotopy groups and

singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...

groups tensored with the rational numbers, ignoring

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

elements and simplifying certain calculations.

Kleinian groups

Sullivan and William Thurston generalized

Lipman Bers Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also ...

' density conjecture from singly degenerate Kleinian surface groups to all finitely generated

Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...

s in the late 1970s and early 1980s. The conjecture states that every finitely generated Kleinian group is an algebraic limit of

geometrically finite In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite group ...

Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in 2011 and 2012 respectively.

Conformal and quasiconformal mappings

The Connes–Donaldson–Sullivan–Teleman index theorem is an extension of the Atiyah–Singer index theorem to

quasiconformal In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D ...

manifolds due to a joint paper by

Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...

and Sullivan in 1989 and a joint paper by

Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...

, Sullivan, and Nicolae Teleman in 1994. In 1987, Sullivan and Burton Rodin proved Thurston's conjecture about the approximation of the Riemann map by circle packings.

String topology

Sullivan and Moira Chas started the field of string topology, which examines algebraic structures on the homology of free loop spaces. They developed the Chas–Sullivan product to give a partial singular homology analogue of the cup product from

singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

. String topology has been used in multiple proposals to construct

topological quantum field theories In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...

in mathematical physics.

Dynamical systems

In 1975, Sullivan and Bill Parry introduced the topological

Parry–Sullivan invariant In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical systems. It provides a partial classificati ...

for flows in one-dimensional dynamical systems. In 1985, Sullivan proved the no-wandering-domain theorem. This result was described by mathematician Anthony Philips as leading to a "revival of holomorphic dynamics after 60 years of stagnation."

Awards and honors

* 1971 Oswald Veblen Prize in Geometry * 1981 Prix Élie Cartan, French Academy of Sciences * 1983 Member, National Academy of Sciences * 1991 Member, American Academy of Arts and Sciences * 1994 King Faisal International Prize for Science * 2004

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 scienc ...

* 2006 Steele Prize for lifetime achievement * 2010 Wolf Prize in Mathematics, for "his contributions to algebraic topology and conformal dynamics" * 2012 Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

* 2014 Balzan Prize in Mathematics (pure or applied) * 2022 Abel Prize

Personal life

Sullivan is married to fellow mathematician Moira Chas.

References

External links

* *
Sullivan's homepage at CUNY

Sullivan's homepage at Stony Brook University

Dennis Sullivan
International Balzan Prize Foundation {{DEFAULTSORT:Sullivan, Dennis 1941 births 20th-century American mathematicians 21st-century American mathematicians Abel Prize laureates City University of New York faculty Dynamical systems theorists Graduate Center, CUNY faculty Fellows of the American Mathematical Society Homotopy theory Living people Mathematicians from Michigan Members of the United States National Academy of Sciences National Medal of Science laureates Princeton University alumni Recipients of the Great Cross of the National Order of Scientific Merit (Brazil) Rice University alumni Stony Brook University faculty Topologists Wolf Prize in Mathematics laureates