Dennis Sullivan
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Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
,
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University of New York and is a distinguished professor 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 Houston ( ) is the List of cities in Texas by population, most populous city in the U.S. state of Texas and in the Southern United States. Located in Southeast Texas near Galveston Bay and the Gulf of Mexico, it is the county seat, seat of ...
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, according to which, in his own words: He received his
Bachelor of Arts A Bachelor of Arts (abbreviated B.A., BA, A.B. or AB; from the Latin ', ', or ') is the holder of a bachelor's degree awarded for an undergraduate program in the liberal arts, or, in some cases, other disciplines. A Bachelor of Arts deg ...
degree from Rice University in 1963. He obtained his
Doctor of Philosophy A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of Postgraduate education, graduate study and original resear ...
from
Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
in 1966 with his thesis, ''Triangulating homotopy equivalences'', under the supervision of William Browder.


Career

Sullivan worked at the
University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands and Warwickshire, England. The university was founded in 1965 as part of ...
on a
NATO The North Atlantic Treaty Organization (NATO ; , OTAN), also called the North Atlantic Alliance, is an intergovernmental organization, intergovernmental Transnationalism, transnational military alliance of 32 Member states of NATO, member s ...
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 university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...
from 1967 to 1969 and then a Sloan Fellow at
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
from 1969 to 1973. He was a visiting scholar at the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
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 (also with a small theoretical biology g ...
(IHÉS) in 1974. In 1981, he became the Albert Einstein Chair in Science (Mathematics) at the Graduate Center of the City University of New York 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 An ...
, particularly for classifying high-dimensional manifolds. His thesis work was focused on the '' Hauptvermutung''. 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 ''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 spaces and given the compact-open topology, is weakly contractible. Sullivan's conjecture was also first presented in his 1970 notes. Sullivan and Daniel Quillen (independently) created rational homotopy theory in the late 1960s and 1970s. It examines "rationalizations" of simply connected
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s with homotopy groups and singular homology groups tensored with the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s, ignoring torsion elements and simplifying certain calculations.


Kleinian groups

Sullivan and William Thurston generalized Lipman Bers' 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 (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s in the late 1970s and early 1980s. The conjecture states that every finitely generated Kleinian group is an algebraic limit of geometrically finite 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 manifolds due to a joint paper by Simon Donaldson and Sullivan in 1989 and a joint paper by Alain Connes, 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. String topology has been used in multiple proposals to construct topological quantum field theories in mathematical physics.


Dynamical systems

In 1975, Sullivan and Bill Parry introduced the topological Parry–Sullivan invariant 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 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 method, scientific research. It was at the forefron ...
*1983 Member,
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
*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 science, behavior ...
*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 *2014 Balzan Prize in Mathematics (pure or applied) *2022 Abel Prize


Personal life

Sullivan is married to fellow mathematician Moira Chas.


See also

* Assembly map * Double bubble conjecture * Flexible polyhedron * Formal manifold * Loch Ness monster surface * Normal invariant * Ring lemma * Rummler–Sullivan theorem * Ruziewicz problem


References


External links

* *
Sullivan's homepage at the City University of New YorkSullivan's homepage at Stony Brook UniversityDennis Sullivan
International Balzan Prize Foundation {{DEFAULTSORT:Sullivan, Dennis 1941 births Living people 20th-century American mathematicians 21st-century American mathematicians Abel Prize laureates Dynamical systems theorists CUNY Graduate Center faculty Fellows of the American Mathematical Society Homotopy theory 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 American topologists Wolf Prize in Mathematics laureates