Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, 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 at Stony Brook University.
Dennis Parnell Sullivan
February 12, 1941
|Education||Rice University (BA)|
Princeton University (MA, PhD)
|Institutions||Stony Brook University|
City University of New York
|Thesis||Triangulating Homotopy Equivalences (1966)|
|Doctoral advisor||William Browder|
|Doctoral students||Harold Abelson|
Curtis T. McMullen
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:
[A]ny surface topologically like a balloon, and no matter what shape—a banana or the statue of David by Michelangelo—could be placed on to a perfectly round sphere so that the stretching or squeezing required at each and every point is the same in all directions at each such point.
He received his Bachelor of Arts degree from Rice in 1963. He obtained his Doctor of Philosophy from Princeton University in 1966 with his thesis, Triangulating homotopy equivalences, under the supervision of William Browder.
Sullivan worked at the University of Warwick on a NATO Fellowship from 1966 to 1967. He was a Miller Research Fellow at the University of California, Berkeley from 1967 to 1969 and then a Sloan Fellow at Massachusetts Institute of Technology from 1969 to 1973. He was a visiting scholar at the Institute for Advanced Study 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 (IHÉS) in 1974. In 1981, he became the Albert Einstein Chair in Science (Mathematics) at the Graduate Center, 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.
Along with Browder and his other students, Sullivan was an early adopter of surgery theory, 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 spaces with homotopy groups and singular homology groups tensored with the rational numbers, ignoring torsion elements and simplifying certain calculations.
Sullivan and William Thurston generalized Lipman Bers' density conjecture from singly degenerate Kleinian surface groups to all finitely generated Kleinian groups 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.
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.
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.
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."
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