Sullivan was born in Port Huron, Michigan, on February 12, 1941.^{[1]}^{[2]} His family moved to Houston soon afterwards.^{[1]}^{[2]}
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.^{[2]}^{[3]} 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.^{[4]}
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.^{[6]}^{[8]} In 1981, he became the Albert Einstein Chair in Science (Mathematics) at the Graduate Center of the City University of New York^{[9]} and reduced his duties at the IHÉS to a half-time appointment.^{[1]} He joined the mathematics faculty at Stony Brook University in 1996^{[6]} and left the IHÉS the following year.^{[6]}^{[8]}
Along with Browder and his other students, Sullivan was an early adopter of surgery theory, particularly for classifying high-dimensional manifolds.^{[2]}^{[3]}^{[1]} His thesis work was focused on the Hauptvermutung.^{[1]}
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"^{[11]} (or localized), a procedure hitherto applied to the algebraic constructs made from them.^{[3]}^{[12]}
Sullivan and William Thurston generalized Lipman Bers' density conjecture from singly degenerate Kleinian surface groups to all finitely generatedKleinian groups in the late 1970s and early 1980s.^{[17]}^{[18]} 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.^{[17]}^{[18]}
Sullivan and Moira Chas started the field of string topology, which examines algebraic structures on the homology of free loop spaces.^{[22]}^{[23]} They developed the Chas–Sullivan product to give a partial singular homology analogue of the cup product from singular cohomology.^{[22]}^{[23]} String topology has been used in multiple proposals to construct topological quantum field theories in mathematical physics.^{[24]}
In 1975, Sullivan and Bill Parry introduced the topological Parry–Sullivan invariant for flows in one-dimensional dynamical systems.^{[25]}^{[26]}
In 1985, Sullivan proved the no-wandering-domain theorem.^{[3]} This result was described by mathematician Anthony Philips as leading to a "revival of holomorphic dynamics after 60 years of stagnation."^{[1]}
^ ^{a}^{b}Cohen, Ralph Louis; Jones, John D. S.; Yan, Jun (2004). "The loop homology algebra of spheres and projective spaces". In Arone, Gregory; Hubbuck, John; Levi, Ran; Weiss, Michael (eds.). Categorical decomposition techniques in algebraic topology: International Conference in Algebraic Topology, Isle of Skye, Scotland, June 2001. Birkhäuser. pp. 77–92.