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. Kleinian groups 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. 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 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 2006 Steele Prize for lifetime achievement 2010 Wolf Prize in Mathematics, for "his