2014 Balzan Prize for Mathematics (pure/applied)
For his major contributions to topology and the theory of dynamical systems, opening new perspectives for generations to come. For his exceptional results in many fields of mathematics, such as topology, geometry, the theory of Kleinian groups, analysis and number theory.
Dennis Sullivan is Professor at the State University of New York at Stony Brook and the City University of New York. He took his doctorate at Princeton, and worked at Warwick, MIT and the Institut des Hautes Études Scientifiques in Paris before returning to the United States.
Sullivan is an exceptional mathematician who has pioneered exciting new areas, in particular, in topology and in dynamics. His contributions have made a major impact and will continue to be exploited by many future generations of mathematicians.
Algebraic topology is the study of topology using algebraic tools. The theory is particularly useful when algebra allows a complete description of topology.
At the beginning of his career, Sullivan was one of the main proponents of (mathematical!) surgery theory. He made a fundamental contribution to the Hauptvermutung, which concerns the different ways of triangulating space. He also obtained a complete classification of simply connected high-dimensional manifolds, for which the homotopy type is known. His article “Genetics of homotopy theory and the Adams conjecture” represents a remarkable step forward.
Later, he developed (along with Quillen) rational homotopy theory, which is one of the mathematical gems of the twentieth century. A purely algebraic structure – Sullivan’s minimal model – makes it possible to completely reconstruct the rational homotopy type of a space. “Infinitesimal computations in topology” is one of the most important texts on algebraic topology in the twentieth century, of similar stature to Poincaré’s seminal paper “Analysis Situs”.
In the second part of his career, Sullivan transformed the theory of dynamical systems. The study of complex dynamics, initiated by Fatou and Julia at the beginning of the twentieth century, had been neglected, until Sullivan applied and developed useful, “quasiconformal” tools from harmonic analysis. The theory was revolutionary. Among his other work on this theme, his “Non wandering theorem” with its magnificent proof is particularly notable.
Sullivan has an overall, unitary vision of mathematics. For example, the concept of The Sullivan Dictionary provides parallels for theories that might seem far apart, for example, the dynamics of rational fractions and Kleinian groups. At the end of the 1970s, the physicists Coullet-Tresser and Feigenbaum conceptualized the phenomenon of universality in the transition towards chaos. Sullivan succeeded in placing the problem in an appropriate context, thanks in particular to analogies with non-Euclidean geometry. The proofs for general forms of the renormalisation conjectures followed the pattern of Sullivan’s universality results on the period doubling cascade.
He is currently working on topological field theory and formalism in string theory. Once again, he is concerned with understanding the nature of space through the power of algebra. For example, with Moira Chas, he developed the field of ‘string topology’. His universal vision of mathematics has led to an interest in fluid dynamics, to which he takes an extremely original approach.
Above and beyond his results and discoveries, Sullivan has a unique talent for animating research and inspiring enthusiasm in young people. Sullivan has discovered vast territories, most of which remain to be explored. His influence on the community of mathematicians has been enormous.