John Milnor

John Milnor
Name	John Milnor
Caption	Milnor in 2007
Birth date	20 February 1931
Birth place	Orange, New Jersey, U.S.
Fields	Mathematics
Workplaces	Princeton University, University of California, Los Angeles, Institute for Advanced Study, Stony Brook University
Alma mater	Princeton University
Doctoral advisor	Ralph Fox
Doctoral students	Tadashi Tokieda, Laurent Siebenmann, Jonathan Sondow
Known for	Exotic spheres, Fary–Milnor theorem, Milnor–Thurston kneading theory, Milnor map, Milnor number, Milnor's theorem, Milnor–Wood inequality
Awards	Fields Medal (1962), National Medal of Science (1967), Leroy P. Steele Prize (1982, 2004, 2011), Wolf Prize in Mathematics (1989), Abel Prize (2011), American Mathematical Society Fellow

John Milnor is an American mathematician renowned for his profound and diverse contributions to several fields, including differential topology, K-theory, and dynamical systems. Awarded the Fields Medal in 1962 for his work on the Poincaré conjecture in higher dimensions and the discovery of exotic spheres, his career has been marked by a unique ability to uncover deep, unexpected phenomena. His later work, recognized with honors like the Abel Prize and the Wolf Prize in Mathematics, continues to influence areas from geometric topology to complex dynamics.

Early life and education

Born in Orange, New Jersey, he demonstrated exceptional mathematical talent from a young age, winning the Westinghouse Science Talent Search in 1947. He entered Princeton University as an undergraduate, where he quickly came under the influence of prominent mathematicians like Solomon Lefschetz and Ralph Fox. Completing his undergraduate degree in 1951, he remained at Princeton University for his doctoral studies, earning his Ph.D. in 1954 under the supervision of Ralph Fox with a dissertation on isotopy of links, a topic in knot theory.

Mathematical contributions

His early breakthroughs came in differential topology, where he constructed the first examples of exotic spheres, seven-dimensional manifolds that are homeomorphic but not diffeomorphic to the standard 7-sphere; this work was pivotal in the development of surgery theory. He made fundamental contributions to the study of the Poincaré conjecture, proving it for dimensions greater than four, and introduced important invariants like the Milnor number in singularity theory. In algebraic K-theory, he defined Milnor K-theory, and his later research delved into holomorphic dynamics, where the Milnor–Thurston kneading theory for interval maps and his seminal work on the multibrot set became foundational.

Major awards and honors

His exceptional research has been recognized with the highest honors in mathematics. He received the Fields Medal in 1962 at the International Congress of Mathematicians in Stockholm. The National Medal of Science was awarded to him by President Lyndon B. Johnson in 1967. He is a three-time recipient of the Leroy P. Steele Prize from the American Mathematical Society and was awarded the Wolf Prize in Mathematics in 1989. In 2011, he was a co-recipient of the prestigious Abel Prize for his pioneering discoveries in topology, geometry, and algebra.

Academic career

He spent the majority of his early career at Princeton University, where he was appointed a professor in 1960. After a period at the Institute for Advanced Study, he joined the faculty of the University of California, Los Angeles in 1967. In 1970, he moved to Stony Brook University, helping to build its mathematics department into a major research center. He served as director of the Institute for Mathematical Sciences at Stony Brook University and remains an active Professor Emeritus, continuing to mentor students and produce influential research.

Selected publications

His written work is noted for its exceptional clarity and depth. Key monographs include *Morse Theory* (1963), a classic text on the subject, and *Lectures on the h-cobordism theorem* (1965), which helped disseminate Stephen Smale's proof of the Poincaré conjecture. Other influential books are *Singular Points of Complex Hypersurfaces* (1968), *Dynamics in One Complex Variable* (1990), and *Topology from the Differentiable Viewpoint* (1965). His collected works were published by the American Mathematical Society and Publish or Perish Press.

Personal life

He is married to mathematician Dusa McDuff, a noted contributor to symplectic geometry and a fellow professor at Stony Brook University. Known for his modest and unassuming demeanor, he is an avid photographer and has a deep interest in the visual arts. His intellectual legacy is characterized by an enduring curiosity, having made landmark contributions across disparate areas of pure mathematics over a career spanning more than six decades.

Category:American mathematicians Category:Fields Medal winners Category:Abel Prize laureates