He announced a proof of Arnold diffusion for nearly integrable Hamiltonian systems with three degrees of freedom. He prepared the technique, formulated a proper concept of genericity and made some important progresses towards its solution. 6. In a series of papers,Mather, John N. "Commutators of diffeomorphisms: II." Commentarii Mathematici Helvetici 50.1 (1975): 33-40. he proved that for certain regularity r, depending on the dimension of the smooth manifold M, the group Diff(M, r) is perfect, i.e. equal to its own commutator subgroup, where Diff(M, r) is the group of C^r diffeomorphisms of a smooth manifold M that are isotopic to the identity through a compactly supported C^r'' isotopy. He also constructed counterexamples where the regularity-dimension condition is violated. Mather was one of the three editors of the Annals of Mathematics Studies series published by Princeton University Press. He was a member of the National Academy of Sciences beginning in 1988. He received the John J. Carty Award of the National Academy of Sciences in 1978 (for pure mathematics) and the George David Birkhoff Prize in applied mathematics in 2003. He also received the Brazilian Order of Scientific Merit in 2000 and the Brouwer Medal from the Royal Dutch Mathematical Society in 2014. See also List of members of the National Academy of Sciences References External links Mather notes on Topological Stability (on the Princeton University website, pdf file) John Mather bibliography on the Princeton University website (pdf file) Obituary on the Princeton University website 1942 births 2017 deaths Members of the United States National Academy of Sciences 20th-century American mathematicians 21st-century American mathematicians Princeton University faculty Recipients of the Great Cross of the National Order of Scientific Merit (Brazil) Brouwer Medalists Harvard University alumni Princeton University alumni Institute for Advanced Study visiting scholars Dynamical systems theorists People from Los Angeles Mathematicians from California