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John N. Mather

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John N. Mather
Mather at Oberwolfach in 2005
Born
John Norman Mather

(1942-06-09)June 9, 1942
Los Angeles, California, U.S.
DiedJanuary 28, 2017(2017-01-28) (aged 74)
Princeton, New Jersey, U.S.
Alma materHarvard University
Princeton University
Known forSmooth functions
Topologically stratified space
Aubry–Mather theory
Mather theory
AwardsJohn J. Carty Award for the Advancement of Science (1978)
National Order of Scientific Merit (Brazil) (2000)
George David Birkhoff Prize (2003)
Brouwer Medal (2014)
Scientific career
FieldsMathematics
InstitutionsInstitut des Hautes Études Scientifiques
Harvard University
Princeton University
Doctoral advisorJohn Milnor
Doctoral studentsGiovanni Forni (1993)
Vadim Kaloshin (2001)
Alfonso Sorrentino (2008)

John Norman Mather (June 9, 1942 – January 28, 2017) was a mathematician at Princeton University known for his work on singularity theory and Hamiltonian dynamics. He was descended from Atherton Mather (1663–1734), a cousin of Cotton Mather. His early work dealt with the stability of smooth mappings between smooth manifolds of dimensions n (for the source manifold N) and p (for the target manifold P). He determined the precise dimensions (n,p) for which smooth mappings are stable with respect to smooth equivalence by diffeomorphisms of the source and target (i.e., infinitely differentiable coordinate changes).[1]

Mather also proved the conjecture of the French topologist René Thom that under topological equivalence smooth mappings are generically stable: the subset of the space of smooth mappings between two smooth manifolds consisting of the topologically stable mappings is a dense subset in the smooth Whitney topology. His notes on the topic of topological stability are still a standard reference on the topic of topologically stratified spaces.[2]

In the 1970s, Mather switched to the field of dynamical systems. He made the following main contributions to dynamical systems that deeply influenced the field.

1. He introduced the concept of Mather spectrum and gave a characterization of Anosov diffeomorphisms.[3]

2. Jointly with Richard McGehee, he gave an example of collinear four-body problem which has initial conditions leading to solutions that blow up in finite time. This was the first result that made the Painlevé conjecture plausible.[4]

3. He developed a variational theory for the globally action minimizing orbits for twist maps (convex Hamiltonian systems of two degrees of freedom), along the line of the work of George David Birkhoff, Marston Morse, Gustav A. Hedlund, et al. This theory is now known as Aubry–Mather theory.[5][6]

4. He developed the Aubry–Mather theory in higher dimensions, a theory which is now called Mather theory.[7][8][9] This theory turned out to be deeply related to the viscosity solution theory of Michael G. Crandall, Pierre-Louis Lions et al. for Hamilton–Jacobi equation. The link was revealed in the weak KAM theory of Albert Fathi.[10]

5. He announced a proof of Arnold diffusion for nearly integrable Hamiltonian systems with three degrees of freedom.[11] He prepared the technique, formulated a proper concept of genericity and made some important progresses towards its solution.

6. In a series of papers,[12][13] 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.[14]

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)[15] 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

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References

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  1. ^ Mather, J. N. "Stability of C∞ mappings. VI: The nice dimensions". Proceedings of Liverpool Singularities-Symposium, I (1969/70), Lecture Notes in Math., Vol. 192, Springer, Berlin (1971), 207–253.
  2. ^ Mather, John "Notes on topological stability." ``Bulletin of the American Math. Soc. (N. S.) 49 (2012), no. 4, 475-506.
  3. ^ Mather, John N. "Characterization of Anosov diffeomorphisms." Indagationes Mathematicae (Proceedings). Vol. 71. North-Holland, 1968.
  4. ^ Mather, John N., and Richard McGehee. "Solutions of the collinear four body problem which become unbounded in finite time." Dynamical systems, theory and applications. Springer Berlin Heidelberg, 1975. 573–597.
  5. ^ Mather, John, and Giovanni Forni. "Action minimizing orbits in hamiltomian systems." Transition to chaos in classical and quantum mechanics (1994): 92–186.
  6. ^ Bangert, Victor. "Mather sets for twist maps and geodesics on tori." Dynamics reported. Vieweg+ Teubner Verlag, 1988. 1–56.
  7. ^ Mather, John N. "Action minimizing invariant measures for positive definite Lagrangian systems", Mathematische Zeitschrift 207.1 (1991): 169–207.
  8. ^ Mather, John N. "Variational construction of connecting orbits." Annales de l'Institut Fourier, Vol. 43. No. 5. 1993.
  9. ^ Sorrentino, Alfonso "Action-minimizing methods in Hamiltonian dynamics: an introduction to Aubry–Mather theory", Mathematical Notes Series Vol. 50 (Princeton University Press), 128 pp., ISBN 9780691164502, 2015.
  10. ^ Fathi, Albert. "Weak KAM theorem in Lagrangian dynamics preliminary version number 10", Cambridge University Press (2008).
  11. ^ J.N. Mather, Arnold diffusion. I: Announcement of results, Journal of Mathematical Sciences, Vol. 124, No. 5, 2004
  12. ^ Mather, John N. "Commutators of diffeomorphisms." Commentarii Mathematici Helvetici 49.1 (1974): 512-528.
  13. ^ Mather, John N. "Commutators of diffeomorphisms: II." Commentarii Mathematici Helvetici 50.1 (1975): 33-40.
  14. ^ Mather, John N. "Commutators of diffeomorphisms, III: a group which is not perfect." Commentarii Mathematici Helvetici 60.1 (1985): 122-124.
  15. ^ "John J. Carty Award for the Advancement of Science". Archived from the original on 2015-02-28.
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