John Norman Mather (June 9, 1942 – January 28, 2017) was a mathematician at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

known for his work on

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

and Hamiltonian dynamics. He was descended from Atherton Mather (1663–1734), a cousin of

Cotton Mather Cotton Mather (; February 12, 1663 – February 13, 1728) was a New England Puritan clergyman and a prolific writer. Educated at Harvard College, in 1685 he joined his father Increase as minister of the Congregationalist Old North Meeting H ...

. His early work dealt with the stability of smooth mappings between

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

s 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

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

s of the source and target (i.e., infinitely differentiable coordinate changes). Mather also proved the conjecture of the French

topologist In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he becam ...

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. 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 diffeomorphism In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...

s. 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. 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. 4. He developed the Aubry–Mather theory in higher dimensions, a theory which is now called Mather theory. This theory turned out to be deeply related to the

viscosity solution In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). ...

theory of

Michael G. Crandall Michael Grain Crandall (born November 29, 1940, in Baton Rouge, Louisiana) is an American mathematician, specializing in differential equations. Mathematical career In 1962 Crandall earned a baccalaureate in engineering physics from Universit ...

Pierre-Louis Lions Pierre-Louis Lions (; born 11 August 1956) is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 199 ...

et al. for

Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...

. The link was revealed in the weak KAM theory of

Albert Fathi Albert Fathi (born 27 October 1951, in Egypt) is an Egyptian-French mathematician. He specializes in dynamical systems and is currently a professor at the Georgia Institute of Technology. Fathi attended the ''Collège des frères Lasalle'' in Ca ...

. 5. He announced a proof of

Arnold diffusion In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusio ...

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, 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, John N. "Commutators of diffeomorphisms, III: a group which is not perfect." Commentarii Mathematici Helvetici 60.1 (1985): 122-124. Mather was one of the three editors of the Annals of Mathematics Studies series published by

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

. He was a member of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...

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.

References

External links

Mather notes on Topological Stability
(on the Princeton University website,

pdf Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

file)
John Mather bibliography
on the Princeton University website (

file) *
Obituary
on the Princeton University website {{DEFAULTSORT:Mather, John Norman 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

See also

References

External links