This article, based on the 1999 Josiah Willard Gibbs Lecture, discusses mathematical models for the rhythms that are observed in certain kinds of brain activity, as well as transitions between different rhythms (pp. 6)
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Some classical nonlinear PDEs have solutions that are asymptotic to nontrivial linear combinations of solitary waves. The authors show how to make the study of these equations more systematic by introducing actions of loop groups on the spaces of solutions. (pp. 17)
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Arthur gives a short introduction to the work of Harish-Chandra and then provides hints of the nature of the Langlands Program, offering a sense of the role played by Harish-Chandra's work in the Langlands Program. (pp. 26)
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