The PDE Seminar (MA876) this semester will devoted to a study of so-called ``hard implicit function theorems''. The most famous of these types of theorems are Nash's embedding theorem and the Kolmogorov-Arnold-Moser (KAM) theorem of classical dynamics. We'll start with a review of the classical implicit function theorem and then move on to eventually cover both of these classical extensions.

Grading: All students enrolled in the course will be expected to give at least two lectures on topics related to the course. Your grade will be based on these lectures.

References:

Schwartz, J. T. Nonlinear Functional Analysis, Gordon and Breach (1969).

Hamilton, R. S. The inverse function theorem of Nash and Moser, Bull. AMS vol. 7 pp. 65-163 (1982).

Alvarez-Samaniego, B. and Lannes, D. A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations arXiv:math/0701681v1

Saint-Raymond, X. A simple Nash-Moser implicit function theorem, L'Enseignement Mathematique, vol. 35, pp. 217-226 (1989).

Poppenberg, M. Smooth solutions for a class of nonlinear parabolic evolution equations J. London Math. Soc. vol. 61, pp. 216-244 (2000).

Wayne, C. E. An introduction to KAM theory Lectures in Applied Math. vol. 31 pp. 3-30 (1996).

de la Llave, R. A tutorial on KAM theory Proc. Symposia in Pure Math. vol. 69, pp. 175-288 (2001).

My office hours this spring are Monday 2-3, Tuesday 2-3 and Thursday, 10-11. My office is 242 MCS, phone number 3-1495.