Foreword

The two volumes Nonlinear Functional Analysis and Its Applications, published

in the series Proceedings of Symposia in Pure Mathematics (vol. 45, parts 1 and

2), are the result of the thirty-first Summer Research Institute of the American

Mathematical Society held at the University of California at Berkeley from July

11 to July 29, 1983. This institute was partially supported by a grant from the

National Science Foundation, and organized by an Organizing Committee con-

sisting of Haim Brezis, Felix Browder (Chairman), Tosio Kato, J.-L. Lions, Louis

Nirenberg, and Paul Rabinowitz.

The purpose of the institute was to present and develop research on an

international basis in nonlinear functional analysis and its applications, especially

in the study of boundary value problems for nonlinear partial differential

equations and corresponding problems in geometry and mathematical physics.

Major topics which were covered in a series of expository lectures as well as

research talks included: Minimax methods in the calculus of variations, existence

theory for variational problems without compactness, theories of degree of

mapping, inverse function theorems of Nash-Moser type, nonlinear semigroup

theory, nonlinear equations of evolution, nonlinear problems of control theory,

periodic solutions of Hamiltonian systems, generalizations of the Morse theory,

nonlinear partial differential equations in gauge field theory, the theory of

Feigenbaum cascades, the study of the Navier-Stokes equations, nonlinear el-

liptic equations in differential geometry, and a variety of topics concerning

nonlinear elliptic boundary value and eigenvalue problems, bifurcation theory,

nonlinear hyperbolic equations, nonlinear conservation laws, nonlinear Hamilto-

nian-Jacobi equations, and an even wider variety of physical applications.

There were 13 series of expository lectures totaling 39 hours of lectures which

summarized main directions and methods in current research. In addition, there

were 115 one-hour research lectures.

A total of 203 mathematicians registered for the Institute, twenty of whom were

students. The international character of the Institute is reflected in the national

origins of the participants. Twenty-two countries not in North America were

represented by the following numbers of participants: Africa (1), Australia (3),

Belgium (1), Brazil (1), China (4), Czechoslovakia (1), England (4), France (22),

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