This book demonstrates the development of the regularity theory of solutions of fully nonlinear elliptic equations. Caffarelli and Cabr 'e offer a detailed presentation of all techniques needed to extend the classic Schauder and Calder 'on-Zygmund regularity theory for linear elliptic equations to a fully nonlinear context. The authors present key ideas and prove all results needed for the theory of viscosity solutions of nonlinear equations, and regularity theory for convex fully nonlinear equations and for fully nonlinear equations with variable coefficients. This book is suitable as a text...

Deals with the geometry of paths, the equations of the paths being a generalization of those of geodesics by the process. This book begins with a definition of covariant differentiation which involves functions $L DEGREESi_$ of the coordinates, the law connecting the corresponding functions in any two coordinate systems being funda

This monograph is dedicated to the systematic exposition of the whole variety of topics related to quantum cohomology. The subject first originated in theoretical physics (quantum string theory) and has continued to develop extensively. The author's approach to quantum cohomology is based on the notion of the Frobenius manifold. The first part of the book is devoted to this notion and its extensive interconnections with algebraic formalism of operads, differential equations, perturbations, and geometry. In the second part of the book, the author describes the construction of quantum...

Presents the theory of systems of partial differential equations and the theory of Pfaffian systems so as to exhibit the relations between them. In presenting the theory of Pfaffian systems, the author develops, in detail, the theories of Grassmann algebras and rings with differentiation.

Features lattice theory and portrays its structure and indicates some of its most interesting applications.

Presents a comprehensive study of the algebraic theory of quadratic forms, from classical theory to the developments, including results and proofs. Written from the viewpoint of algebraic geometry, this book includes the theory of quadratic forms over fiel

The main topic of this book is the deep relation between the spacings betweens zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. The book draws on, and gives accessible accounts of, many disparate areas of mathematics.

Part of "Colloquium Series", this book presents systematic treatment of orthogonal polynomials.

Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. The development of analysis in general and of the theory of partial differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides not only...

This volume presents a systematic and unified study of geometric nonlinear functional analysis. This area has its classical roots in the beginning of the 20th century and is a very active research area, having close connections to geometric measure theory, probability, classical analysis, combinatorics, and Banach space theory. The main theme of the book is the study of uniformly continuous and Lipschitz functions between Banach spaces (for example, differentiability, stability, approximation, existence of extensions and fixed points). This study leads naturally also to the classification of...