Mathematical Notes^{39}
Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Series Editors
The Mathematical Notes series features informal notes from developing courses and seminars on advanced mathematical topics.

The description for this book, Lectures on Vector Bundles over Riemann Surfaces. (MN6), Volume 6, will be forthcoming.

These notes describe a general procedure for calculating the Betti numbers of the projective quotient varieties that geometric invariant theory associates to reductive group actions on nonsingular complex projective varieties. These...

The interplay between analysis on Lie groups and the theory of special functions is well known. This monograph deals with the case of the Heisenberg group and the related expansions in terms of Hermite, special Hermite, and Laguerre...

The second edition of Introduction to Partial Differential Equations, which originally appeared in the Princeton series Mathematical Notes, serves as a text for mathematics students at the intermediate graduate level. The goal is to...

An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The...

There was a special year devoted to the topic of several complex variables at the MittagLeffler Institute in Stockholm, Sweden, and this volume contains the resulting survey papers and research papers. The work covers a broad spectrum...

The theory of Dmodules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence...

Written by Arthur Ogus on the basis of notes from Pierre Berthelot's seminar on crystalline cohomology at Princeton University in the spring of 1974, this book constitutes an informal introduction to a significant branch of algebraic...

This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather...

The classical uniformization theorem for Riemann surfaces and its recent extensions can be viewed as introducing special pseudogroup structures, affine or projective structures, on Riemann surfaces. In fact, the additional structures...

The purpose of this book is to provide a selfcontained account, accessible to the nonspecialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup structures.
Originally published in 1981.
The... 
Many of the operators one meets in several complex variables, such as the famous Lewy operator, are not locally solvable. Nevertheless, such an operator L can be thoroughly studied if one can find a suitable relative parametrixan...

The theory of pseudodifferential operators (which originated as singular integral operators) was largely influenced by its application to function theory in one complex variable and regularity properties of solutions of elliptic...

Kenneth Brakke studies in general dimensions a dynamic system of surfaces of no inertial mass driven by the force of surface tension and opposed by a frictional force proportional to velocity. He formulates his study in terms of...

Encompassing both introductory and more advanced research material, these notes deal with the author's contributions to stochastic processes and focus on Brownian motion processes and its derivative white noise.
Originally published in... 
Mathematical Notes, 29
Originally published in 1983.
The Princeton Legacy Library uses the latest printondemand technology to again make available previously outofprint books from the distinguished backlist of Princeton University... 
This book presents a development of the basic facts about harmonic analysis on local fields and the ndimensional vector spaces over these fields. It focuses almost exclusively on the analogy between the local field and Euclidean cases...

The ∂̄ Neumann problem is probably the most important and natural example of a nonelliptic boundary value problem, arising as it does from the CauchyRiemann equations. It has been known for some time how to prove solvability and...

This book is meant to give an account of recent developments in the theory of Plateau's problem for parametric minimal surfaces and surfaces of prescribed constant mean curvature ("Hsurfaces") and its analytical framework. A...

This book builds upon the revolutionary discovery made in 1974 that when one passes from function f to a function J of paths joining two points A1≠A1 the connectivities R1 of the domain of f can be replaced by connectivities R1 over...

Based on a series of lectures given by HarishChandra at the Institute for Advanced Study in 19711973, this book provides an introduction to the theory of harmonic analysis on reductive padic groups.
Originally published in 1979.
The... 
This book is a sequel to Lectures on Complex Analytic Varieties: The Local Paranwtrization Theorem (Mathematical Notes 10, 1970). Its unifying theme is the study of local properties of finite analytic mappings between complex analytic...

This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but...

This volume offers a systematic treatment of certain basic parts of algebraic geometry, presented from the analytic and algebraic points of view. The notes focus on comparison theorems between the algebraic, analytic, and continuous...

Mathematical No/ex, 27
Originally published in 1981.
The Princeton Legacy Library uses the latest printondemand technology to again make available previously outofprint books from the distinguished backlist of Princeton University... 
In the spring of 1985, A. Casson announced an interesting invariant of homology 3spheres via constructions on representation spaces. This invariant generalizes the Rohlin invariant and gives surprising corollaries in lowdimensional...

This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. Paula Tretkoff emphasizes those finite covers that are free quotients of the...

John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as...

The recent introduction of the SeibergWitten invariants of smooth fourmanifolds has revolutionized the study of those manifolds. The invariants are gaugetheoretic in nature and are close cousins of the muchstudied SU(2)invariants...

Diffusive motiondisplacement due to the cumulative effect of irregular fluctuationshas been a fundamental concept in mathematics and physics since Einstein's work on Brownian motion. It is also relevant to understanding various...

This book provides a comprehensive and uptodate introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology...

This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in Orsay from 1976 to 1977, it covers...

This book introduces a new class of nonassociative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely...

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the wellknown...

Addressing physicists and mathematicians alike, this book discusses the finite dimensional representation theory of sl(2), both classical and quantum. Covering representations of U(sl(2)), quantum sl(2), the quantum trace and color...

This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of...

The object of this monograph is to give an exposition of the realvariable theory of Hardy spaces (HP spaces). This theory has attracted considerable attention in recent years because it led to a better understanding in Rn of such...

Based on lectures in Erevan, this exposition of ergodic theory contains a rich collection of examples well chosen to introduce the reader to the main themes of the subject. Topics discussed include existence of invariant measures...

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor...