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Introduces the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators and spectral theory of self-adjoint operators. This work presents the theorems and methods of abstract functional analysis and applications of these methods to Banach algebras and theory of unbounded self-adjoint operators.

Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauss curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.

Deals with the topic of geometric configurations of points and lines. This book presents the history of the topic, with its surges and declines since its beginning in 1876. It covers the advances in the field since the revival of interest in geometric configurations.

Markov processes are among the most important stochastic processes for both theory and applications. This book develops the general theory of these processes, and applies this theory to various special examples.

Offers a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. This title introduces a class of stratified spaces, so-called stratifolds. It derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality.

A companion volume to ""Graduate Algebra: Commutative View"", this book presents a unified approach to many important topics, such as group theory, ring theory, Lie algebras, and gives conceptual proofs of many basic results of noncommutative algebra.

Representation theory plays important roles in geometry, algebra, analysis, and mathematical physics. This book presents an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. It is suitable for a year-long graduate course.

Presents an integrated development of the theory of several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. This text includes a treatment of the local theory.

Presents a basic collection of principles, techniques, and applications needed to conduct independent research in gauge theory and its use in geometry and topology. This title includes self-contained computations of the Seiberg-Witten invariants of most simply connected algebraic surfaces using Witten's factorization method.

Presents an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. This book includes such topics as Riemann surfaces, holomorphic functions of several variables, classification and deformation of singularities, and fundamentals of differential topology.

Presents two essential and apparently unrelated subjects that include, microlocal analysis and the theory of pseudo-differential operators, a basic tool in the study of partial differential equations and in analysis on manifolds; and the Nash-Moser theorem, that is fundamentally important in geometry, dynamical systems, and nonlinear PDE.

Fourier analysis encompasses a variety of perspectives and techniques. This book presents the real variable methods of Fourier analysis introduced by Calderon and Zygmund. It includes topics such as the Hardy-Littlewood maximal function and the Hilbert transform. It also covers the study of singular integral operators and multipliers.

Presents main theorems about class fields of algebraic number fields. This book features calculations with quadratic fields that show the use of the norm residue symbol.

Probability theory has become a convenient language and a useful tool in many areas of modern analysis. This book intends to explore part of this connection concerning the relations between Brownian motion on a manifold and analytical aspects of differential geometry. It begins with a review of stochastic differential equations on Euclidean space.

Intended for graduate students who have completed standard courses in linear algebra and abstract algebra, this title provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles.

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. This book treats the spectral theory of automorphic forms as the study of the space $L^2 (H\Gamma)$ and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $H$.

Describes the properties of harmonic and caloric measures in Lipschitz domains, a relation between parallel surfaces and elliptic equations, monotonicity formulas and rigidity. This book is suitable for graduate students and researchers interested in partial differential equations.

Symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups. This book presents an introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra.

Suitable for instructors for use as a companion volume to the book, ""A Modern Theory of Integration"" (""AMS Graduate Studies in Mathematics"" series, Volume 32).

Presents a comprehensive exposition of the theory of linear operators on Banach spaces and Banach lattices. This book offers a presentation that includes various developments in operator theory and draws together results that are spread over the vast literature. It contains over 600 exercises to help students master the material developed.

Presents an introduction to functional analysis and the initial fundamentals of $C^*$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide an account of the introductory portions of this important and technically difficult subject.

Contains lectures that focus on fundamentals of the modern theory of linear elliptic and parabolic equations in Holder spaces. This title shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations.

Suitable for a first year graduate course on global analysis, this title proves the basics of Fourier transforms, Sobolev theory and interior regularity at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex.

Deals with finite-alphabet stationary processes, which are important in physics, engineering, and data compression. This book gives a careful presentation of the many models for stationary finite-alphabet processes that have been developed in probability theory, ergodic theory, and information theory.

Covers the key topics of numerical methods. This work covers topics including interpolation, the fast Fourier transform, iterative methods for solving systems of linear and nonlinear equations, numerical methods for solving ODEs, numerical methods for matrix eigenvalue problems, approximation theory and computer arithmetic.

There are incredibly rich connections between classical analysis and number theory. This title uncovers interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. It is suitable for a graduate course or for independent reading.

Presents an account of noncommutative Noetherian rings. This book covers the major developments from the 1950s, stemming from Goldie's theorem and onward, including applications to group rings, enveloping algebras of Lie algebras, PI rings, differential operators, and localization theory.