Bøger i Graduate Studies in Mathematics serien

This new edition, now in two parts, has been significantly reorganized and many sections have been rewritten. This first part, designed for a first year of graduate algebra, consists of two courses: Galois theory and Module theory.

Providing a self-contained treatment of persistence theory that is accessible to graduate students, this monograph includes chapters on infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat.

Recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results. This book collects these results, both old and new, and organises them into a natural foundation for all aspects of dynamical systems theory.

Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry.

$\textrm{C}^*$-approxmation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important such as: nuclearity, exactness, quasidiagonality, and local reflexivity.

Analyzes how and when objects like polyhedra or manifolds embed in a given higher-dimensional manifold. This book offers description of the various classic examples of wild embeddings. It provides details of the codimension-three proofs, including the requisite piecewise linear tools.

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.

Presents an introduction to various topics in harmonic analysis. This book discusses Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem).

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.

Treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann manifolds, isometric actions, and symplectic and Poisson geometry. This book covers a wide range of topics of modern coordinate-free differential geometry.

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.

Presents an introduction to algebraic number theory, and the rudiments of algebraic geometry. This book includes such topics as transfer and character theory of finite groups, modules over artinian rings, modules over Dedekind domains, and transcendental field extensions.

Offers a treatment of measure and integration that begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. This text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales.

A survey of asymptotic methods set in the applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. It is suitable for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering.

Presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds. This book unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory.

Defines simplicial homology and cohomology, with various examples and applications. This book introduces Kolmogorov-Alexander multiplication in cohomology. It deals with the applications of simplicial homology and cohomology to obstruction theory, in particular, to characteristic classes of vector bundles.

Presents complex algebraic curves. This book covers a range of topics from conics and cubics to theta functions, Jacobians, and questions of moduli. It is suitable for those preparing a course on the topic of complex curves.

Presents the concepts and tools to solve problems of analysis, and shows how to apply these concepts to research in a variety of areas. This work covers the min-max principle, semi-classical approximation, coherent states and Lieb-Thirring inequalities. It also covers Sobolev Inequalities, including the Nash and Log Sobolev inequalities.

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.

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.

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.