Bøger i Graduate Studies in Mathematics serien

Algebraic number theory is one of the most refined creations in mathematics. This book presents the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory.

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.

Presents an introduction to differential geometry through differential forms, emphasizing their applications in various areas of mathematics and physics. This work focuses on Stokes' theorem, the classical integral formulas and their applications to harmonic functions and topology.

Focuses on linear equations of first and second order. An important feature of his treatment is that the majority of the techniques are applicable more generally. In particular, the author emphasizes a priori estimates throughout the text, even for those equations that can be solved explicitly.

Suitable for students in mathematics, physics, engineering, and computer science, this book explores contemporary approximation theory. It covers such topics as projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, and thin-plate splines.

Offers a focused point of view on the differential geometry of curves and surfaces. This monograph treats the Gauss - Bonnet theorem and discusses the Euler characteristic. It also covers Alexandrov's theorem on embedded compact surfaces in R3 with constant mean curvature.

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.

Presents a comprehensive introduction to research on the applications of graph theory to real-world networks such as web graph. This book discusses both models of the web graph and algorithms for searching the web.

Offers an introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. This book includes topics such as the nondegeneracy of the ground state, spectra of atoms, and scattering theory. It is suited for self-study and includes exercises.

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.

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.

Covers classical modular forms. This book is suitable to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations.

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.

For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. This book examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold.

Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. This book aims to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the 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.

Takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. This book covers Riemann surfaces, with complex charts and meromorphic functions taking center stage.

Centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semi simple Lie groups. This book culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups.

Understanding and working with the models of financial markets requires a sound knowledge of the mathematical tools and ideas from which they are built. Covering the topic of financial modelling and optimization, this book helps readers obtain a self-contained introduction to the underlying mathematics. It is suitable as a graduate textbook.

Presents the fundamentals of non-archimedean dynamics, including a unified exposition of Rivera-Letelier's classification theorem, as well as results on wandering domains, repelling periodic points, and equilibrium measures. The Berkovich projective line is developed from the ground up, as are relevant results in non-archimedean analysis.

Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers. With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a course on functional analysis for beginning graduate students.

Presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized.

This is the second part of the new edition of Advanced Modern Algebra. The material has been significantly reorganised and sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra.

Presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. Azumaya algebras, the henselization of local rings, and Galois theory are introduced and treated. Interwoven throughout these applications is the important notion of etale algebras.

Offers an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces.

Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry.