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Lectures on Differential Galois Theory
Differential Galois theory studies solutions of differential equations over a differential base field. In much the same way that ordinary Galois theory is the theory of field extensions generated by solutions of (one variable) polynomial equations, differential Galois theory looks at the nature of the differential field extension generated by the solution of differential equations. An additional feature is that the corresponding differential Galois groups (of automorphisms of the extension fixing the base and commuting with the derivation) are algebraic groups. This book deals with the differential Galois theory of linear homogeneous differential equations, whose differential Galois groups are algebraic matrix groups. In addition to providing a convenient path to Galois theory, this approach also leads to the constructive solution of the inverse problem of differential Galois theory for various classes of algebraic groups. Providing a self-contained development and many explicit examples, this book provides a unique approach to differential Galois theory and is suitable as a textbook at the advanced graduate level.
Partition Problems in Topology
This book presents results on the case of the Ramsey problem for the uncountable: When does a partition of a square of an uncountable set have an uncountable homogeneous set? This problem most frequently appears in areas of general topology, measure theory, and functional analysis. Building on his solution of one of the two most basic partition problems in general topology, the ``S-space problem,'' the author has unified most of the existing results on the subject and made many improvements and simplifications. The first eight sections of the book require basic knowldege of naive set theory at the level of a first year graduate or advanced undergraduate student. The book may also be of interest to the exclusively set-theoretic reader, for it provides an excellent introduction to the subject of forcing axioms of set theory, such as Martin's axiom and the Proper forcing axiom.
Recent Developments in Integrable Systems and Riemann-Hilbert Problems
This volume is a collection of papers presented at a special session on integrable systems and Riemann-Hilbert problems. The goal of the meeting was to foster new research by bringing together experts from different areas. Their contributions to the volume provide a useful portrait of the breadth and depth of integrable systems. Topics covered include discrete Painleve equations, integrable nonlinear partial differential equations, random matrix theory, Bose-Einstein condensation, spectral and inverse spectral theory, and last passage percolation models. In most of these articles, the Riemann-Hilbert problem approach plays a central role, which is powerful both analytically and algebraically. The book is intended for graduate students and researchers interested in integrable systems and its applications.
Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group
This volume presents a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the $q$-Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems. The main results are proved by showing that the Iwahori-Hecke algebras and $q$-Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and s...
Differential Algebra and Related Topics
Differential algebra explores properties of solutions of systems of (ordinary or partial, linear or non-linear) differential equations from an algebraic point of view. It includes as special cases algebraic systems as well as differential systems with algebraic constraints. This algebraic theory of Joseph F Ritt and Ellis R Kolchin is further enriched by its interactions with algebraic geometry, Diophantine geometry, differential geometry, model theory, control theory, automatic theorem proving, combinatorics, and difference equations. Differential algebra now plays an important role in computational methods such as symbolic integration and symmetry analysis of differential equations. These ...
Separable Algebras
This book 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. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups. The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.
Invariant Theory
This volume contains the proceedings of the AMS Special Session on Invariant Theory, held in Denton, Texas in the fall of 1986; also included are several invited papers in this area. The purpose of the conference was to exchange ideas on recent developments in algebraic group actions on algebraic varieties. The papers fall into three main categories: actions of linear algebraic groups; flag manifolds and invariant theory; and representation theory and invariant theory. This book is likely to find a wide audience, for invariant theory is connected to a range of mathematical fields, such as algebraic groups, algebraic geometry, commutative algebra, and representation theory.
Lectures on Harmonic Analysis
This book demonstrates how harmonic analysis can provide penetrating insights into deep aspects of modern analysis. It is both an introduction to the subject as a whole and an overview of those branches of harmonic analysis that are relevant to the Kakeya conjecture. The usual background material is covered in the first few chapters: the Fourier transform, convolution, the inversion theorem, the uncertainty principle and the method of stationary phase. However, the choice of topics is highly selective, with emphasis on those frequently used in research inspired by the problems discussed in the later chapters. These include questions related to the restriction conjecture and the Kakeya conjec...
Complex Proofs of Real Theorems
Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, ``The shortest and best way between two truths of the real domain often passes through the imaginary one.'' Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics. Topics discussed include weighted approximation on the line, Muntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction the...
Arithmetic Noncommutative Geometry
Arithmetic Noncommutative Geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties. Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry. With a foreword written by Yuri Manin and a brief introduction to noncommutative geometry, this book offers a comprehensive account of the cross fertilization between two important areas, noncommutative geometry and number theory. It is suitable for graduate students and researchers interested in these areas.
