Offers introductions to the advances in 3 significant areas of development in commutative algebra and its applications. This book is based on courses at the Winter School on Commutative Algebra and Applications held in Barcelona: 'Tight Closure and Vector

The Hilbert scheme $X n] $ of a surface $X$ describes collections of $n$ (not necessarily distinct) points on $X$. More precisely, it is the moduli space for $0$-dimensional subschemes of $X$ of length $n$. It has been realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory-even theoretical physics. The discussion in the book reflects this feature of Hilbert schemes. For example, a construction of the representation of the infinite dimensional Heisenberg...

J -holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in 1985. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. This book presents the first coherent and full account of the theory of J -holomorphic curves, the details of which are presently scattered in various research papers. The first half of the book is an expository account of the field, explaining the main technical aspects. McDuff and Salamon give complete proofs of Gromov's compactness theorem for spheres and of...

This volume is based on the author's lecture courses to algebraists at Munich and at Goteborg. He presents a unified approach from the point of view of Frobenius algebras/extensions. The book is intended for graduate students and research mathematicians working in algebra and topology.

Developments in topology and analysis have led to the creation of new lines of investigation in differential geometry. The 2000 Barrett Lectures present the background, context and main techniques of three such lines by means of surveys by researchers.

Coarse geometry is the study of spaces (particularly metric spaces) from a ''large scale'' point of view, so that two spaces that look the same from a great distance are actually equivalent. This point of view is effective because it is often true that the relevant geometric properties of metric spaces are determined by their coarse geometry. Two examples of important uses of coarse geometry are Gromov's beautiful notion of a hyperbolic group and Mostow's proof of his famous rigidity theorem. The first few chapters of the book provide a general perspective on coarse structures. Even when only...

Presents a self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the $q$-Schur algebras. This book intends to classify the blocks and the simple modules of both algebras. It also includes a chapter containing a survey of the advances and open problems.