**Author**: Francis Borceux

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 341

**View:** 966

Develops Galois theory in a more general context, emphasizing category theory.

This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabelian categories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers, specifically those interested in Galois theory and Hopf algebras and their categorical unification.

This volume contains a selection of papers presented at the 10th and 11th Meeting of the Association for Mathematics of Language, held in Los Angeles, CA, USA in July 2007 and in Bielefeld, Germany, in August 2009.The 19 revised papers presented together with 3 invited speeches were carefully selected from numerous submissions. The papers in this collection reflect a wide range of theoretical topics relating to language and computation including papers on the intersection of computational complexity, formal language theory, proof theory, and logic, as well as phonology, lexical semantics, syntax and typology.

Galois connections provide the order- or structure-preserving passage between two worlds of our imagination - and thus are inherent in hu man thinking wherever logical or mathematical reasoning about cer tain hierarchical structures is involved. Order-theoretically, a Galois connection is given simply by two opposite order-inverting (or order preserving) maps whose composition yields two closure operations (or one closure and one kernel operation in the order-preserving case). Thus, the "hierarchies" in the two opposite worlds are reversed or transported when passing to the other world, and going forth and back becomes a stationary process when iterated. The advantage of such an "adjoint situation" is that information about objects and relationships in one of the two worlds may be used to gain new information about the other world, and vice versa. In classical Galois theory, for instance, properties of permutation groups are used to study field extensions. Or, in algebraic geometry, a good knowledge of polynomial rings gives insight into the structure of curves, surfaces and other algebraic vari eties, and conversely. Moreover, restriction to the "Galois-closed" or "Galois-open" objects (the fixed points of the composite maps) leads to a precise "duality between two maximal subworlds".

This IMA Volume in Mathematics and its Applications TOWARDS HIGHER CATEGORIES contains expository and research papers based on a highly successful IMA Summer Program on n-Categories: Foundations and Applications. We are grateful to all the participants for making this occasion a very productive and stimulating one. We would like to thank John C. Baez (Department of Mathematics, University of California Riverside) and J. Peter May (Department of Ma- ematics, University of Chicago) for their superb role as summer program organizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of the IMA Markus Keel, Deputy Director of the IMA v PREFACE DEDICATED TO MAX KELLY, JUNE 5 1930 TO JANUARY 26 2007. This is not a proceedings of the 2004 conference “n-Categories: Fo- dations and Applications” that we organized and ran at the IMA during the two weeks June 7–18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. We also thank the IMA sta? for a magni?cent job. There has been a great deal of work in higher c- egory theory since then, but we still feel that it is not yet time to o?er a volume devoted to the main topic of the conference.

This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations.

Assuming little technical background, the author presents the strong analogies between these two concepts starting at an elementary level.

This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realisation of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.