This is the proceedings of the meeting entitled "The 12th MSJ International Research Institute of the Mathematical Society of Japan 2003". The papers cover several important topics in Singularity theory. Especially some of them are survey on motivic integrations, Thom polynomials, complex analytic singularity theory, generic differential geometry etc.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

The book offers a comprehensive introduction to Leavitt path algebras (LPAs) and graph C*-algebras. Highlighting their significant connection with classical K-theory-which plays an important role in mathematics and its related emerging fields-this book allows readers from diverse mathematical backgrounds to understand and appreciate these structures. The articles on LPAs are mostly of an expository nature and the ones dealing with K-theory provide new proofs and are accessible to interested students and beginners of the field. It is a useful resource for graduate students and researchers working in this field and related areas, such as C*-algebras and symbolic dynamics.

The Grothendieck-Teichmuller group was defined by Drinfeld in quantum group theory with insights coming from the Grothendieck program in Galois theory. The ultimate goal of this book is to explain that this group has a topological interpretation as a group of homotopy automorphisms associated to the operad of little 2-discs, which is an object used to model commutative homotopy structures in topology. This volume gives a comprehensive survey on the algebraic aspects of this subject. The book explains the definition of an operad in a general context, reviews the definition of the little discs operads, and explains the definition of the Grothendieck-Teichmuller group from the viewpoint of the theory of operads. In the course of this study, the relationship between the little discs operads and the definition of universal operations associated to braided monoidal category structures is explained. Also provided is a comprehensive and self-contained survey of the applications of Hopf algebras to the definition of a rationalization process, the Malcev completion, for groups and groupoids. Most definitions are carefully reviewed in the book; it requires minimal prerequisites to be accessible to a broad readership of graduate students and researchers interested in the applications of operads.

Ce livre des Elements de mathematique est consacre a la Topologie algebrique. Les quatre premiers chapitres presentent la theorie des revetements d'un espace topologique et du groupe de Poincare. On construit le revetement universel d'un espace connexe pointe delacable et on etablit l'equivalence de categories entre revetements de cet espace et actions du groupe de Poincare. On demontre une version generale du theoreme de van Kampen exprimant le groupoide de Poincare d'un espace topologique comme un coegalisateur de diagrammes de groupoides. Dans de nombreuses situations geometriques, on en deduit une presentation explicite du groupe de Poincare.

The field of differential topology underwent a dramatic development period between 1955 and 1965. This collection of articles written by one of the creators of this field contains not only original papers, but also previously unpublished expository lectures. It includes commentary by the author, filling in some of the historical context, and outlining subsequent developments. It includes a rich bibliography of newer and older papers, providing a wider and deeper understanding of the subject. It also outlines the actual state of the art, and provides an index that will allow the reader to browse easily through the book. Of particular interest are the articles related to the existence of exotic differentiable structures on spheres, the achievement for which J. Milnor was awarded the Fields Medal in 1962.

A fundamental element of the study of 3-manifolds is Thurston's remarkable geometrization conjecture, which states that the interior of every compact 3-manifold has a canonical decomposition into pieces that have geometric structures. In most cases, these structures are complete metrics of constant negative curvature, that is to say, they are hyperbolic manifolds. The conjecture has been proved in some important cases, such as Haken manifolds and certain types of fibered manifolds. The influence of Thurston's hyperbolization theorem on the geometry and topology of 3-manifolds has been tremendous. This book presents a complete proof of the hyperbolization theorem for 3-manifolds that fiber over the circle, following the plan of Thurston's original (unpublished) proof, though the double limit theorem is dealt with in a different way. The book should be suitable for graduate students with a background in modern techniques of low-dimensional topology and will also be of interest to researchers in geometry and topology. This is the English translation of a volume originally published in 1996 by the Societe Mathematique de France.

This book is a course in representation theory of semisimple groups, automorphic forms and the relations between these two subjects written by some of the world's leading experts in these fields. It is based on the 1996 instructional conference of the International Centre for Mathematical Sciences in Edinburgh. The book begins with an introductory treatment of structure theory and ends with an essay by Robert Langlands on the current status of functoriality. All papers are intended to provide overviews of the topics they address, and the authors have supplied extensive bibliographies to guide the reader who wants more detail.The aim of the articles is to treat representation theory with two goals in mind: to help analysts make systematic use of Lie groups in work on harmonic analysis, differential equations, and mathematical physics and to provide number theorists with the representation-theoretic input to Wiles' proof of Fermat's Last Theorem. This book features discussion of representation theory from many experts' viewpoints; treatment of the subject from the foundations through recent advances; discussion of the analogies between analysis of cusp forms and analysis on semisimple symmetric spaces, which have been at the heart of research breakthroughs for 40 years; and, extensive bibliographies.

This is the second of two volumes on the qualitative theory of foliations. For this volume, the authors have selected three special topics: analysis on foliated spaces, characteristic classes of foliations, and foliated manifolds. Each of these is an example of deep interaction between foliation theory and some other highly-developed area of mathematics. In all cases, the authors present useful, in-depth introductions, which lead to further study using the extensive available literature. This comprehensive volume has something to offer a broad spectrum of readers: from beginners to advanced students to professional researchers. It contains exercises and many illustrations. The book would make an elegant supplementary text for a topics course at the advanced graduate level. ""Foliations I"" is Volume 23 in the AMS series, ""Graduate Studies in Mathematics"".

This volume touches upon some significant themes that have arisen in the fields of operator algebras, low-dimensional topology and mathematical physics. Gathering together a collection of papers from a NATO-sponsored Advance Research Workshop, its intention is to reveal that these seemingly diverse fields are now inextricably interconnected. While operator algebras and mathematical physics have enjoyed a mutually beneficial relationship dating back to the time of von Neumann, the link between operator algebras and low-dimensional topology only emerged with the work of Vaughan Jones.