Peter McMullen, Egon Schulte

Abstract regular polytopes are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book will interest researchers and graduate students in discrete geometry, combinatorics and group theory.

Edited by David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin Shen

Finsler geometry generalises Riemannian geometry in the same sense that Banach spaces generalise Hilbert spaces. This book presents an expository account of seven important topics in Riemann--Finsler geometry, ones which have recently undergone significant development but have not had a detailed pedagogical treatment elsewhere. Each article will open the door to an active area of research, and is suitable for a special topics course in graduate-level differential geometry. The contributors consider issues related to volume, geodesics, curvature, complex differential geometry and parametrised jet bundles, and include a variety of instructive examples.

John McCleary

Spectral sequences are among the most elegant, most powerful, and most complicated methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. It introduces the algebraic foundations in an accessible manner, starting from informal calculations, to give the novice a familiarity with the range of applications possible with spectral sequences. This second edition contains a new chapter on the Bockstein spectral sequence and an updated treatment of other topological sequences. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Allen Hatcher

In most mathematics departments at major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology. This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. The book presents the basic material of the subject: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. A unique feature is the inclusion of many optional topics for which elementary expositions are hard to find. Researchers and students alike will welcome this aspect of the book.

Shigeru Mukai, Translated by W. M. Oxbury

Incorporated in this volume are the first two books in Mukai's series on moduli theory. The notion of a moduli space is central to geometry. However, its influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. An accurate account of Mukai's influential Japanese texts, this translation will be a valuable resource for researchers and graduate students working in a range of areas.

Hal Schenck

The interplay between algebra and geometry is a beautiful area of mathematical investigation. Recent advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. Suitable for graduate students, the objective of this book is to bring advanced algebra to life with lots of examples. The first chapters provide an introduction to commutative algebra and connections to geometry. The rest of the book then focuses on three active areas of contemporary algebra: Homological Algebra, Algebraic Combinatorics and Algebraic Topology, and Algebraic Geometry.

Károly Böröczky, Jr

Finite arrangements of convex bodies were intensively investigated in the second half of the 20th century. Connections to many other subjects were made, including crystallography, the local theory of Banach spaces, and combinatorial optimisation. This book, the first one dedicated solely to the subject, provides an in-depth state-of-the-art discussion of the theory of finite packings and coverings by convex bodies. It contains various new results and arguments, besides collecting those scattered around in the literature, and provides a comprehensive treatment of problems whose interplay was not clearly understood before.

Claire Voisin, Translated by Leila Schneps

This is a modern introduction to Kaehlerian geometry and Hodge structure. It starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory and culminates with the Hodge decomposition theorem. The book is is completely self-contained and can be used by students, while its content gives an up-to-date account of Hodge theory and complex algebraic geometry. The text is complemented by exercises which provide useful results in complex algebraic geometry.

Claire Voisin, Translated by Leila Schneps

The second volume of this modern and unique account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. The main results of the second part are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part deals with the relationships between Hodge theory and algebraic cycles. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in algebraic and differential geometry.

David Mumford, Caroline Series, David Wright

Felix Klein rediscovered in mathematics an idea from Eastern philosophy: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour. The whole Universe was mirrored in each pearl. For a century this idea, practically impossible to represent by hand, barely existed outside the imagination of mathematicians. In the 1980s the authors embarked on the first computer exploration of Klein's vision. Join them on the path from simple mathematics to computer programs that generate these images related to ideas at the forefront of knowledge.

Isaac Chavel

Classical isoperimetric inequalities in the plane relate the area of a domain to the length of its boundary and, in space, the volume to the area of its boundary. This advanced introduction emphasizes the subject's variety of ideas, techniques, and applications. The author discusses inequalities in Euclidean and Riemannian geometry, methods of classical differential geometry and elementary modern geometric measure, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on Riemannian manifolds. Requiring only a basic course in differential geometry, this book will appeal to graduate students and researchers alike in differential geometry, analysis, and related subjects.

Bill Casselman

This practical introduction to the techniques needed to produce mathematical illustrations of high quality is suitable for anyone with a modest acquaintance with coordinate geometry. The author combines a completely self-contained step-by-step introduction to the graphics programming language PostScript with advice on what goes into good mathematical illustrations, chapters showing how good graphics can be used to explain mathematics, and a treatment of all the mathematics needed to make such illustrations. The many small simple graphics projects can also be used in courses in geometry, graphics, or general mathematics. Code for many of the illustrations is included, and can be downloaded from the book's web site: www.math.ubc.ca/~cass/graphics/manual. Mathematicians, scientists, engineers, and even graphic designers seeking help in creating technical illustrations need look no further.

Peter R. Cromwell

Polyhedra have cropped up in many different guises throughout recorded history. In modern times, polyhedra and their symmetries have been cast in a new light by combinatorics and group theory. This book comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics. The author strikes a balance between covering the historical development of the theory surrounding polyhedra, and presenting a rigorous treatment of the mathematics involved. It is attractively illustrated with dozens of diagrams to illustrate ideas that might otherwise prove difficult to grasp. Historians of mathematics, as well as those more interested in the mathematics itself, will find this unique book fascinating.

János Kollár, Karen E. Smith, Alessio Corti

Arising from a summer school course taught by János Kollár, this book develops the modern theory of rational and nearly rational varieties at a level that will particularly suit graduate students. There are numerous examples and exercises, all of which are accompanied by fully worked out solutions, that will make this book ideal as the basis of a graduate course. It will act as a valuable reference for researchers whilst helping graduate students to reach the point where they can begin to tackle contemporary research problems.

Fritz Gesztesy, Helge Holden

The Korteweg-de Vries (KdV) equation, the AKNS equation, the nonlinear Schrödinger equation, the sine-Gordon equation, the Thirring system, and the Camassa-Holm equation are all completely integrable nonlinear partial differential equations that permit special classes of solutions. This book offers a detailed treatment of the class of algebro-geometric solutions and their representations in terms of Riemann theta functions. It provides a rigorous and self-contained presentation at the graduate level. Background material is succinctly presented in appendices and detailed notes to each chapter, and an exhaustive bibliography enhances the main text.

Edited by Silvio Levy

The German mathematician Felix Klein discovered in 1879 that the surface that we now call the Klein quartic has many remarkable properties. Since then, mathematicians have discovered that the same object comes up in different guises in many areas of mathematics, from complex analysis and geometry to number theory. This volume explores the rich tangle of properties and theories surrounding this multiform object. It includes expository and research articles by renowned mathematicians in different fields and a beautifully illustrated essay by the mathematical sculptor Helaman Ferguson. The book closes with the first English translation of Klein's seminal article on this surface.