Books

The Clay Mathematics Institute publishes two book series with the American Mathematical Society: the Clay Mathematics Monographs and the Clay Mathematics Proceedings. CMI has also produced several videos.

Ricci Flow and the Poincaré Conjecture
Authors: John Morgan and Gang Tian

For over 100 years the Poincaré Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the seven Clay Millennium Problems in Mathematics. In 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincaré Conjecture in the affirmative.

Analytic Number Theory: A Tribute to Gauss and Dirichlet
Editors: William Duke and Yuri Tschinkel

Articles in this volume are based on talks given at the Gauss-Dirichlet Conference held in Göttingen on June 20-24, 2005. The conference commemorated the 150th anniversary of the death of C.-F. Gauss and the 200th anniversary of the birth of J.-L. Dirichlet.

Surveys in Noncommutative Geometry
Editors: Nigel Higson and John Roe

In June 2000, the Clay Mathematics Institute organized an Instructional Symposium on Noncommutative Geometry in conjunction with the AMS-IMS-SIAM Joint Summer Research Conference. These events were held at Mount Holyoke College in Massachusetts from June 18 to 29, 2000. The Instructional Symposium consisted of several series of expository lectures which were intended to introduce key topics in noncommutative geometry to mathematicians unfamiliar with the subject. Those expository lectures have been edited and are reproduced in this volume.

Floer Homology, Gauge Theory, and Low Dimensional Topology
Editors: David Ellwood, Peter Ozsváth, András Stipsicz and Zoltán Szabó

This volume is based on lecture courses and advanced seminars given at the 2004 Clay Mathematics Institute at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary. Several of the authors have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art introduction to current research, covering material from Heegaard Floer homology, contact geometry, smooth four—manifold topology, and symplectic four—manifolds.

Lecture notes on Motivic Cohomology
Authors: Carlo Mazza, Vladimir Voevodsky and Charles Weibel

This book provides an account of the triangulated theory of motives. Its purpose is to introduce Motivic Cohomology, develop its main properties and, finally, to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, étale cohomology and Chow groups.

The Millennium Prize Problems
Editors: James Carlson, Arthur Jaffe and Andrew Wiles

On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert delivered the famous lecture in which he described twenty-three problems that were to play an influential role in future mathematical research. A century later, on May 24, 2000, at a meeting at the Collge de France, the Clay Mathematics Institute announced the creation of a US$7 million prize fund for the solution of seven important classic problems that have resisted solution. The prize fund is divided equally among the seven problems. There is no time limit for their solution.

Harmonic Analysis, The Trace Formula, And Shimura Varieties
Editors: James Arthur, David Ellwood and Robert Kottwitz

The modern theory of automorphic forms, embodied in what has come to be known as the Langlands program, is an extraordinary unifying force in mathematics. It proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations. These "reciprocity laws", conjectured by Langlands, are still largely unproved. However, their capacity to unite large areas of mathematics insures that they will be a central area of study for years to come.The goal of this volume is to provide an entry point into this exciting and challenging field.

Global Theory of Minimal Surfaces
Editor: David Hoffman

In the Summer of 2001, the Mathematical Sciences Research Institute (MSRI) hosted the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces. During that time, MSRI became the world center for the study of minimal surfaces: 150 mathematicians--undergraduates, post-doctoral students, young researchers, and world experts--participated in the most extensive meeting ever held on the subject in its 250-year history. The unusual nature of the meeting made it possible to put together this collection of expository lectures and specialized reports, giving a panoramic view of a vital subject presented by leading researchers in the field.

Strings and Geometry
Editors: Michael Douglas, Jerome Gauntlett and Mark Gross

This volume is the proceedings of the 2002 Clay Mathematics Institute School on Geometry and String Theory. This month-long program was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England, and was organized by both mathematicians and physicists: A. Corti, R. Dijkgraaf, M. Douglas, J. Gauntlett, M. Gross, C. Hull, A. Jaffe and M. Reid. The early part of the school had many lectures that introduced various concepts of algebraic geometry and string theory with a focus on improving communication between these two fields. During the latter part of the program there were also a number of research level talks.

Mirror Symmetry
Editors: Cumrun Vafa and Eric Zaslow

This book is a product of a month-long school on mirror symmetry that CMI held at Pine Manor College in Brookline, Massachusetts in the Spring of 2000. The aim of the book is to provide a pedagogical introduction to the field of mirror symmetry from both a mathematical and physical perspective. After covering the relevant background material, the main part of the monograph is devoted to the proof of mirror symmetry from various viewpoints. More advanced topics are also discussed. In particular, topological strings at higher genera and the notion of holomorphic anomaly.

Strings 2001
Editors: Atish Dabholkar, Sunil Mukhi and Spenta Wadia

String theory, sometimes called the "Theory of Everything", has the potential to provide answers to key questions involving quantum gravity, black holes, supersymmetry, cosmology, singularities and the symmetries of nature. This multi-authored book summarizes the latest results across all areas of string theory from the perspective of world-renowned experts, including Michael Green, David Gross, Stephen Hawking, John Schwarz, Edward Witten and others.