David Ellwood

Emma Previato

David A. Ellwood

This collection of cutting-edge articles on vector bundles and related topics originated from a CMI workshop, held in October 2006, that brought together a community indebted to the pioneering work of P. E. Newstead, visiting the United States for the first time since the 1960s. Moduli spaces of vector bundles were then in their infancy, but are now, as demonstrated by this volume, a powerful tool in symplectic geometry, number theory, mathematical physics, and algebraic geometry. In fact, the impetus for this volume was to offer a sample of the vital convergence of techniques and fundamental progress, taking place in moduli spaces at the outset of the twenty-first century. This volume contains contributions by J. E. Andersen and N. L. Gammelgaard (Hitchin's projectively flat connection and Toeplitz operators), M. Aprodu and G. Farkas (moduli spaces), D. Arcara and A. Bertram (stability in higher dimension), L. Jeffrey (intersection cohomology), J. Kamnitzer (Langlands program), M. Lieblich (arithmetic aspects), P. E. Newstead (coherent systems), G. Pareschi and M. Popa (linear series on Abelian varieties), and M. Teixidor i Bigas (bundles over reducible curves). These articles do require a working knowledge of algebraic geometry, symplectic geometry and functional analysis, but should appeal to practitioners in a diversity of fields. No specialization should be necessary to appreciate the contributions, or possibly to be stimulated to work in the various directions opened by these path-blazing ideas; to mention a few, the Langlands program, stability criteria for vector bundles over surfaces and threefolds, linear series over abelian varieties and Brauer groups in relation to arithmetic properties of moduli spaces.

Clay Mathematics Institute Workshop on Moduli Spaces of Vector Bundles, with a View Towards Coherent Sheaves, October 6-11, 2006, Cambridge, Massachusetts

Grassmannians, Moduli Spaces and Vector Bundles

Igor Rodnianski

Gigliola Staffilani

Jared Wunsch

This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zürich, Switzerland. The lectures were designed for graduate students and mathematicians within five years of the Ph.D., and the main focus of the program was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave and Schrödinger equations) but also in the high energy or semi-classical limits of elliptic problems. The three main courses focused primarily on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrödinger and wave equations, and evolution problems in general relativity. These major topics were supplemented by several mini-courses reporting on the derivation of effective evolution equations from microscopic quantum dynamics; on wave maps with and without symmetries; on quantum N-body scattering, diffraction of waves, and symmetric spaces; and on nonlinear Schrödinger equations at critical regularity. Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of technical machinery. Moreover, the treatment in this volume emphasizes common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive commutator arguments. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

Evolution Equations

Herwig Hauser

Shigefumi Mori

Josef Schicho

Resolution of Singularities has long been considered as being a difficult to access area of mathematics. The more systematic and simpler proofs that have appeared in the last few years in zero characteristic now give us a much better understanding of singularities. They reveal the aesthetics of both the logical structure of the proof and the various methods used in it. The present volume is intended for readers who are not yet experts but always wondered about the intricacies of resolution. As such, it provides a gentle and quite comprehensive introduction to this amazing field. The book may tempt the reader to enter more deeply into a topic where many mysteries--especially the positive characteristic case--await to be disclosed. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

The Resolution of Singular Algebraic Varieties

The Shock of America is based on the proposition that whenever Europeans of the last 100 years or more contemplated those margins of their experience where change occurred, there, sooner or later, they would find America. How Europeans have come to terms over the decades with this dynamic force in their midst, and what these terms were, is the story at the heart of this text. Masses of Europeans have been enthralled by the real or imaginary prospects coming out of the USA. Important minorities were at times deeply upset by them. Sometime the roles were reversed or shaken up. But nobody could be indifferent for long. Inspiration, provocation, myth, menace, model: all these categories and many more have been deployed to try to cope with the Americans. Attitudes and stereotypes have emerged, intellectual resources have been mobilised, positions and policies developed; all trying to explain and deal with the kind of radiant modernity America built over the course of the twentieth century. David Ellwood combines political, economic, and cultural themes, suggesting that American mass culture has provided the United States with a uniquely effective link between power and influence over time. The book is structured in three parts; a separation based on the proposition that America's influence as an unavoidable force for or against innovation was visible most conspicuously after Europe's three greatest military-political conflicts of the contemporary era: the Great War, World War II, and the Cold War. It concludes with the emotional upsurge in Europe which greeted the arrival of Obama on the world scene, suggesting that in spite of all the disappointments and frictions of the years, the US still retained its privileged place as a source of inspiration for the future across the Western world.

Europe and the Challenge of the Century

The Shock of America

An ambitious, original book describing a century of Europe coping with America: its inventions, personalities, films, armies, business, and politics. These decades reveal how much emotional energy Europeans invested in finding their own ways to reconcile tradition and modernity under the pressure of the ever-evolving American challenge.

David Ellwood

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