Mathematical Notes

Olivier Druet is Researcher at CNRS, Ecole Normale Supérieure de Lyon. Emmanuel Hebey is Professor at Université de Cergy-Pontoise. Frédéric Robert is Associate Professor at Université de Nice Sophia-Antipolis. Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side. A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary. Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields. "This is an important and original work. It develops critical new ideas and methods for the analysis of elliptic PDEs on compact manifolds, especially in the framework of the Yamabe equation, critical Sobolev embedding, and blow-up techniques. This volume will have an important influence on current research."-William Beckner, University of Texas at Austin

Richard M. Weiss is William Walker Professor of Mathematics at Tufts University. He is the author of The Structure of Spherical Buildings (Princeton) and the coauthor (with Jacques Tits) of Moufang Polygons. He received a Humboldt Research Prize in 2004. This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E6, E7, and E8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra. Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms. "[T]here have been many results on finite generalized quadrangles that fill a well-needed gap in the mathematical literature, but this monograph is much deeper and will enable progress to be made in a difficult technical area where some exotic forms of algebraic groups have hitherto been little understood."---Robert L. Devaney, Bulletin of the AMS

Eduardo Cattani is professor of mathematics at the University of Massachusetts, Amherst. Fouad El Zein is a researcher at the Institut de Mathématiques de Jussieu, Université de Paris VII. Phillip A. Griffiths is former director and professor emeritus of mathematics at the Institute for Advanced Study in Princeton. Lê Dũng Tráng is professor emeritus of mathematics at the Aix-Marseille Université. This book provides a comprehensive and up-to-date introduction to Hodge theory-one of the central and most vibrant areas of contemporary mathematics-from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and doesn't require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch-Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures. The contributors include Patrick Brosnan, James Carlson, Eduardo Cattani, François Charles, Mark Andrea de Cataldo, Fouad El Zein, Mark L. Green, Phillip A. Griffiths, Matt Kerr, Lê Dũng Tráng, Luca Migliorini, Jacob P. Murre, Christian Schnell, and Loring W. Tu. "Charles and Schnell's chapter beautifully surveys the theory of absolute Hodge classes, giving in particular a complete proof of Deligne's theorem on absolute Hodge classes on abelian varieties. . . . A welcome addition to the literature and should be useful to both graduate students and researchers working in Hodge theory."---Dan Petersen, MathSciNet

Alfonso Sorrentino is associate professor of mathematics at the University of Rome "Tor Vergata" in Italy. He holds a PhD in mathematics from Princeton University. John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach-known as Aubry-Mather theory-singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, Alfonso Sorrentino first focuses on the core questions the theory aims to answer-notably the destiny of broken invariant KAM tori and the onset of chaos-and describes how it can be viewed as a natural counterpart of KAM theory. He achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. Sorrentino then describes the whole theory and its subsequent developments and applications in their full generality. Shedding new light on John Mather's revolutionary ideas, this book is certain to become a foundational text in the modern study of Hamiltonian systems.

Paula Tretkoff is professor of mathematics at Texas A&M University and director of research in the National Center for Scientific Research (CNRS) at the Université de Lille 1, France. This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. Paula Tretkoff emphasizes those finite covers that are free quotients of the complex two-dimensional ball. Tretkoff also includes background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function. The material in this book began as a set of lecture notes, taken by Tretkoff, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded by Hirzebruch and Tretkoff over a number of years. In this book, Tretkoff has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers. Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry. "A very welcome addition to the literature and is recommended for anyone interested in the theory under discussion."---Daniel Greb, MathSciNet