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K-theory and Noncommutative Geometry
Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging family of mathematical structures and their invariants: topological spaces, rings, algebraic varieties and operator algebras are the dominant examples. The invariants range from characteristic classes in cohomology, determinants of matrices, Chow groups of varieties, as well as traces and indices of elliptic operators. Thus K-theory is notable for its connections with other branches of mathematics. Noncommutative geometry develops tools which allow one to think of noncommutative algebras in the same footing as commutative ones: as algebras of functions on (noncommutative) spaces. The algebras in quest...
Resolution of Singularities
In September 1997, the Working Week on Resolution of Singularities was held at Obergurgl in the Tyrolean Alps. Its objective was to manifest the state of the art in the field and to formulate major questions for future research. The four courses given during this week were written up by the speakers and make up part I of this volume. They are complemented in part II by fifteen selected contributions on specific topics and resolution theories. The volume is intended to provide a broad and accessible introduction to resolution of singularities leading the reader directly to concrete research problems.
Functional Analysis on the Eve of the 21st Century
A four-day conference, "Functional Analysis on the Eve of the Twenty First Century," was held at Rutgers University, New Brunswick, New Jersey, from October 24 to 27, 1993, in honor of the eightieth birthday of Professor Israel Moiseyevich Gelfand. He was born in Krasnye Okna, near Odessa, on September 2, 1913. Israel Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped to shape our understanding of the term "functional analysis" itself, as has the celebrated journal Functional Analysis and Its Applications, which he edited for many years. Functional analysis appeared at the beginning of the ...
$K$-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras
Volume 1 of two - also available in a two volume set.
The Arnoldfest
This volume presents articles originating from invited talks at an exciting international conference held at The Fields Institute in Toronto celebrating the sixtieth birthday of the renowned mathematician, Vladimir Arnold. Experts from the world over--including several from "Arnold's school"--gave illuminating talks and lively poster sessions. The presentations focused on Arnold's main areas of interest: singularity theory, the theory of curves, symmetry groups, dynamical systems, mechanics, and related areas of mathematics. The book begins with notes of three lectures by V. Arnold given in the framework of the Institute's Distinguished Lecturer program. The topics of the lectures are: (1) From Hilbert's Superposition Problem to Dynamical Systems (2) Symplectization, Complexification, and Mathematical Trinities (3) Topological Problems in Wave Propagation Theory and Topological Economy Principle in Algebraic Geometry. Arnold's three articles include insightful comments on Russian and Western mathematics and science. Complementing the first is Jurgen Moser's "Recollections", concerning some of the history of KAM theory.
Introduction to Arakelov Theory
Arakelov introduced a component at infinity in arithmetic considerations, thus giving rise to global theorems similar to those of the theory of surfaces, but in an arithmetic context over the ring of integers of a number field. The book gives an introduction to this theory, including the analogues of the Hodge Index Theorem, the Arakelov adjunction formula, and the Faltings Riemann-Roch theorem. The book is intended for second year graduate students and researchers in the field who want a systematic introduction to the subject. The residue theorem, which forms the basis for the adjunction formula, is proved by a direct method due to Kunz and Waldi. The Faltings Riemann-Roch theorem is proved without assumptions of semistability. An effort has been made to include all necessary details, and as complete references as possible, especially to needed facts of analysis for Green's functions and the Faltings metrics.
Russian Mathematicians In The 20th Century
In the 20th century, many mathematicians in Russia made great contributions to the field of mathematics. This invaluable book, which presents the main achievements of Russian mathematicians in that century, is the first most comprehensive book on Russian mathematicians. It has been produced as a gesture of respect and appreciation for those mathematicians and it will serve as a good reference and an inspiration for future mathematicians. It presents differences in mathematical styles and focuses on Soviet mathematicians who often discussed “what to do” rather than “how to do it”. Thus, the book will be valued beyond historical documentation.The editor, Professor Yakov Sinai, a distin...
Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties
This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. This book is co-published with the Centre de Recherches Mathématiques.
