University Lecture Series, Volume: 52
2010; 125 pp; softcover
ISBN-13: 978-0-8218-4964-4
Expected publication date is January 30, 2010.
The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well.
This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.
Graduate students and research mathematicians interested in algebraic geometry and Hodge theory.
Basic definitions
Basic results
Syzygy schemes
The conjectures of Green and Green-Lazarsfeld
Koszul cohomology and the Hilbert scheme
Koszul cohomology of a $K3$ surface
Specific versions of the syzygy conjectures
Applications
Bibliography
Index
2010; 348 pp; hardcover
ISBN-13: 978-0-8218-4368-0
Expected publication date is February 11, 2010.
Jacques Hadamard, among the greatest mathematicians of the twentieth century, made signal contributions to a number of fields. But his mind could not be confined to the upper reaches of mathematical thought. He also produced a massive two-volume work, on plane and solid geometry, for pre-college teachers in the French school system.
In those books, Hadamard's style invites participation. His exposition is minimal, providing only the results necessary to support the solution of the many elegant problems he poses afterwards. That is, the problems interpret the text in the way that harmony interprets melody in a well-composed piece of music.
The present volume offers solutions to the problems in the first part of Hadamard's work (Lessons in Geometry. I. Plane Geometry, Jacques Hadamard, Amer. Math. Soc. (2008)), and can be viewed as a reader's companion to that book. It requires of the reader only the background of high school plane geometry, which Lessons in Geometry provides. The solutions strive to connect the general methods given in the text with intuitions that are natural to the subject, giving as much motivation as possible as well as rigorous and formal solutions. Ideas for further exploration are often suggested, as well as hints for classroom use.
This book will be of interest to high school teachers, gifted high school students, college students, and those mathematics majors interested in geometry.
High school students and teachers, undergraduate and graduate students interested in geometry.
Solutions and comments for problems in book I
Solutions and comments for problems in book II
Solutions and comments for problems in book III
Solutions and comments for complements to book III
Solutions and comments for problems in book IV
Graduate Studies in Mathematics, Volume: 111
2010; 176 pp; hardcover
ISBN-13: 978-0-8218-4938-5
Expected publication date is February 28, 2010.
In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture. Furthermore, various convergence theorems have been established.
This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold whose sectional curvatures all lie in the interval (1,4] is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H. E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen.
This text originated from graduate courses given at ETH Zurich and Stanford University, and is directed at graduate students and researchers. The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required.
Graduate students and research mathematicians interested in differential geometry and topology of manifolds.
A survey of sphere theorems in geometry
Hamilton's Ricci flow
Interior estimates
Ricci flow on $S^2$
Pointwise curvature estimates
Curvature pinching in dimension 3
Preserved curvature conditions in higher dimensions
Convergence results in higher dimensions
Rigidity results
Convergence of evolving metrics
Results from complex linear algebra
Problems
Bibliography
Index
Graduate Studies in Mathematics, Volume: 112
2010; approx. 408 pp; hardcover
ISBN-13: 978-0-8218-4904-0
Expected publication date is February 27,
Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences, and many other disciplines.
This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the Karush-Kuhn-Tucker theory of nonlinear programming in Banach spaces.
The exposition begins with control problems with linear equation, quadratic cost function and control constraints. To make the book self-contained, basic facts on weak solutions of elliptic and parabolic equations are introduced. Principles of functional analysis are introduced and explained as they are needed. Many simple examples illustrate the theory and its hidden difficulties. This start to the book makes it fairly self-contained and suitable for advanced undergraduates or beginning graduate students.
Advanced control problems for nonlinear partial differential equations are also discussed. As prerequisites, results on boundedness and continuity of solutions to semilinear elliptic and parabolic equations are addressed. These topics are not yet readily available in books on PDEs, making the exposition also interesting for researchers.
Alongside the main theme of the analysis of problems of optimal control, Troltzsch also discusses numerical techniques. The exposition is confined to brief introductions into the basic ideas in order to give the reader an impression of how the theory can be realized numerically. After reading this book, the reader will be familiar with the main principles of the numerical analysis of PDE-constrained optimization.
Graduate students and research mathematicians interested in optimal control theory and PDEs.
Introduction and examples
Linear-quadratic elliptic control problems
Linear-quadratic parabolic control problems
Optimal control of semilinear elliptic equations
Optimal control of semilienar parabolic equations
Optimization problems in Banach spaces
Supplementary results on partial differential equations
Bibliography
Index
Contemporary Mathematics, Volume: 505
2010; 249 pp; softcover
ISBN-13: 978-0-8218-4770-1
Expected publication date is February 13, 2010.
This volume contains the Proceedings of the 8th International Conference on Harmonic Analysis and Partial Differential Equations, held in El Escorial, Madrid, Spain, on June 16-20, 2008.
Featured in this book are papers by Steve Hoffmann and Carlos Kenig, which are based on two mini-courses given at the conference. These papers present topics of current interest, which assume minimal background from the reader, and represent state-of-the-art research in a useful way for young researchers. Other papers in this volume cover a range of fields in Harmonic Analysis and Partial Differential Equations and, in particular, illustrate well the fruitful interplay between these two fields.
Graduate students and research mathematicians interested in harmonic analysis and partial differential equations.
A. Cohen, W. Dahmen, and R. DeVore -- Instance optimal decoding by thresholding in compressed sensing
S. Hofmann -- Local $T(b)$ theorems and applications in PDE
C. E. Kenig -- The global behavier of solutions to critical nonlinear dispersive and wave equations
P. Auscher and J. M. Martell -- Weighted norm inequalities, off-diagonal estimates and elliptic operators
J. Bennett -- Heat-flow monotonicity related to some inequalities in euclidean analysis
A. Carbery -- A uniform sublevel set estimate
P. Auscher, A. Axelsson, and A. McIntosh -- On a quadratic estimate related to the Kato conjecture and boundary value problems
C. Muscalu -- Flag paraproducts
J. Ortega-Cerda and B. Pridhnani -- The Polya-Tchebotarov problem
M. T. Lacey, S. Petermichl, J. C. Pipher, and B. D. Wick -- Iterated Riesz commutators: a simple proof of boundedness
G. Garrigos and A. Seeger -- A mixed norm variant of Wolff's inequality for paraboloids
S. Thangavelu -- On the unreasonable effectiveness of Gutzmer's formula
L. Vega -- Bilinear virial identities and oscillatory integrals
E. Hernandez, H. ?iki?, G. Weiss, and E. Wilson -- On the properties of the integer translates of a square integrable function