Werner Ballmann:

Lectures on Kahler Manifolds

ISBN 978-3-03719-025-8
July 2006, 182 pages, softcover, 17.0 cm x 24.0 cm.

These notes are based on lectures the author held at the University of Bonn and the Erwin-Schrodinger-Institute in Vienna. The aim is to give a thorough introduction to the theory of Kahler manifolds with special emphasis on the differential geometric side of Kahler geometry. The exposition starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of Kahler manifolds. The more advanced topics are the cohomology of Kahler manifolds, Calabi conjecture, Gromov's Kahler hyperbolic spaces, and the Kodaira embedding theorem. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture. There are appendices on Chern?Weil theory, symmetric spaces, and L2-cohomology.

Preface

Table of contents

Reto Muller (ETH Zurich):

Differential Harnack Inequalities and the Ricci Flow

ISBN 978-3-03719-030-2
August 2006, 100 pages, softcover, 17.0 cm x 24.0 cm.

The classical Harnack inequalities play an important role in the study of parabolic partial differential equations. The idea of finding a differential version of such a classical Harnack inequality goes back to Peter Li and Shing Tung Yau, who introduced a pointwise gradient estimate for a solution of the linear heat equation on a manifold which leads to a classical Harnack type inequality if being integrated along a path. Their idea has been successfully adopted and generalized to (nonlinear) geometric heat flows such as mean curvature flow or Ricci flow; most of this work was done by Richard Hamilton. In 2002, Grisha Perelman presented a new kind of differential Harnack inequality which involves both the (adjoint) linear heat equation and the Ricci flow. This led to a completely new approach to the Ricci flow that allowed interpretation as a gradient flow which maximizes different entropy functionals. This approach forms the main analytic core of Perelman's attempt to prove the Poincare conjecture. It is, however, of completely independent interest and may as well prove useful in various other areas, such as, for instance, the theory of Kahler manifolds.

The goal of this book is to explain this analytic tool in full detail for the two examples of the linear heat equation and the Ricci flow. It begins with the original Li--Yau result, presents Hamilton's Harnack inequalities for the Ricci flow, and ends with Perelman's entropy formulas and space-time geodesics.

The text is a self-contained, modern introduction to the Ricci flow and the analytic methods to study it. It is primarily addressed to students who have a basic introductory knowledge of analysis and of Riemannian geometry and who are attracted to further study in geometric analysis. No previous knowledge of differential Harnack inequalities or the Ricci flow is required.

Table of contents

Alexander Fedotov - Frederic Klopp

Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrodinger operators

Memoires de la SMF 104 (2006), vi+105 pages

Resume :

Effet tunnel faiblement resonant pour des operateurs de Schrodinger quasi-periodiques adiabatiques
Cet article est consacre a l'etude du spectre d'une famille d'operateurs quasi-periodiques obtenus comme perturbations adiabatiques d'un operateur periodique fixe. Nous montrons que, dans certaines regions d'energies, la perturbation entraine des phenomenes de resonance similaires a ceux observes dans le cas de deux puits. Ces effets s'observent autant sur la geometrie du spectre que sur sa nature. En particulier, on peut observer un entrelacement de types spectraux i.e. une alternance entre du spectre singulier et du spectre absolument continu. Un autre phenomene observe est l'apparition d'ilots de spectre absolument continu dans du spectre singulier dus aux resonances.

Mots clefs : Equations de Schrodinger quasi-periodique, double puis resonnant, spectre purement ponctuel, spectre absolument continu, methode BKW complexe, matrice de monodromie

Abstract:

In this paper, we study spectral properties of the one dimensional periodic Schrodinger operator with an adiabatic quasi-periodic perturbation. We show that in certain energy regions the perturbation leads to resonance effects related to the ones observed in the problem of two resonating quantum wells. These effects affect both the geometry and the nature of the spectrum. In particular, they can lead to the intertwining of sequences of intervals containing absolutely continuous spectrum and intervals containing singular spectrum. Moreover, in regions where all of the spectrum is expected to be singular, these effects typically give rise to exponentially small ``islands'' of absolutely continuous spectrum.

Key words: Quasi-periodic Schrodinger equation, two resonating wells, pure point spectrum, absolutely continuous spectrum, complex WKB method, monodromy matrix

ISBN : 978-2-85629-188-7

Zuming Feng, Melanie Matchett Wood, and Cecil Rousseau, Editors

USA & International Mathematical Olympiads 2005

The Mathematical Olympiad books, covering the USA Mathematical Olympiad (USAMO) and the International Mathematical Olympiad (IMO), have been published annually by the MAA American Mathematics Competitions since 1976. This is the sixth volume in that series published by the MAA in its Problem Book series.

Through the AMC contests and the IMO, young gifted mathematicians are identified and recognized while they are still in secondary school. Participation in these competitions provides them with the chance to measure themselves against other exceptional students from all over the world.

This work was prepared by Zuming Feng, Melanie Wood, the Leader and Deputy Leader of the 2004 USA IMO team, and by Cecil Rousseau, the chair of the USAMO Committee. In addition to presenting their own carefully written solutions to the problems, Zuming and Po-Shen provide remarkable solutions developed by the examination committees, contestants, and experts, during or after the contests. They also provide a detailed report of the 2000?2004 USAMO/IMO results and a comprehensive guide to other materials that emphasize advanced problem-solving. This collection of excellent problems and beautiful solutions is a valuable companion for students who wish to develop their interest in mathematics outside the school curriculum and to deepen their knowledge of mathematics.

ISBN: 10: 0883858231
100 pp., Paperbound, 2006
Series:Problem Books

Compiled and augmented by Harold Reiter

The Contest Problem Book VII
American Mathematics Competitions 1995-2000 Contests

This is the seventh book of problems and solutions from the Mathematics Competitions (published by the MAA.)
Contest Problem Book VIIchronicles 275 problems from the American Mathematics Contests (AMC 12 and AMC 10 for the years 1995 through 2000, including the 50th Anniversary AHSME issued in 1999. Twenty-three additional problems with solutions are included. A Problem Index classifies the 275 problems in to the following subject areas: Algebra, Complex Numbers, Discrete Mathematics (including Counting Problems), Logic, and Discrete Probability, Geometry (including Three Dimensional Geometry), Number Theory (including Divisibility, Representation, and Modular Arithmetic), Statistics, and Trigonometry.

For over 50 years many excellent exams have been prepared by individuals throughout our mathematical community in the hope that all secondary school students will have an opportunity to participate in these problem solving and enriching mathematics experiences. The American Mathematics Contests are intended for everyone from the average student at a typical school who enjoys mathematics to the very best student at the most special school.

ISBN:0-88385-821-5
200 pp., Paperbound, 2006
Series:Anneli Lax New Mathematical Library