Rekha R. Thomas, University of Washington, Seattle, WA

Lectures in Geometric Combinatorics

Student Mathematical Library, Volume: 33
2006; 143 pp; softcover

ISBN-10: 0-8218-4140-8
ISBN-13: 978-0-8218-4140-2
Expected publication date is July 29, 2006.

This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the state polytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Grobner bases of toric ideals and other methods from commutative algebra.

The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes.

Table of Contents

Edited by: Vestislav Apostolov, Universite du Quebec a Montreal, QC, Canada, Andrew Dancer and Nigel Hitchin, Mathematical Institute, Oxford, England, and McKenzie Wang, McMaster University, Hamilton, ON, Canada

Perspectives in Riemannian Geometry

CRM Proceedings & Lecture Notes, Volume: 40
2006; 248 pp; softcover
ISBN-10: 0-8218-3852-0
ISBN-13: 978-0-8218-3852-5
Expected publication date is July 15, 2006.

Special geometries as well as the relation between curvature and topology have always been of interest to differential geometers. More recently, these topics have turned out to be of use in physical problems related to string theory as well. This volume provides a unique and thorough survey on the latest developments on Riemannian geometry, special geometrical structures on manifolds, and their interactions with other fields such as mathematical physics, complex analysis, and algebraic geometry.

This volume presents ten papers written by participants of the "Short Program on Riemannian Geometry," a workshop held at the CRM in Montreal in 2004. It will be a valuable reference for graduate students and research mathematicians alike.

Readership

Graduate students and research mathematicians interested in differential geometry.

Table of Contents

M. T. Anderson -- Topics in comformally compact Einstein metrics
O. Biquard -- Cauchy-Riemann 3-manifolds and Einstein fillings
C. P. Boyer and K. Galicki -- Sasakian geometry and Einstein metrics on spheres
R. L. Bryant -- Second order families of special Lagrangian 3-folds
A. Dancer and M. Y. Wang -- Einstein equations, superpotentials and convex polytopes
L. David and P. Gauduchon -- The Bochner-flat geometry of weighted projective spaces
K. Grove -- Aspects of comparison geometry
N. Hitchin -- Low-dimensional geometry--A variational approach
C. LeBrun -- Twistors, holomorphic disks, and Riemann surfaces with boundary
A. Nabutovsky -- Combinatorics of the spaces of Riemannian structures and logic phenomena of Euclidean quantum gravity

Kiyosi Ito, Kyoto University, Japan

Essentials of Stochastic Processes

Translations of Mathematical Monographs, Volume: 231
2006; 171 pp; hardcover
ISBN-10: 0-8218-3898-9
ISBN-13: 978-0-8218-3898-3
Expected publication date is July 9, 2006.

This book is an English translation of Kiyosi Ito's monograph published in Japanese in 1957. It gives a unified and comprehensive account of additive processes (or Levy processes), stationary processes, and Markov processes, which constitute the three most important classes of stochastic processes. Written by one of the leading experts in the field, this volume presents to the reader lucid explanations of the fundamental concepts and basic results in each of these three major areas of the theory of stochastic processes.

With the requirements limited to an introductory graduate course on analysis (especially measure theory) and basic probability theory, this book is an excellent text for any graduate course on stochastic processes.

Kiyosi Ito is famous throughout the world for his work on stochastic integrals (including the Ito formula), but he has made substantial contributions to other areas of probability theory as well, such as additive processes, stationary processes, and Markov processes (especially diffusion processes), which are topics covered in this book. For his contributions and achievements, he has received, among others, the Wolf Prize, the Japan Academy Prize, and the Kyoto Prize.

Readership

Graduate students and research mathematicians interested in stochastic processes.

Table of Contents

Edited by: David A. Ellwood, Clay Mathematics Institute, Cambridge, MA, Peter S. Ozsvath, Columbia University, New York, NY, Andras I. Stipsicz, Renyi Institute, Budapest, Hungary, and Zoltan Szabo, Princeton University, NJ

Floer Homology, Gauge Theory, and Low-Dimensional Topology

Clay Mathematics Proceedings, Volume: 5
2006; approx. 280 pp; softcover
ISBN-10: 0-8218-3845-8
ISBN-13: 978-0-8218-3845-7
Expected publication date is August 4, 2006.

Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these equations have come from mathematical physics, and play an important role in the description of the electro-weak and strong nuclear forces. The use of gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980s, and was revolutionized by the introduction of the Seiberg-Witten equations in the mid-1990s. Since the birth of the subject, it has retained its close connection with symplectic topology. The analogy between these two fields of study was further underscored by Andreas Floer's construction of an infinite-dimensional variant of Morse theory that applies in two a priori different contexts: either to define symplectic invariants for pairs of Lagrangian submanifolds of a symplectic manifold, or to define topological invariants for three-manifolds, which fit into a framework for calculating invariants for smooth four-manifolds. "Heegaard Floer homology", the recently-discovered invariant for three- and four-manifolds, comes from an application of Lagrangian Floer homology to spaces associated to Heegaard diagrams. Although this theory is conjecturally isomorphic to Seiberg-Witten theory, it is more topological and combinatorial in flavor and thus easier to work with in certain contexts. The interaction between gauge theory, low-dimensional topology, and symplectic geometry has led to a number of striking new developments in these fields. The aim of this volume is to introduce graduate students and researchers in other fields to some of these exciting developments, with a special emphasis on the very fruitful interplay between disciplines.

This volume is based on lecture courses and advanced seminars given at the 2004 Clay Mathematics Institute Summer School at the Alfred Renyi Institute of Mathematics in Budapest, Hungary. Several of the authors have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art introduction to current research, covering material from Heegaard Floer homology, contact geometry, smooth four-manifold topology, and symplectic four-manifolds.

Table of Contents

Heegaard Floer homology and knot theory

P. S. Ozsvath and Z. Szabo -- An introduction to Heegaard Floer homology
P. S. Ozsvath and Z. Szabo -- Lectures on Heegaard Floer homology
H. Goda -- Circle valued Morse theory for knots and links

Floer homologies and contact structures

J. B. Etnyre -- Lectures on open book decompositions and contact structures
A. I. Stipsicz -- Contact surgery and Heegaard Floer theory
P. Lisca and A. I. Stipsicz -- Ozsvath-Szabo invariants and contact surgery
T. Ekholm -- Double points of exact Lagrangian immersions and Legendrian contact homology

Symplectic 4-manifolds and Seiberg-Witten invariants

R. Fintushel -- Knot surgery revisited
R. J. Stern -- Will we ever classify simply-connected smooth 4-manifolds?
J. Park -- A note on symplectic 4-manifolds with $b_{2}^+ =1$ and $K^2 \geq 0$
T.-J. Li -- The Kodaira dimension of symplectic 4-manifolds
D. Auroux -- Symplectic 4-manifolds, singular plane curves, and isotopy problems
I. Smith -- Monodromy, vanishing cycles, knots and the adjoint quotient