Contemporary Mathematics, Volume: 492
2009; approx. 129 pp; softcover
ISBN-13: 978-0-8218-4663-6
Expected publication date is August 30, 2009.
Developmental biology and tumour growth are two important areas of current research where mathematics increasingly provides powerful new techniques and insights. The unfolding complexity of living structures from egg to embryo gives rise to a number of difficult quantitative problems that are ripe for mathematical models and analysis. Understanding this early development process involves the study of pattern formation, which mathematicians view through the lens of dynamical systems. This book addresses several issues in developmental biology, including Notch signalling pathway integration and mesenchymal compartment formation.
Tumour growth is one of the primary challenges of cancer research. Its study requires interdisciplinary approaches involving the close collaboration of mathematicians, biologists and physicians. The summer school addressed angiogenesis, modelling issues arising in radiotherapy, and tumour growth viewed from the individual cell and the relation to a multiphase-fluid flow picture of that process.
This book is suitable for researchers, graduate students, and advanced undergraduates interested in mathematical methods of developmental biology or tumour growth.
This book is copublished by the Real Sociedad Matematica Espanola and the American Mathematical Society.
Advanced undergraduate students, graduate students, and research mathematicians intersted in the mathematical methods of developmental biology.
*B. Alsina, A. L. Garcia de Lomana, J. Villa-Freixa, and F. Giraldez --
Developmental biology and mathematics: the rules of an embryo
*C. G. Arques and M. Torres -- From lineage to shape: modeling dorsal-ventral
specification in the developing mouse limb
*R. Fior and D. Henrique -- Notch-mathics
*T. Alarcon -- Modelling tumour-induced angiogenesis: a review of individual-based
models and multiscale approaches
*A. Cappuccio, M. A. Herrero, and L. Nunez -- Tumour radiotherapy and its
mathematical modelling
*J. Galle and L. Preziosi -- Multiphase and individual cell-based models
of tumour growth
2009; 324 pp; softcover
ISBN-13: 978-0-8218-4814-2
Expected publication date is September 27, 2009.
This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. The selected problems do not require advanced mathematics, making this book accessible to a variety of readers.
Mathematical recreations offer a rich playground for both amateur and professional mathematicians. Believing that creative stimuli and aesthetic considerations are closely related, great mathematicians from ancient times to the present have always taken an interest in puzzles and diversions. The goal of this book is to show that famous mathematicians have all communicated brilliant ideas, methodological approaches, and absolute genius in mathematical thoughts by using recreational mathematics as a framework. Concise biographies of many mathematicians mentioned in the text are also included.
The majority of the mathematical problems presented in this book originated in number theory, graph theory, optimization, and probability. Others are based on combinatorial and chess problems, while still others are geometrical and arithmetical puzzles.
This book is intended to be both entertaining as well as an introduction to various intriguing mathematical topics and ideas. Certainly, many stories and famous puzzles can be very useful to prepare classroom lectures, to inspire and amuse students, and to instill affection for mathematics.
General readers and undergraduate students interested in puzzles and recreational mathematics.
*Recreational mathematics
*Arithmetics
*Number theory
*Geometry
*Tiling and packing
*Physics
*Combinatorics
*Probability
*Graphs
*Chess
*Miscellany
*Method of continued fractions for solving Pell's equations
*Geometrical inversion
*Some basic facts from graph theory
*linear difference equations with constant coefficients
*Biographies
*Bibliography
*Name index
Clay Mathematics Monographs, Volume: 4
2009; approx. 684 pp; hardcover
ISBN-13: 978-0-8218-3848-8
Expected publication date is September 17,
Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In 1985, the introduction of Calabi-Yau manifolds into physics as a way to compactify ten-dimensional space-time has led to exciting cross-fertilization between physics and mathematics, especially with the discovery of mirror symmetry in 1989. A new string revolution in the mid-1990s brought the notion of branes to the forefront. As foreseen by Kontsevich, these turned out to have mathematical counterparts in the derived category of coherent sheaves on an algebraic variety and the Fukaya category of a symplectic manifold.
This has led to exciting new work, including the Strominger-Yau-Zaslow conjecture, which used the theory of branes to propose a geometric basis for mirror symmetry, the theory of stability conditions on triangulated categories, and a physical basis for the McKay correspondence. These developments have led to a great deal of new mathematical work.
One difficulty in understanding all aspects of this work is that it requires being able to speak two different languages, the language of string theory and the language of algebraic geometry. The 2002 Clay School on Geometry and String Theory set out to bridge this gap, and this monograph builds on the expository lectures given there to provide an up-to-date discussion including subsequent developments. A natural sequel to the first Clay monograph on Mirror Symmetry, it presents the new ideas coming out of the interactions of string theory and algebraic geometry in a coherent logical context. We hope it will allow students and researchers who are familiar with the language of one of the two fields to gain acquaintance with the language of the other.
The book first introduces the notion of Dirichlet brane in the context of topological quantum field theories, and then reviews the basics of string theory. After showing how notions of branes arose in string theory, it turns to an introduction to the algebraic geometry, sheaf theory, and homological algebra needed to define and work with derived categories. The physical existence conditions for branes are then discussed and compared in the context of mirror symmetry, culminating in Bridgeland's definition of stability structures, and its applications to the McKay correspondence and quantum geometry. The book continues with detailed treatments of the Strominger-Yau-Zaslow conjecture, Calabi-Yau metrics and homological mirror symmetry, and discusses more recent physical developments.
This book is suitable for graduate students and researchers with either a physics or mathematics background, who are interested in the interface between string theory and algebraic geometry.
Graduate students and research mathematicians interested in mathematical aspects of quantum field theory, in particular string theory and mirror symmetry.
*Overview and physical background
*D-branes and K-theory in 2D topological field theory
*Open strings and Dirichlet branes
*Representation theory, homological algebra and geometry
*Dirichlet branes and stability conditions
*The Strominger-Yau-Zaslow picture of mirror symmetry
*Metric aspects of Calabi-Yau manifolds
*The mathematics of homological mirror symmetry
*Bibliography
*Index
University Lecture Series, Volume: 49
2009; approx. 257 pp; softcover
ISBN-13: 978-0-8218-4756-5
Expected publication date is September 26, 2009.
Mathematics has been called the science of order. The subject is remarkably
good for generalizing specific cases to create abstract theories. However,
mathematics has little to say when faced with highly complex systems, where
disorder reigns. This disorder can be found in pure mathematical arenas,
such as the distribution of primes, the 3n+1 conjecture, and class field
theory.
The purpose of this book is to provide examples--and rigorous proofs--of the complexity law:
(1) discrete systems are either simple or they exhibit advanced pseudorandomness;
(2) a priori probabilities often exist even when there is no intrinsic symmetry.
Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics.
This book considers randomness and complexity. The traditional approach to complexity--computational complexity theory--is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity.
The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly self-contained.
Graduate students and research mathematicians interested in discrete mathematics, combinatorics, and number theory.
Reading the shadows on the wall and formulating a vague conjecture *Complex
systems
*Collecting data: Apparent randomness of digit sequences
*Collecting data: More randomness in number theory
*Laplace and the principle of insufficient reason
*Collecting proofs for the SLG conjecture
More evidence for the SLG conjecture: Exact solutions in real game theory
*Ramsey theory and games
*Pratice session (I): More on Ramsey games and strategies
*Practice session (II): Connectivity games and more strategies
*What kind of games*
*Exact solutions of games: Understanding via the equiprobability postulate
*Equiprobability postulate with constraints (endgame policy)
*Constraints and threshold clustering
*Threshold clustering and a few bold conjectures
New evidence: Games and graphs, the surplus, and the square root law *Yet
another simplification: Sparse hypergraphs and the surplus
*Is surplus the right concept* (I)
*Is surplus the right concept* (II)
*Working with a game-theoretic partition function
*An attempt to save the variance
*Proof of theorem 1: Combining the variance with an exponential sum
*Proof of theoem 2: The upper bound
*Conclusion (I): More on theorem 1
*Conclusion (II): Beyond the SLG conjecture
*Dictionary of phrases and concepts
*References
Student Mathematical Library, Volume: 51
2009; approx. 315 pp; softcover
ISBN-13: 978-0-8218-2138-1
Expected publication date is September 13, 2009.
This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a careful introduction to the theory of numerical methods with a novel approach to the analysis of errors and stability of various numerical solution algorithms. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject.
Undergraduate and graduate students interested in ordinary differential equations and numerical methods.
*Introduction
*Differential equations and their solutions
*Linear differential equations
*Second-order ODE and the calculus of variations
*Newtonian mechanics
*Numerical methods
*Linear algebra and analysis
*The magic of iteration
*Vector fields as differential operators
*Coordinate systems and canonical forms
*Parametrized curves and arclength
*Smoothness with respect to initial conditions
*Canonical form for linear operators
*Runge-Kutta Methods
*Multistep methods
*Iterative interpolation and its error
*Bibliography
*Index
Graduate Studies in Mathematics, Volume: 106
2009; 474 pp; hardcover
ISBN-13: 978-0-8218-3697-2
Expected publication date is September 11, 2009.
A topological embedding is a homeomorphism of one space onto a subspace
of another. The book analyzes how and when objects like polyhedra or manifolds
embed in a given higher-dimensional manifold. The main problem is to determine
when two topological embeddings of the same object are equivalent in the
sense of differing only by a homeomorphism of the ambient manifold. Knot
theory is the special case of spheres smoothly embedded in spheres; in
this book, much more general spaces and much more general embeddings are
considered. A key aspect of the main problem is taming: when is a topological
embedding of a polyhedron equivalent to a piecewise linear embedding* A
central theme of the book is the fundamental role played by local homotopy
properties of the complement in answering this taming question.
The book begins with a fresh description of the various classic examples of wild embeddings (i.e., embeddings inequivalent to piecewise linear embeddings). Engulfing, the fundamental tool of the subject, is developed next. After that, the study of embeddings is organized by codimension (the difference between the ambient dimension and the dimension of the embedded space). In all codimensions greater than two, topological embeddings of compacta are approximated by nicer embeddings, nice embeddings of polyhedra are tamed, topological embeddings of polyhedra are approximated by piecewise linear embeddings, and piecewise linear embeddings are locally unknotted. Complete details of the codimension-three proofs, including the requisite piecewise linear tools, are provided. The treatment of codimension-two embeddings includes a self-contained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimension-one embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties.
Graduate students and research mathematicians interested in geometric topology.
*Prequel
*Tame and knotted embeddings
*Wild and flat embeddings
*Engulfing, cellularity, and embedding dimension
*Trivial-range embeddings
*Codimension-three embeddings
*Codimension-two embeddings
*Codimension-one embeddings
*Codimension-zero embeddings
*Bibliography
*Selected symbols and abbreviations
*Index