IRMA Lectures in Mathematics and Theoretical Physics Vol. 11
ISBN 978-3-03719-029-6
May 2007, 802 pages, hardcover, 17.0 cm x 24.0 cm.
The Teichmuller space of a surface was introduced by O. Teichmuller in the 1930s. It is a basic tool in the study of Riemann's moduli space and of the mapping class group. These objects are fundamental in several fields of mathematics including algebraic geometry, number theory, topology, geometry, and dynamics.
The original setting of Teichmuller theory is complex analysis. The work of Thurston in the 1970s brought techniques of hyperbolic geometry in the study of Teichmuller space and of its asymptotic geometry. Teichmuller spaces are also studied from the point of view of the representation theory of the fundamental group of the surface in a Lie group G, most notably G?=?PSL(2,?) and G?=?PSL(2,?). In the 1980s, there evolved an essentially combinatorial treatment of the Teichmuller and moduli spaces involving techniques and ideas from high-energy physics, namely from string theory. The current research interests include the quantization of Teichmuller space, the Weil?Petersson symplectic and Poisson geometry of this space as well as gauge-theoretic extensions of these structures. The quantization theories can lead to new invariants of hyperbolic 3-manifolds.
The purpose of this handbook is to give a panorama of some of the most important aspects of Teichmuller theory. The handbook should be useful to specialists in the field, to graduate students, and more generally to mathematicians who want to learn about the subject. All the chapters are self-contained and have a pedagogical character. They are written by leading experts in the subject.
Contents
EMS Monographs in Mathematics
ISBN 978-3-03719-036-4
May 2007, 212 pages, hardcover, 16.5 cm x 23.5 cm.
Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of model spaces are the hyperbolic spaces (in the sense of Gromov), for which the asymptotic geometry is nicely encoded in the boundary at infinity.
??In the first part of this book, in analogy with the concepts of classical hyperbolic geometry, the authors provide a systematic account of the basic theory of Gromov hyperbolic spaces. These spaces have been studied extensively in the last twenty years, and have found applications in group theory, geometric topology, Kleinian groups, as well as dynamics and rigidity theory. In the second part of the book, various aspects of the asymptotic geometry of arbitrary metric spaces are considered. It turns out that the boundary at infinity approach is not appropriate in the general case, but dimension theory proves useful for finding interesting results and applications.
??The text leads concisely to some central aspects of the theory. Each chapter concludes with a separate section containing supplementary results and bibliographical notes. Here the theory is also illustrated with numerous examples as well as relations to the neighboring fields of comparison geometry and geometric group theory.
??The book is based on lectures the authors presented at the Steklov Institute in St. Petersburg and the University of Zurich. It addressed to graduate students and researchers working in geometry, topology, and geometric group theory.
Contents
Zurich Lectures in Advanced Mathematics
ISBN 978-3-03719-035-7
September 2007, 388 pages, softcover, 17.0 cm x 24.0 cm.
Quantitative Risk Management (QRM) has become a field of research of considerable importance to numerous areas of application, including insurance, banking, energy, medicine, reliability. Mainly motivated by examples from insurance and finance, the authors develop a theory for handling multivariate extremes. The approach borrows ideas from portfolio theory and aims at an intuitive approach in the spirit of the Peaks over Thresholds method. The point of view is geometric. It leads to a probabilistic description of what in QRM language may be referred to as a high risk scenario: the conditional behaviour of risk factors given that a large move on a linear combination (portfolio, say) has been observed. The theoretical models which describe such conditional extremal behaviour are characterized and their relation to the limit theory for coordinatewise maxima is explained.
The first part is an elegant exposition of coordinatewise extreme value theory; the second half develops the more basic geometric theory. Besides a precise mathematical deduction of the main results, the text yields numerous discussions of a more applied nature. A twenty page preview introduces the key concepts; the extensive introduction provides links to financial mathematics and insurance theory.
The book is based on a graduate course on point processes and extremes. It could form the basis for an advanced course on multivariate extreme value theory or a course on mathematical issues underlying risk. Students in statistics and finance with a mathematical, quantitative background are the prime audience. Actuaries and risk managers involved in data based risk analysis will find the models discussed in the book stimulating. The text contains many indications for further research.
Contents
This book contains the first systematic exposition of the global and local theory of dynamics equivariant with respect to a (compact) Lie group. Aside from general genericity and normal form theorems on equivariant bifurcation, it describes many general families of examples of equivariant bifurcation and includes a number of novel geometric techniques, in particular, equivariant transversality. This important book forms a theoretical basis of future work on equivariant reversible and Hamiltonian systems.
This book also provides a general and comprehensive introduction to codimension one equivariant bifurcation theory. In particular, it includes the bifurcation theory developed with Roger Richardson on subgroups of reflection groups and the Maximal Isotropy Subgroup Conjecture. A number of general results are also given on the global theory. Introductory material on groups, representations and G-manifolds are covered in the first three chapters of the book. In addition, a self-contained introduction of equivariant transversality is given, including necessary results on stratifications as well as results on equivariant jet transversality developed by Edward Bierstone.
Contents:
Groups
Group Actions and Representations
Smooth G-manifolds
Equivariant Bifurcation Theory: Steady State Bifurcation
Equivariant Bifurcation Theory: Dynamics
Equivariant Transversality
Applications of G-transversality to Bifurcation Theory I
Equivariant Dynamics
Dynamical Systems on G-manifolds
Applications of G-transversality to Bifurcation Theory II
Readership: Academics and graduate students in pure and applied mathematics.
492pp Pub. date: Sept 2007
ISBN 978-1-86094-828-2
1-86094-828-6
ISBN: 9781584888680
ISBN-10: 1584888687
Publisher: Chapman and Hall/CR
Publication Date: 07/20/2007
Pages: 298
Trim Size: 6-1/8 x 9-1/4
Binding(s): Hardback
About the Title
Since the first athletic events found a fan base, sports and statistics have always maintained a tight and at times mythical relationship. As a way to relay the telling of a game's drama and attest to the prodigious powers of the heroes involved, those reporting on the games tallied up the numbers that they believe best described the action and best defined the winning edge. However, they may not have always counted the right numbers. Many of our hallowed beliefs about sports statistics have long been fraught with misnomers. Whether it concerns Scottish football or American baseball, the most revered statistics often have little to do with any winning edge.
Covering an international collection of sports, Statistical Thinking in Sports provides an accessible survey of current research in statistics and sports, written by experts from a variety of arenas. Rather than rely on casual observation, they apply the rigorous tools of statistics to re-examine many of those concepts that have gone from belief to fact, based mostly on the repetition of their claims. Leaving assumption behind, these researchers take on a host of tough questions-
Is a tennis player only as good as his or her first serve?
Is there such a thing as home field advantage?
Do concerns over a decline in soccer's competitive balance have any merit?
What of momentum-is its staying power any greater than yesterday's win?
And what of pressure performers? Are there such creatures or ultimately, does every performer fall back to his or her established normative?
Investigating a wide range of international team and individual sports, the book considers the ability to make predictions, define trends, and measure any number of influences. It is full of interesting and useful examples for those teaching introductory statistics. Although the articles are aimed at general readers, the serious researcher in sports statistics will also find the articles of value and highly useful as starting points for further research.