Fields Institute Communications, Volume: 60
2011; 288 pp; hardcover
ISBN-13: 978-0-8218-5226-2
Expected publication date is March 31, 2011.
This volume is a collection of papers on number theory which evolved out of the workshop WIN--Women In Numbers, held November 2-7, 2008, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. It includes articles showcasing outcomes from collaborative research initiated during the workshop as well as survey papers aimed at introducing graduate students and recent PhDs to important research topics in number theory.
The contributions in this volume span a wide range of topics in arithmetic geometry and algebraic, algorithmic, and analytic number theory. Clusters of papers center around the four topics of moduli spaces and Shimura curves, curves and Jacobians over finite fields, Galois covers of function fields in positive characteristic, and zeta functions of graphs, with a fifth group of three individual articles on modular forms, Iwasawa theory, and Galois representations, respectively.
The workshop and this volume are part of a broader WIN initiative, whose goals are to highlight and increase the research activities of women in number theory and to train female graduate students in number theory and related fields.
Graduate students and research mathematicians interested in number theory.
Part I. Moduli spaces and Shimura curves
H. Grundman -- Hilbert modular variety compositions
P. Bayer -- Contributions to Shimura curves
H. Grundman, J. Johnson-Leung, K. Lauter, A. Salerno, B. Viray, and E. Wittenborn -- Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory
E. Mantovan -- l-adic etale cohomology of PEL type Shimura varieties with non-trivial coefficients
Part II. Curves and Jacobians over finite fields
J. Balakrishnan, J. Belding, S. Chisholm, K. Eisentrager, K. E. Stange, and E. Teske -- Pairings on hyperelliptic curves
A. Bucur, C. David, B. Feigon, and M. Lalin -- Biased statistics for traces of cyclic p-fold covers over finite fields
L. Berger, J. L. Hoelscher, Y. Lee, J. Paulhus, and R. Scheidler -- The
ell-rank structure of a global function field
Part III. Galois covers of function fields in positive characteristic
R. Pries and K. Stevenson -- A survey of Galois theory of curves in characteristic p
I. I. Bouw -- Covers of the affine line in positive characteristic with prescribed ramification
L. Gruendken, L. Hall-Seelig, B.-H. Im, E. Ozman, R. Pries, and K. Stevenson -- Semi-direct Galois covers of the affine line
Part IV. Zeta functions of graphs
A. Terras -- Looking into a graph theory mirror of number theoretic zetas
W.-C. W. Li -- Zeta functions of group based graphs and complexes
B. Malmskog and M. Manes -- Ramified covers of graphs and the Ihara zeta functions of certain ramified covers
Part V. Other topics
S. A. Garthwaite, L. Long, H. Swisher, and S. Treneer -- Zeros of classical Eisenstein series and recent developments
S. Ramdorai -- On the mu-invariant in Iwasawa theory
S. Arias-de-Reyna and N. Vila -- Galois representations and the tame inverse Galois problem
Mathematical Surveys and Monographs, Volume: 170
2011; 251 pp; hardcover
ISBN-13: 978-0-8218-5288-0
Expected publication date is March 26, 2011.
Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorious difficulties of renormalization have made quantum field theory very inaccessible for mathematicians. This book provides complete mathematical foundations for the theory of perturbative quantum field theory, based on Wilson's ideas of low-energy effective field theory and on the Batalin-Vilkovisky formalism. As an example, a cohomological proof of perturbative renormalizability of Yang-Mills theory is presented.
An effort has been made to make the book accessible to mathematicians who have had no prior exposure to quantum field theory. Graduate students who have taken classes in basic functional analysis and homological algebra should be able to read this book.
Graduate students and research mathematicians interested in quantum field theory and mathematical physics.
Introduction
Theories, Lagrangians and counterterms
Field theories on mathbb{R}^n
Renormalizability
Gauge symmetry and the Batalin-Vilkovisky formalism
Renormalizability of Yang-Mills theory
Asymptotics of graph integrals
Nuclear spaces
Bibliography
Mathematical Surveys and Monographs, Volume: 172
2011; 347 pp; hardcover
ISBN-13: 978-0-8218-5336-8
Expected publication date is March 30, 2011.
The book provides an outline and modern overview of the classification of the finite simple groups. It primarily covers the "even case", where the main groups arising are Lie-type (matrix) groups over a field of characteristic 2. The book thus completes a project begun by Daniel Gorenstein's 1983 book, which outlined the classification of groups of "noncharacteristic 2 type".
However, this book provides much more. Chapter 0 is a modern overview of the logical structure of the entire classification. Chapter 1 is a concise but complete outline of the "odd case" with updated references, while Chapter 2 sets the stage for the remainder of the book with a similar outline of the "even case". The remaining six chapters describe in detail the fundamental results whose union completes the proof of the classification theorem. Several important subsidiary results are also discussed. In addition, there is a comprehensive listing of the large number of papers referenced from the literature. Appendices provide a brief but valuable modern introduction to many key ideas and techniques of the proof. Some improved arguments are developed, along with indications of new approaches to the entire classification--such as the second and third generation projects--although there is no attempt to cover them comprehensively.
The work should appeal to a broad range of mathematicians--from those who just want an overview of the main ideas of the classification, to those who want a reader's guide to help navigate some of the major papers, and to those who may wish to improve the existing proofs.
Graduate students and research mathematicians interested in classification of finite simple groups.
Background and overview
Introduction
Overview: The classification of groups of Gorenstein-Walter type
Overview: The classification of groups of characteristic 2 type
Outline of the classification of groups of characteristic 2 type
e(G)leq2: The classification of quasithin groups
e(G)=3: The classification of rank 3 groups
e(G)geq4: The pretrichotomy and trichotomy theorems
The classification of groups of standard type
The classification of groups of GF(2) type
The final contradiction: Eliminating the Uniqueness Case
Appendices
Some background material related to simple groups
Overview of some techniques used in the classification
References and index
References used for both GW type and characteristic 2 type
References mainly for GW type
References used primarily for characteristic 2 type
Expository references mentioned
Index
Graduate Studies in Mathematics, Volume: 121
2011; approx. 320 pp; hardcover
ISBN-13: 978-0-8218-5323-8
Expected publication date is April 15, 2011.
Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces.
This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science.
The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.
Graduate students and research mathematicians interested in the theory of minimal surfaces.
The beginning of the theory
Curvature estimates and consequences
Weak convergence, compactness and applications
Existence results
Min-max constructions
Embedded solutions of the Plateau problem
Minimal surfaces in three-manifolds
The structure of embedded minimal surfaces
Exercises
Bibliography
Index
AMS Chelsea Publishing, Volume: 241
312 pp; hardcover
ISBN-13: 978-0-8218-5328-3
Expected publication date is April 16, 2011.
This highly flexible text is organized into two parts: Part I is suitable for a one-semester course at the first-year graduate level, and the book as a whole is suitable for a full-year course.
Part I treats the theory of measure and integration over abstract measure spaces. Prerequisites are a familiarity with epsilon-delta arguments and with the language of naive set theory (union, intersection, function). The fundamental theorems of the subject are derived from first principles, with details in full. Highlights include convergence theorems (monotone, dominated), completeness of classical function spaces (Riesz-Fischer theorem), product measures (Fubini's theorem), and signed measures (Radon-Nikodym theorem).
Part II is more specialized; it includes regular measures on locally compact spaces, the Riesz-Markoff theorem on the measure-theoretic representation of positive linear forms, and Haar measure on a locally compact group. The group algebra of a locally compact group is constructed in the last chapter, by an especially transparent method that minimizes measure-theoretic difficulties. Prerequisites for Part II include Part I plus a course in general topology.
Graduate students interested in teaching and learning the theory of measure and integration.
Measures
Measurable functions
Sequences of measurable functions
Integrable functions
Convergence theorems
Product measures
Finite signed measures
Integration over locally compact spaces
Integration over locally compact groups
References and notes
Bibliography
Index