Contemporary Mathematics, Volume: 556
2011; 217 pp; softcover
ISBN-13: 978-0-8218-5228-6
Expected publication date is November 20, 2011.
This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010 in Banff, Ontario, Canada.
This volume contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and touch upon core topics such as homotopy methods for approximating complex solutions, robust floating point methods for clusters of roots, and speed-ups for counting real solutions. Vital related topics such as circuit complexity, random polynomials over local fields, tropical geometry, and the theory of fewnomials, amoebae, and coamoebae are treated as well. Recent advances on Smale's 17th Problem, which deals with numerical algorithms that approximate a single complex solution in average-case polynomial time, are also surveyed.
Graduate students and research mathematicians interested in algorithms in algebraic geometry.
M. Avendano and A. Ibrahim -- Multivariate ultrametric root counting
D. J. Bates, J. D. Hauenstein, and A. J. Sommese -- A parallel endgame
C. Beltran and L. M. Pardo -- Efficient polynomial system solving by numerical methods
B. Grenet, E. L. Kaltofen, P. Koiran, and N. Portier -- Symmetric determinantal representation of formulas and weakly skew circuits
T.-L. Lee and T.-Y. Li -- Mixed volume computation in solving polynomial systems
A. Leykin -- A search for an optimal start system for numerical homotopy continuation
M. Nisse -- Complex tropical localization, and coamoebas of complex algebraic hypersurfaces
O. Bastani, C. J. Hillar, D. Popov, and J. M. Rojas -- Randomization, sums of squares, near-circuits, and faster real root counting
K. Rusek, J. Shakalli, and F. Sottile -- Dense fewnomials
Z. Zeng -- The numerical greatest common divisor of univariate polynomials
History of Mathematics, Volume: 38
2011; approx. 448 pp; hardcover
ISBN-13: 978-0-8218-4464-9
Expected publication date is December 4, 2011.
The theory of complex dynamics, whose roots lie in 19th-century studies of the iteration of complex function conducted by K?nigs, Schoder, and others, flourished remarkably during the first half of the 20th century, when many of the central ideas and techniques of the subject developed. This book by Alexander, Iavernaro, and Rosa paints a robust picture of the field of complex dynamics between 1906 and 1942 through detailed discussions of the work of Fatou, Julia, Siegel, and several others.
A recurrent theme of the authors' treatment is the center problem in complex dynamics. They present its complete history during this period and, in so doing, bring out analogies between complex dynamics and the study of differential equations, in particular, the problem of stability in Hamiltonian systems. Among these analogies are the use of iteration and problems involving small divisors which the authors examine in the work of Poincare and others, linking them to complex dynamics, principally via the work of Samuel Lattes, in the early 1900s, and Jurgen Moser, in the 1960s.
Many details will be new to the reader, such as a history of Lattes functions (functions whose Julia set equals the Riemann sphere), complex dynamics in the United States around the time of World War I, a survey of complex dynamics around the world in the 1920s and 1930s, a discussion of the dynamical programs of Fatou and Julia during the 1920s, and biographical material on several key figures. The book contains graphical renderings of many of the mathematical objects the authors discuss, including some of the intriguing fractals Fatou and Julia studied, and concludes with several appendices by current researchers in complex dynamics which collectively attest to the impact of the work of Fatou, Julia, and others upon the present-day study.
Graduate students and research mathematicians interested in complex dynamics, complex analysis, and the history of mathematics.
Preliminaries
A complex dynamics primer
Introduction: Dynamics of a complex history
Iteration and differential equations I: The Poincare connection
Iteration and differential equations II: Small divisors
The core (1906-1920)
Early overseas results: The United States
The road to the Grand Prix des Sciences Mathematiques
Works written for the Grand Prix
Iteration in Italy
The giants fall
After-maths (1920-1942)
Branching out: Fatou and Julia in the 1920s
The German wave
Siegel, the center problem, and KAM theory
Iteratin' around the globe
Tying the future to the past
Report on the Grand Prix des Sciences Mathematiques in 1918
A history of normal families
Singular lines of analytic functions
Kleinian groups
Curves of Julia
Progress in Julia's extension of Schwarz's lemma
The Denjoy-Wolff theorem
Dynamics of self-maps of the unit disc
Koebe and uniformization
Permutable maps in the 1920s
The last 60 years in permutable maps
Understanding Julia sets of entire maps
Fatou: A biographical sketch
Gaston Julia: A biographical sketch
Selected biographies
Remarks on computer graphics
Glossary
Bibliography
Index
Proceedings of Symposia in Pure Mathematics, Volume: 83
2011; approx. 357 pp; hardcover
ISBN-13: 978-0-8218-5195-1
Expected publication date is December 16, 2011.
Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physical systems. Quantum field theory (QFT) has proven to be a rich source of ideas for mathematics for a long time. However, fundamental questions such as "What is a QFT?" did not have satisfactory mathematical answers, especially on spaces with arbitrary topology, fundamental for the formulation of perturbative string theory.
This book contains a collection of papers highlighting the mathematical foundations of QFT and its relevance to perturbative string theory as well as the deep techniques that have been emerging in the last few years.
The papers are organized under three main chapters: Foundations for Quantum Field Theory, Quantization of Field Theories, and Two-Dimensional Quantum Field Theories. An introduction, written by the editors, provides an overview of the main underlying themes that bind together the papers in the volume.
Graduate students and research mathematicians interested in mathematical aspects of quantum field theory.
H. Sati and U. Schreiber -- Introduction
Foundations for quantum field theory
J. E. Bergner -- Models for (infty,n)-categories and the cobordism hypothesis
I. Weiss -- From operads to dendroidal sets
A. Davydov, L. Kong, and I. Runkel -- Field theories with defects and the centre functor
Quantization of field theories
F. Paugam -- Homotopical Poisson reduction of gauge theories
J. Distler, D. S. Freed, and G. W. Moore -- Orientifold precis
Two-dimensional quantum field theories
A. Kapustin and N. Saulina -- Surface operators in 3d topological field theory and 2d rational conformal field theory
L. Kong -- Conformal field theory and a new geometry
Y. Soibelman -- Collapsing conformal field theories, spaces with non-negative Ricci curvature and non-commutative geometry
S. Stolz and P. Teichner -- Supersymmetric field theories and generalized cohomology
C. L. Douglas and A. G. Henriques -- Topological modular forms and conformal nets
Contemporary Mathematics,Volume: 557
2011; 388 pp; softcover
ISBN-13: 978-0-8218-5246-0
Expected publication date is December 4, 2011.
This volume contains the proceedings of the conference on Representation Theory and Mathematical Physics, in honor of Gregg Zuckerman's 60th birthday, held October 24-27, 2009, at Yale University.
Lie groups and their representations play a fundamental role in mathematics, in particular because of connections to geometry, topology, number theory, physics, combinatorics, and many other areas. Representation theory is one of the cornerstones of the Langlands program in number theory, dating to the 1970s. Zuckerman's work on derived functors, the translation principle, and coherent continuation lie at the heart of the modern theory of representations of Lie groups. One of the major unsolved problems in representation theory is that of the unitary dual. The fact that there is, in principle, a finite algorithm for computing the unitary dual relies heavily on Zuckerman's work.
In recent years there has been a fruitful interplay between mathematics and physics, in geometric representation theory, string theory, and other areas. New developments on chiral algebras, representation theory of affine Kac-Moody algebras, and the geometric Langlands correspondence are some of the focal points of this volume. Recent developments in the geometric Langlands program point to exciting connections between certain automorphic representations and dual fibrations in geometric mirror symmetry.
Graduate students and research mathematicians interested in representation theory and connections between representation theory and mathematical physics.
Expository papers
R. A. Herb and P. J. Sally, Jr. -- The Plancherel formula, the Plancherel theorem, and the Fourier transform of orbital integrals
T. Kobayashi -- Branching problems of Zuckerman derived functor modules
B. H. Lian, A. R. Linshaw, and B. Song -- Chiral equivariant cohomology of spheres
Research papers
J. Adams -- Computing global characters
D. M. Barbasch and P. E. Trapa -- Stable combinations of special unipotent representations
E. Dan-Cohen and I. Penkov -- Levi components of parabolic subalgebras of finitary Lie algebras
H. Garland -- On extending the Langlands-Shahidi method to arithmetic quotients of loop groups
M. W. Hero, J. F. Willenbring, and L. K. Williams -- The measurement of quantum entanglement and enumeration of graph coverings
D. Lu and R. Howe -- The dual pair (O_{p,q}, Owidetilde{Sp}_{2,2}) and Zuckerman translation
B. Kostant and N. Wallach -- On the algebraic set of singular elements in a complex simple Lie algebra
A. G. Lisi -- An explicit embedding of gravity and the standard model in E_8
G. Lusztig -- From groups to symmetric spaces
G. Lusztig -- Study of antiorbital complexes
S. D. Miller and W. Schmid -- Adelization of automorphic distributions and mirabolic Eisenstein series
I. Penkov and V. Serganova -- Categories of integrable sl(infty)-, o(infty)-, sp(infty)-modules
S. Sahi -- Binomial coefficients and Littlewood-Richardson coefficients for interpolation polynomials and Macdonald polynomials
B. Speh -- Restriction of some representations of U(p,q) to a symmetric subgroup
Graduate Studies in Mathematics, Volume: 129
2012; approx. 373 pp; hardcover
ISBN-13: 978-0-8218-4694-0
Expected publication date is January 7, 2012.
This text emphasizes rigorous mathematical techniques for the analysis of boundary value problems for ODEs arising in applications. The emphasis is on proving existence of solutions, but there is also a substantial chapter on uniqueness and multiplicity questions and several chapters which deal with the asymptotic behavior of solutions with respect to either the independent variable or some parameter. These equations may give special solutions of important PDEs, such as steady state or traveling wave solutions. Often two, or even three, approaches to the same problem are described. The advantages and disadvantages of different methods are discussed.
The book gives complete classical proofs, while also emphasizing the importance of modern methods, especially when extensions to infinite dimensional settings are needed. There are some new results as well as new and improved proofs of known theorems. The final chapter presents three unsolved problems which have received much attention over the years.
Graduate students and research mathematicians interested in ODEs and PDEs.
Introduction
An introduction to shooting methods
Some boundary value problems for the Painleve transcendents
Periodic solutions of a higher order system
A linear example
Homoclinic orbits of the FitzHugh-Nagumo equations
Singular perturbation problems--rigorous matching
Asymptotics beyond all orders
Some solutions of the Falkner-Skan equation
Poiseuille flow: Perturbation and decay
Bending of a tapered rod; variational methods and shooting
Uniqueness and multiplicity
Shooting with more parameters
Some problems of A. C. Lazer
Chaotic motion of a pendulum
Layers and spikes in reaction-diffusion equations, I
Uniform expansions for a class of second order problems
Layers and spikes in reaction-diffusion equations, II
Three unsolved problems
Bibliography
Index
Graduate Studies in Mathematics, Volume: 130
2011; 164 pp; hardcover
ISBN-13: 978-0-8218-7287-1
Expected publication date is January 7, 2012.
This book provides a concise yet comprehensive and self-contained introduction to Grobner basis theory and its applications to various current research topics in commutative algebra. It especially aims to help young researchers become acquainted with fundamental tools and techniques related to Grobner bases which are used in commutative algebra and to arouse their interest in exploring further topics such as toric rings, Koszul and Rees algebras, determinantal ideal theory, binomial edge ideals, and their applications to statistics.
The book can be used for graduate courses and self-study. More than 100 problems will help the readers to better understand the main theoretical results and will inspire them to further investigate the topics studied in this book.
Graduate students and research mathematicians interested in Groebner bases.
Polynomial rings and ideals
Grobner bases
First applications
Grobner bases for modules
Grobner bases of toric ideals
Selected applications in commutative algebra and combinatorics
Bibliography
Index