University Lecture Series, Volume: 45
2008; 203 pp; softcover
ISBN-13: 978-0-8218-4468-7
Expected publication date is September 19, 2008.
In recent decades, p-adic geometry and p-adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations. The Arizona Winter School 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject.
Following invaluable introductions by John Tate and Vladimir Berkovich, two pioneers of non-archimedean geometry, Brian Conrad's chapter introduces the general theory of Tate's rigid analytic spaces, Raynaud's view of them as the generic fibers of formal schemes, and Berkovich spaces. Samit Dasgupta and Jeremy Teitelbaum discuss the p-adic upper half plane as an example of a rigid analytic space, and give applications to number theory (modular forms and the p-adic Langlands program). Matthew Baker offers a detailed discussion of the Berkovich projective line and p-adic potential theory on that and more general Berkovich curves. Finally, Kiran Kedlaya discusses theoretical and computational aspects of p-adic cohomology and the zeta functions of varieties. This book will be a welcome addition to the library of any graduate student and researcher who is interested in learning about the techniques of p-adic geometry.
Graduate students and research mathematicians interested in number theory and algebraic geometry.
V. Berkovich -- Non-archimedean analytic geometry: first steps
B. Conrad -- Several approaches to non-archimedean geometry
S. Dasgupta and J. Teitelbaum -- The p-adic upper half plane
M. Baker -- An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves
K. S. Kedlaya -- p-adic cohomology: from theory to practice
Mathematical Surveys and Monographs,Volume: 149
2008; 325 pp; hardcover
ISBN-13: 978-0-8218-4705-3
Expected publication date is September 25, 2008.
Quantum field theory has been a great success for physics, but it is difficult for mathematicians to learn because it is mathematically incomplete. Folland, who is a mathematician, has spent considerable time digesting the physical theory and sorting out the mathematical issues in it. Fortunately for mathematicians, Folland is a gifted expositor.
The purpose of this book is to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures, in a form that will be comprehensible to mathematicians. Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties.
The book begins with a review of classical physics and quantum mechanics, then proceeds through the construction of free quantum fields to the perturbation-theoretic development of interacting field theory and renormalization theory, with emphasis on quantum electrodynamics. The final two chapters present the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions.
Graduate students and research mathematicans interested in mathematical physics, specifically, quantum field theory.
American Mathematical Society Translations--Series 2, Volume: 224
2008; 284 pp; hardcover
ISBN-13: 978-0-8218-4674-2
Expected publication date is September 14, 2008.
This volume contains a selection of papers based on presentations given in 2006-2007 at the S. P. Novikov Seminar at the Steklov Mathematical Institute in Moscow. Novikov's diverse interests are reflected in the topics presented in the book. The articles address topics in geometry, topology, and mathematical physics. The volume is suitable for graduate students and researchers interested in the corresponding areas of mathematics and physics.
Graduate students and research mathematicians interested in geometry, topology, and mathematical physics.
A. V. Alexeevski and S. M. Natanzon -- Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups
V. M. Buchstaber and S. Terzi? -- Equivariant complex structures on homogeneous spaces and their cobordism classes
B. Dubrovin -- On universality of critical behaviour in Hamiltonian PDEs
M. Feigin and A. P. Veselov -- On the geometry of $\vee$-systems
P. G. Grinevich and I. A. Taimanov -- Spectral conservation laws for periodic nonlinear equations of the Melnikov type
S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernandez -- An equivariant version of the monodromy zeta function
A. Hamilton and A. Lazarev -- Symplectic $\mathcal{A}_\infty$-algebras and string topology operations
H. M. Khudaverdian and T. T. Voronov -- Differential forms and odd symplectic geometry
I. Krichever and T. Shiota -- Abelian solutions of the KP equation
A. Y. Maltsev -- Deformations of the Whitham systems in the almost linear case
O. I. Mokhov -- Frobenius manifolds as a special class of submanifolds in pseudo-Euclidean spaces
M. V. Pavlov -- Integrability of the Gibbons-Tsarev system
A. K. Pogrebkov -- 2D toda chain and associated commutator identity
O. K. Sheinman -- On certain current algebras related to finite-zone integration
Contemporary Mathematics, Volume: 466
2008; approx. 190 pp; softcover
ISBN-13: 978-0-8218-4267-6
Expected publication date is October 4, 2008.
This volume is a collection of research papers presented at the program on Moving Interface Problems and Applications in Fluid Dynamics, which was held between January 8 and March 31, 2007 at the Institute for Mathematical Sciences (IMS) of the National University of Singapore. The topics discussed include modeling and simulations of biological flow coupled to deformable tissue/elastic structure, shock wave and bubble dynamics and various applications including biological treatments with experimental verification, multi-medium flow or multi-phase flow and various applications including cavitation/supercavitation, detonation problems, Newtonian and non-Newtonian fluid, and many other areas. Readers can benefit from some recent research results in these areas.
Graduate students and research mathematicians interested in numerical analysis, computational fluid dynamics, and mathematical biology.
R. Dillon, M. Owen, and K. Painter -- A single-cell-based model of multicellular growth using the immersed boundary method
J. Hua, P. Lin, and J. F. Stene -- Numerical simulation of gas bubbles rising in viscous liquids at high Reynolds number
K. Ito and Z. Qiao -- A high order finite difference scheme for the Stokes equations
S. Jiang and G. Ni -- An efficient $\gamma$-model BGK scheme for multicomponent flows on unstructured meshes
D. V. Le, B. C. Khoo, and Z. Li -- An implicit-forcing immersed interface method for the incompressible Navier-Stokes equations
A. Naber, C. Liu, and J. J. Feng -- The nucleation and growth of gas bubbles in a Newtonian fluid: An energetic variational phase field approach
X. Pan -- Critical fields of liquid crystals
J. Palacios and G. Tryggvason -- The transient motion of buoyant bubbles in a vertical Couette flow
X. S. Wang -- Issues of immersed boundary/continuum methods
H. Xie, K. Ito, Z. Li, and J. Toivanen -- A finite element method for interface problems with locally modified triangulations
SMF/AMS Texts and Monographs, Volume: 15
2008; 149 pp; softcover
ISBN-13: 978-0-8218-4413-7
Expected publication date is October 18, 2008.
Hamiltonian systems began as a mathematical approach to the study of mechanical systems. As the theory developed, it became clear that the systems that had a sufficient number of conserved quantities enjoyed certain remarkable properties. These are the completely integrable systems. In time, a rich interplay arose between integrable systems and other areas of mathematics, particularly topology, geometry, and group theory.
This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations.
Audin has included many examples and exercises. Most of the exercises build on the material in the text. None of the important proofs have been relegated to the exercises. Many of the examples are classical, rather than abstract.
This book would be suitable for a graduate course in Hamiltonian systems.
Graduate students interested in Hamiltonian and integrable systems.
Introduction to integrable systems
Action-angle variables
Integrability and Galois groups
An introduction to Lax equations
Appendix A: What one needs to know about differential Galois theory
Appendix B: What one needs to know about algebraic curves
Bibliography
Index
*
2008; approx. 235 pp; softcover
ISBN-13: 978-0-8218-4375-8
Expected publication date is January 2, 2009.
Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics.
This book stresses the modern approach to the subject and contains much material not previously available in book form. It also discusses the relation between Poncelet's theorem and some aspects of queueing theory and mathematical billiards.
The proof of Poncelet's theorem presented in this book relates it to the theory of elliptic curves and exploits the fact that such curves are endowed with a group structure. The book also treats the real and degenerate cases of Poncelet's theorem. These cases are interesting in themselves, and their proofs require some other considerations. The real case is handled by employing notions from dynamical systems.
The material in this book should be understandable to anyone who has taken the standard courses in undergraduate mathematics. To achieve this, the author has included in the book preliminary chapters dealing with projective geometry, Riemann surfaces, elliptic functions, and elliptic curves. The book also contains numerous figures illustrating various geometric concepts.
Undergraduate and graduate students interested in projective geometry, complex analysis, dynamical systems, and general mathematics.
Introduction
Projective geometry
Basic notions of projective geometry
Conics
Intersection of two conics
Complex analysis
Riemann surfaces
Elliptic functions
The modular function
Elliptic curves
Poncelet and Cayley theorems
Poncelet's theorem
Cayley's theorem
Non-generic cases
The real case of Poncelet's theorem
Related topics
Billiards in an ellipse
Double queues
Supplement
Billiards and the Poncelet theorem
Appendices
Factorization of homogeneous polynomials
Degenerate conics of a conic pencil. Proof of Theorem 4.9
Lifting theorems
Proof of Theorem 11.5
Billiards in an ellipse. Proof of Theorem 13.1
References