CBMS Regional Conference Series in Mathematics, Number: 116
2012; 129 pp; softcover
ISBN-13: 978-0-8218-8979-4
Expected publication date is October 11, 2012.
This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce.
The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.
Graduate students and research mathematicians interested in deformations of algebras, moduli spaces, algebraic geometry, and/or algebraic topology.
Basic notions
Deformations and cohomology
Finer structures of cohomology
The gauge group
The simplicial Maurer-Cartan space
Strongly homotopy Lie algebras
Homotopy invariance and quantization
Brief introduction to operads
L∞-algebras governing deformations
Examples
Index
Bibliography
Pure and Applied Undergraduate Texts, Volume: 18
2012; 398 pp; hardcover
ISBN-13: 978-0-8218-8984-8
Expected publication date is November 1, 2012.
Analysis plays a crucial role in the undergraduate curriculum. Building upon the familiar notions of calculus, analysis introduces the depth and rigor characteristic of higher mathematics courses. Foundations of Analysis has two main goals. The first is to develop in students the mathematical maturity and sophistication they will need as they move through the upper division curriculum. The second is to present a rigorous development of both single and several variable calculus, beginning with a study of the properties of the real number system.
The presentation is both thorough and concise, with simple, straightforward explanations. The exercises differ widely in level of abstraction and level of difficulty. They vary from the simple to the quite difficult and from the computational to the theoretical. Each section contains a number of examples designed to illustrate the material in the section and to teach students how to approach the exercises for that section.
The list of topics covered is rather standard, although the treatment of some of them is not. The several variable material makes full use of the power of linear algebra, particularly in the treatment of the differential of a function as the best affine approximation to the function at a given point. The text includes a review of several linear algebra topics in preparation for this material. In the final chapter, vector calculus is presented from a modern point of view, using differential forms to give a unified treatment of the major theorems relating derivatives and integrals: Green's, Gauss's, and Stokes's Theorems.
At appropriate points, abstract metric spaces, topological spaces, inner product spaces, and normed linear spaces are introduced, but only as asides. That is, the course is grounded in the concrete world of Euclidean space, but the students are made aware that there are more exotic worlds in which the concepts they are learning may be studied.
Undergraduate students interested in real analysis.
The real numbers
Sequences
Continuous functions
The derivative
The integral
Infinite series
Convergence in Euclidean space
Functions on Euclidean space
Differentiation in several variables
Integration in several variables
Vector calculus
Degrees of infinity
Bibliography
Index
University Lecture Series, Volume: 60
2012; approx. 197 pp; softcover
ISBN-13: 978-0-8218-6912-3
Expected publication date is November 18, 2012.
Meeks and Perez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the focus of the account. The proof of the classification depends on the work of many currently active leading mathematicians, thus making contact with much of the most important results in the field. Through the telling of the story of the classification of minimal planar domains, the general mathematician may catch a glimpse of the intrinsic beauty of this theory and the authors' perspective of what is happening at this historical moment in a very classical subject.
This book includes an updated tour through some of the recent advances in the theory, such as Colding-Minicozzi theory, minimal laminations, the ordering theorem for the space of ends, conformal structure of minimal surfaces, minimal annular ends with infinite total curvature, the embedded Calabi-Yau problem, local pictures on the scale of curvature and topology, the local removable singularity theorem, embedded minimal surfaces of finite genus, topological classification of minimal surfaces, uniqueness of Scherk singly periodic minimal surfaces, and outstanding problems and conjectures.
Graduate students and research mathematicians interested in minimal surface theory.
Introduction
Basic results in classical minimal surface theory
Minimal surfaces with finite topology and more than one end
Limits of embedded minimal surfaces without local area or curvature bounds
The structure of minimal laminations of R3
The Ordering Theorem for the space of ends
Conformal structure of minimal surfaces
Uniqueness of the helicoid I: proper case
Embedded minimal annular ends with infinite total curvature
The embedded Calabi-Yau problem
Local pictures, local removable singularities and dynamics
Embedded minimal surfaces of finite genus
Topological aspects of minimal surfaces
Partial results on the Liouville conjecture
The Scherk uniqueness theorem
Calabi-Yau problems
Outstanding problems and conjectures
Bibliography
Proceedings of Symposia in Pure Mathematics, Volume: 84
2012; approx. 337 pp; hardcover
ISBN-13: 978-0-8218-5319-1
Expected publication date is November 9, 2012.
Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry.
In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.
Graduate students and research mathematicians interested in Riemannian geometry and analysis on manifolds.
Expository lectures
D. Borthwick -- Introduction to spectral theory on hyperbolic surfaces
C. Gordon -- Orbifolds and their spectra
A. Uribe and Z. Wang -- A brief introduction to semiclassical analysis
Invited papers
N. Anantharaman and F. Macia -- The dynamics of the Schrodinger flow from the point of view of semiclassical measures
G. Berkolaiko and P. Kuchment -- Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths
J. D. Bouas, S. A. Fulling, F. D. Mera, K. Thapa, C. S. Trendafilova, and J. Wagner -- Investigating the spectral geometry of a soft wall
E. B. Dryden, V. Guillemin, and R. Sena-Dias -- Equivariant inverse spectral problems
C. Gordon, W. Kirwin, D. Schueth, and D. Webb -- Classical equivalence and quantum equivalence of magnetic fields on flat tori
V. Guillemin, A. Uribe, and Z. Wang -- A semiclassical heat trace expansion for the perturbed harmonic oscillator
A. Hassell and A. Barnett -- Estimates on Neumann eigenfunctions at the boundary, and the "method of particular solutions" for computing them
H. Hezari and Z. Wang -- Lower bounds for volumes of nodal sets: An improvement of a result of Sogge-Zelditch
C. Judge -- The nodal set of a finite sum of Maass cusp forms is a graph
T. Kappeler, B. Schaad, and P. Topalov -- Asymptotics of spectral quantities of Schrodinger operators
P. Sarnak -- Recent progress on the quantum unique ergodicity conjecture
I. Wigman -- On the nodal lines of random and deterministic Laplace eigenfunctions
S. Zelditch -- Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I
Graduate Studies in Mathematics, Volume: 142
2012; 187 pp; hardcover
ISBN-13: 978-0-8218-8986-2
Expected publication date is November 14, 2012.
Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemeredi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems.
This book, which is the first monograph in this area, aims to cover all of these topics in a unified manner, as well as to survey some of the most recent developments, such as the application of the theory to count linear patterns in primes. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one's knowledge.
Graduate students and research mathematicians interested in harmonic analysis and number theory.
Higher order Fourier analysis
Related articles
Bibliography
Index
Contemporary Mathematics, Volume: 580
2012; 155 pp; softcover
ISBN-13: 978-0-8218-7553-7
Expected publication date is November 4, 2012.
One of the aims of this conference was to bring together researchers in the field of tropical geometry and its applications, from apparently disparate ends of the spectrum, to foster a mutual understanding and establish a common language which will encourage further developments of the area. This aim is reflected in these articles, which cover areas from automata, through cluster algebras, to enumerative geometry. In addition, two survey articles are included which introduce ideas from researchers on one end of this spectrum to researchers on the other.
This book is intended for graduate students and researchers interested in tropical geometry and integrable systems and the developing links between these two areas.
Graduate students and research mathematicians interested in tropical geometry and its applications to integrable systems.
D. Maclagan -- Introduction to tropical algebraic geometry
R. Inoue and S. Iwao -- Tropical curve and integrable piecewise linear map
F. Block -- Counting algebraic curves with tropical geometry
P. Johnson -- Hurwitz numbers, ribbon graphs, and tropicalization
T. Maeno and Y. Numata -- Sperner property, matroids and finite-dimensional Gorenstein algebras
L. Chekhov and M. Mazzocco -- Block triangular bilinear forms and braid group action
T. Nakanishi -- Tropicalization method in cluster algebras
S. Sergeev -- An application of the max-plus spectral theory to an ultradiscrete analogue of the Lax pair
R. Willox, A. Ramani, J. Satsuma, and B. Grammaticos -- A KdV cellular automaton without integers