Student Mathematical Library, Volume: 64
2012; 301 pp; softcover
ISBN-13: 978-0-8218-7571-1
Expected publication date is June 20, 2012.
The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms "toy geometries", the geometries of Platonic bodies, discrete geometries, and classical continuous geometries.
The text is based on first-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geometry, albeit those lacking this knowledge may refer to a compendium in Chapter 0. Per the author's predilection, the book contains very little regarding the axiomatic approach to geometry (save for a single chapter on the history of non-Euclidean geometry), but two Appendices provide a detailed treatment of Euclid's and Hilbert's axiomatics. Perhaps the most important aspect of this course is the problems, which appear at the end of each chapter and are supplemented with answers at the conclusion of the text. By analyzing and solving these problems, the reader will become capable of thinking and working geometrically, much more so than by simply learning the theory.
Ultimately, the author makes the distinction between concrete mathematical objects called "geometries" and the singular "geometry", which he understands as a way of thinking about mathematics. Although the book does not address branches of mathematics and mathematical physics such as Riemannian and Kahler manifolds or, say, differentiable manifolds and conformal field theories, the ideology of category language and transformation groups on which the book is based prepares the reader for the study of, and eventually, research in these important and rapidly developing areas of contemporary mathematics.
Undergraduates interested in geometry.
About Euclidean geometry
Toy geometries and main definitions
Abstract groups and group presentations
Finite subgroups of $SO(3)$ and the platonic bodies
Discrete subgroups of the isometry group of the plane and tilings
Reflection groups and Coxeter geometries
Spherical geometry
The Poincare disk model of hyperbolic geometry
The Poincare half-plane model
The Cayley-Klein model
Hyperbolic trigonometry and absolute constants
History of non-Euclidean geometry
Projective geometry
"Projective geometry is all geometry"
Finite geometries
The hierarchy of geometries
Morphisms of geometries
Excerpts from Euclid's "Elements"
Hilbert's axioms for plane geometry
Answers & hints
Bibliography
Index
Mathematical Surveys and Monographs,Volume: 181
2012; 367 pp; hardcover
ISBN-13: 978-0-8218-7581-0
Expected publication date is June 2, 2012.
This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras.
Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.
Research mathematicians interested in algebra, in particular, theory of rings and modules.
The Krull-Remak-Schmidt theorem
Semigroups of modules
Dimension zero
Dimension one
Invariant theory
Kleinian singularities and finite CM type
Isoalted singularities and classification in dimension two
The double branched cover
Hypersurfaces with finite CM type
Ascent and descent
Auslander-Buchweitz theory
Totally reflexive modules
Auslander-Reiten theory
Countable Cohen-Macaulay type
The Brauer-Thrall conjectures
Finite CM type in higher dimensions
Bounded CM type
Basics and background
Ramification theory
Bibliography
Index
Contemporary Mathematics, Volume: 569
2012; 202 pp; softcover
ISBN-13: 978-0-8218-5359-7
Expected publication date is June 6, 2012.
This volume is a collection of papers presented at the 11th International Workshop on Real and Complex Singularities, held July 26-30, 2010, in Sao Carlos, Brazil, in honor of David Mond's 60th birthday. This volume reflects the high level of the conference discussing the most recent results and applications of singularity theory. Articles in the first part cover pure singularity theory: invariants, classification theory, and Milnor fibres. Articles in the second part cover singularities in topology and differential geometry, as well as algebraic geometry and bifurcation theory: Artin-Greenberg function of a plane curve singularity, metric theory of singularities, symplectic singularities, cobordisms of fold maps, Goursat distributions, sections of analytic varieties, Vassiliev invariants, projections of hypersurfaces, and linearity of the Jacobian ideal.
Graduate students and research mathematicians interested in singularities in geometry and topology.
J. L. Cisneros-Molina, J. Seade, and J. Snoussi -- Milnor fibrations and the concept of $d$-regularity for analytic map germs
J. C. F. Costa, M. J. Saia, and C. H. Soares Junior -- Bi-Lipschitz $\mathcal{G}$-triviality and Newton polyhedra, $\mathcal{G}=\mathcal{R}, \mathcal{C}, \mathcal{K}, \mathcal{R}_V, \mathcal{C}_V, \mathcal{K}_V$
W. Domitrz and ?. Trebska -- Symplectic $S_{\mu}$ singularities
R. Araujo dos Santos, D. Dreibelbis, and N. Dutertre -- Topology of the real Milnor fiber for isolated singularities
C. Maquera and W. T. Huaraca -- Compact 3-manifolds supporting some $\mathbb{R}^2$-actions
M. Kasedou -- Timelike canal hypersurfaces of spacelike submanifolds in a de Sitter space
D. Lehmann -- Residues in $K$-theory
Y. Mizota and T. Nishimura -- Multicusps
P. Mormul -- Small growth vectors of the compactifications of the contact systems on $J^r(1,1)$
T. Ohmoto -- Vassiliev type invariants for generic mappings, revisited
B. Orefice and J. N. Tomazella -- Sections of analytic variety
S. Saleh -- The Artin-Greenberg function of a plane curve singularity
M. Shubladze -- Singularities with critical locus a complete intersection and trasnversal type $A_1$
Graduate Studies in Mathematics, Volume: 135
2012; approx. 378 pp; hardcover
ISBN-13: 978-0-8218-7576-6
Expected publication date is August 18, 2012.
This book considers evolution equations of hyperbolic and parabolic type. These equations are studied from a common point of view, using elementary methods, such as that of energy estimates, which prove to be quite versatile. The authors emphasize the Cauchy problem and present a unified theory for the treatment of these equations. In particular, they provide local and global existence results, as well as strong well-posedness and asymptotic behavior results for the Cauchy problem for quasi-linear equations. Solutions of linear equations are constructed explicitly, using the Galerkin method; the linear theory is then applied to quasi-linear equations, by means of a linearization and fixed-point technique. The authors also compare hyperbolic and parabolic problems, both in terms of singular perturbations, on compact time intervals, and asymptotically, in terms of the diffusion phenomenon, with new results on decay estimates for strong solutions of homogeneous quasi-linear equations of each type.
This textbook presents a valuable introduction to topics in the theory of evolution equations, suitable for advanced graduate students. The exposition is largely self-contained. The initial chapter reviews the essential material from functional analysis. New ideas are introduced along with their context. Proofs are detailed and carefully presented. The book concludes with a chapter on applications of the theory to Maxwell's equations and von Karman's equations.
Graduate students and research mathematicians interested in partial differential equations.
Functional framework
Linear equations
Quasi-linear equations
Global existence
Asymptotic behavior
Singular convergence
Maxwell's and von Karman's equations
List of function spaces
Bibliography
Index