Graduate Studies in Mathematics, Volume: 69
2009; 376 pp; hardcover
ISBN-13: 978-0-8218-4763-3
Expected publication date is August 6, 2009.
This introductory textbook puts forth a clear and focused point of view on the differential geometry of curves and surfaces. Following the modern point of view on differential geometry, the book emphasizes the global aspects of the subject. The excellent collection of examples and exercises (with hints) will help students in learning the material. Advanced undergraduates and graduate students will find this a nice entry point to differential geometry.
In order to study the global properties of curves and surfaces, it is necessary
to have more sophisticated tools than are usually found in textbooks on
the topic. In particular, students must have a firm grasp on certain topological
theories. Indeed, this monograph treats the Gauss-Bonnet theorem and discusses
the Euler characteristic. The authors also cover Alexandrov's theorem on
embedded compact surfaces in mathbb{R}^3 with constant mean curvature.
The last chapter addresses the global geometry of curves, including periodic
space curves and the four-vertices theorem for plane curves that are not
necessarily convex.
Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry. It is suitable as the text for a first-year graduate course or an advanced undergraduate course.
Undergraduate students, graduate students, and research mathematicians interested in the geometry of curves and surfaces.
Plane and space curves
Surfaces in Euclidean space
The second fundamental form
Separation and orientability
Integration on surfaces
Global extrinsic geometry
Intrinsic geometry of surfaces
The Gauss-Bonnet theorem
Global geometry of curves
Bibliography
Index
DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, Volume: 74
2009; 319 pp; hardcover
ISBN-13: 978-0-8218-4383-3
Expected publication date is August 26, 2009.
Shortest path problems are among the most fundamental combinatorial optimization problems with many applications, both direct and as subroutines. They arise naturally in a remarkable number of real-world settings. A limited list includes transportation planning, network optimization, packet routing, image segmentation, speech recognition, document formatting, robotics, compilers, traffic information systems, and dataflow analysis. Shortest path algorithms have been studied since the 1950's and still remain an active area of research.
This volume reports on the research carried out by participants during the Ninth DIMACS Implementation Challenge, which led to several improvements of the state of the art in shortest path algorithms. The infrastructure developed during the Challenge facilitated further research in the area, leading to substantial follow-up work as well as to better and more uniform experimental standards. The results of the Challenge included new cutting-edge techniques for emerging applications such as GPS navigation systems, providing experimental evidence of the most effective algorithms in several real-world settings.
Graduate students and research mathematicians interested in algorithms and combinatorial optimization problems.
J. L. Santos -- Real-world applications of shortest path algorithms
U. Lauther -- An experimental evaluation of point-to-point shortest path calculation on road networks with precalculated edge-flags
M. Hilger, E. Kohler, R. H. Mohring, and H. Schilling -- Fast point-to-point shortest path computations with arc-flags
D. Delling, M. Holzer, K. Muller, F. Schulz, and D. Wagner -- High-performance multi-level routing
A. V. Goldberg, H. Kaplan, and R. F. Werneck -- Reach for A*: shortest path algorithms with preprocessing
D. Delling, P. Sanders, D. Schultes, and D. Wagner -- Highway hierarchies star
H. Bast, S. Funke, and D. Matijevic -- Ultrafast shortest-path queries via transit nodes
P. Sanders and D. Schultes -- Robust, almost constant time shortest-path queries in road networks
N. Edmonds, A. Breuer, D. Gregor, and A. Lumsdaine -- Single-source shortest paths with the parallel boost graph library
K. Madduri, D. A. Bader, J. W. Berry, and J. R. Crobak -- Parallel shortest path algorithms for solving large-scale instances
D. Ajwani, U. Meyer, and V. Osipov -- Breadth first search on massive graphs
C. L. Barrett, K. Bisset, M. Holzer, G. Konjevod, M. V. Marathe, and D. Wagner -- Engineering label-constrained shortest-path algorithms
C. Demetrescu, A. V. Goldberg
Graduate Studies in Mathematics, Volume: 105
2009; 607 pp; hardcover
ISBN-13: 978-0-8218-4768-8
Expected publication date is August 27, 2009.
Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis.
The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables.
The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces.
The book contains over 200 exercises.
Graduate students and research mathematicians interested in Sobolev spaces, particularly their applications to PDEs.
Mathematical Surveys and Monographs, Volume: 154
2009; approx. 634 pp; hardcover
ISBN-13: 978-0-8218-2681-2
Expected publication date is August 26, 2009.
Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hypersurface type CR-structures and various types of generic distributions. The characteristic feature of parabolic geometries is an equivalent description by a Cartan geometry modeled on a generalized flag manifold (the quotient of a semisimple Lie group by a parabolic subgroup).
Background on differential geometry, with a view towards Cartan connections, and on semisimple Lie algebras and their representations, which play a crucial role in the theory, is collected in two introductory chapters. The main part discusses the equivalence between Cartan connections and underlying structures, including a complete proof of Kostant's version of the Bott-Borel-Weil theorem, which is used as an important tool. For many examples, the complete description of the geometry and its basic invariants is worked out in detail. The constructions of correspondence spaces and twistor spaces and analogs of the Fefferman construction are presented both in general and in several examples. The last chapter studies Weyl structures, which provide classes of distinguished connections as well as an equivalent description of the Cartan connection in terms of data associated to the underlying geometry. Several applications are discussed throughout the text.
Graduate students and research mathematicians interested in parabolic geometry, conformal geometry, almost quaternionic structures, and CR-structures.
Background
Cartan geometries
Semisimple Lie algebras and Lie groups
General theory
Parabolic geometries
A panorama of examples
Distinguished connections and curves
Other prolongation procedures
Tables
Bibliography
Index
Contemporary Mathematics,Volume: 490
2009; 251 pp; softcover
ISBN-13: 978-0-8218-4731-2
Expected publication date is August 9, 2009.
The articles in this volume mainly grew out of talks given at a Conference held at UCLA in January 2008, which honored V. S. Varadarajan on his 70th birthday. The main theme of the Conference was symmetry in mathematics and physics, areas of mathematics and mathematical physics in which Varadarajan has made significant contributions during the past 50 years. Very early in his career he also worked and made significant contributions in the areas of probability and the foundations of quantum mechanics.
Topics covered by the articles in this volume are probability, quantum mechanics, symmetry (broadly interpreted in mathematics and physics), finite and infinite dimensional Lie groups and Lie algebras and their representations, super Lie groups and supergeometry (relatively new but active and important fields at the interface between mathematics and physics), and supersymmetry. The latter topic takes on a special importance since one of the first experiments at the Large Hadron Collider at CERN will be a test of whether supersymmetry exists in the world of elementary particles. A reprint of an exposition of supersymmetry by one of its founders, B. Zumino, appears in this volume.
Graduate students and research mathematicians interested in mathematical physics.
Contemporary Mathematics, Volume: 491
2009; 348 pp; softcover
ISBN-13: 978-0-8218-4651-3
Expected publication date is August 29, 2009.
This volume is an outgrowth of the Sixth Workshop on Lie Theory and Geometry, held in the province of Cordoba, Argentina in November 2007.
The representation theory and structure theory of Lie groups play a pervasive role throughout mathematics and physics. Lie groups are tightly intertwined with geometry and each stimulates developments in the other. The aim of this volume is to bring to a larger audience the mutually beneficial interaction between Lie theorists and geometers that animated the workshop.
Two prominent themes of the representation theoretic articles are Gelfand pairs and the representation theory of real reductive Lie groups. Among the more geometric articles are an exposition of major recent developments on noncompact homogeneous Einstein manifolds and aspects of inverse spectral geometry presented in settings accessible to readers new to the area.
Graduate students and research mathematicians interested in Lie groups and Lie algebras and their applications.
J. Lauret -- Einstein solvmanifolds and nilsolitons
C. U. Sanchez -- Algebraic sets associated to isoparametric submanifolds
L. Ji -- Mostow strong rigidity and nonisomorphism for outer automorphism groups of free groups and mapping class groups
R. J. Miatello and J. P. Rossetti -- Spectral properties of flat manifolds
P. Gilkey -- Heat content, heat trace, and isospectrality
D. Burde, K. Dekimpe, and S. Deschamps -- LR-algebras
C. Benson and G. Ratcliff -- Combinatorial properties of generalized binomial coefficients
C. Benson and G. Ratcliff -- Spherical functions for the action of a finite unitary group on a finite Heisenberg group
S. Gurevich and R. Hadani -- Application of the Weil representation: Diagonalization of the discrete Fourier transform
J. A. Wolf -- Infinite dimensional multiplicity free spaces II: Limits of commutative nilmanifolds
L. Barchini and R. Zierau -- Certain components of Springer fibers: algorithms, examples and applications
E. Galina -- Weighted Vogan diagrams associated to real nilpotent orbits
M. Colarusso -- The Gelfand-Zeitlin integrable system and its action on
generic elements of mathfrak gl(n) and mathfrak so(n)
P. Eberlein and M. Jablonski -- Closed orbits of semisimple group actions and the real Hilbert-Mumford function
N. Andruskiewitsch and F. Fantino -- New techniques for pointed Hopf algebras