Contemporary Mathematics, Volume: 467
2008; 152 pp; softcover
ISBN-13: 978-0-8218-4457-1
This volume contains articles based on talks presented at the Thirteenth Conference of African American Researchers in the Mathematical Sciences (CAARMS), held at Northeastern University and the University of Massachusetts, Boston on June 19-22, 2007. The representation theory of Lie groups and its applications were a major focus of the talks. An overview is included of the recent achievements of the Atlas of Lie Groups Project and the work that remains to be done by the Atlas Project to fully understand the unitary representations of reductive groups. Other articles highlight the significance of scientific computing in Lie Theory and applications of the representation theory of Lie groups to the physics of black holes. This volume contains two research papers not related to Lie Theory: one on the geometry of coisotropic submanifolds of Poisson manifolds and one on recent results on the structure of the set of ultrafilters on the collection of finite subsets of an infinite set. It concludes with a survey of CAARMS12 participants.
Graduate students and research mathematicians interested in representations of Lie groups and related topics.
A. Pantano, A. Paul, and S. A. Salamanca-Riba -- The omega-regular unitary dual of the metaplectic group of rank 2
F. L. Williams -- A resolvent trace formula for the BTZ black hole with conical singularity
A. Wade -- On the geometry of coisotropic submanifolds of Poisson manifolds
A. D. Grainger -- Remarks on ultrafilters on the collection of finite subsets of an infinite set
A. G. Noel -- The atlas of Lie groups and representations: Scope and successes
A. G. Noel -- Maximal tori of reductive centralizers of nilpotents in exceptional complex symmetric spaces: A computational approach
W. A. Massey, D. Raphael, and E. N. Walker -- A survey of CAARMS12 participants
Contemporary Mathematics, Volume: 468
2008; 230 pp; softcover
ISBN-13: 978-0-8218-4459-5
The theme of this volume concerns interactions between group actions and problems in complex analysis.
The first four articles deal with such topics as representation kernels in representation theory, complex automorphisms and holomorphic equivalence of domains, and geometric description of exceptional symmetric domains. The last article is devoted to Seiberg-Witten equations and Taubes correspondence on symplectic 4-manifolds.
Graduate students and research mathematicians interested in several complex variables, representation theory, and theoretical physics.
J. Hilgert -- Reproducing kernels in representation theory
A. Huckleberry -- Actions of complex Lie groups and the Borel-Weil correspondence
J.-J. Loeb -- On complex automorphisms and holomorphic equivalence of domains
G. Roos -- Exceptional symmetric domains
A. Sergeev -- Seiberg-Witten equations and pseudoholomorphic curves
Index
Contemporary Mathematics, Volume: 469
2008; 340 pp; softcover
ISBN-13: 978-0-8218-4286-7
This book presents a collection of articles that cover areas of mathematics related to dynamical systems. The authors are well-known experts who use geometric and probabilistic methods to study interesting problems in the theory of dynamical systems and its applications. Some of the articles are surveys while others are original contributions. The topics covered include: Riemannian geometry, models in mathematical physics and mathematical biology, symbolic dynamics, random and stochastic dynamics.
This book can be used by graduate students and researchers in dynamical systems and its applications.
Graduate students and research mathematicians interested in dynamical systems and geometry.
Contemporary Mathematics, Volume: 470
2008; 286 pp; softcover
ISBN-13: 978-0-8218-4365-9
The power of general purpose computational algebra systems running on personal computers has increased rapidly in recent years. For mathematicians doing research in group theory, this means a growing set of sophisticated computational tools are now available for their use in developing new theoretical results.
This volume consists of contributions by researchers invited to the AMS Special Session on Computational Group Theory held in March 2007. The main focus of the session was on the application of Computational Group Theory (CGT) to a wide range of theoretical aspects of group theory. The articles in this volume provide a variety of examples of how these computer systems helped to solve interesting theoretical problems within the discipline, such as constructions of finite simple groups, classification of p-groups via coclass, representation theory and constructions involving free nilpotent groups. The volume also includes an article by R. F. Morse highlighting applications of CGT in group theory and two survey articles.
Graduate students and researchers interested in various aspects of group theory will find many examples of Computational Group Theory helping research and will recognize it as yet another tool at their disposal.
Graduate students and research mathematicians interested in group theory and computational group theory.
Student Mathematical Library, Volume: 46
2008; approx. 303 pp; softcover
ISBN-13: 978-0-8218-4679-7
Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. At the same time, many of those notions appear in a technically simpler and more graphic form than in their general "natural" settings.
The first, primarily expository, chapter introduces many of the principal actors--the round sphere, flat torus, Mobius strip, Klein bottle, elliptic plane, etc.--as well as various methods of describing surfaces, beginning with the traditional representation by equations in three-dimensional space, proceeding to parametric representation, and also introducing the less intuitive, but central for our purposes, representation as factor spaces. It concludes with a preliminary discussion of the metric geometry of surfaces, and the associated isometry groups. Subsequent chapters introduce fundamental mathematical structures--topological, combinatorial (piecewise linear), smooth, Riemannian (metric), and complex--in the specific context of surfaces.
The focal point of the book is the Euler characteristic, which appears in many different guises and ties together concepts from combinatorics, algebraic topology, Morse theory, ordinary differential equations, and Riemannian geometry. The repeated appearance of the Euler characteristic provides both a unifying theme and a powerful illustration of the notion of an invariant in all those theories.
The assumed background is the standard calculus sequence, some linear algebra, and rudiments of ODE and real analysis. All notions are introduced and discussed, and virtually all results proved, based on this background.
This book is a result of the MASS course in geometry in the fall semester of 2007.
Undergraduate and graduate students interested in broadening their view of geometry and topology.
Various ways of representing surfaces and basic examples
Combinatorial structure and topological classification of surfaces
Differentiable structure on surfaces: Real and complex
Riemannian metrics and geometry of surfaces
Topology and smooth structure revisited
Suggested reading
Hints
Index
Mathematical Surveys and Monographs, Volume: 150
2008; approx. 763 pp; hardcover
ISBN-13: 978-0-8218-4186-0
The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum \mathfrak {gl}_n.
The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum \mathfrak{gl}_n through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature.
Graduate students and research mathematicians interested in quantum groups and finite-dimensional algebras.
Getting started
Quivers and their representations
Representations of quivers
Algebras with Frobenius morphisms
Quivers with automorphisms
Some quantized algebras
Coxeter groups and Hecke algebras
Hopf algebras and universal enveloping algebras
Quantum enveloping algebras
Representations of symmetric groups
Kazhdan-Lusztig combinatorics for Hecke algebras
Cells and representations of symmetric groups
The integral theory of quantum Schur algebras
Ringel-Hall algebras: A realization for the \pm-part
Ringel-Hall algebras
Bases of quantum enveloping algebras of finite type
Green's theorem
The BLM algebra: A realization for quantum \mathfrak{gl}_{n}
Serre relations in quantum Schur algebras
Constructing quantum \mathfrak{gl}_{n} via quantum Schur algebras
Appendices
Varieties and affine algebraic groups
Quantum linear groups through coordinate algebras
Quasi-hereditary and cellular algebras
Bibliography
Index of notation
Index of terminology