AMS/IP Studies in Advanced Mathematics, Volume: 43
2008; 259 pp; hardcover
ISBN-13: 978-0-8218-4675-9
Expected publication date is June 12, 2008.
In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as mathbf{Z} or textrm{SL}(n,mathbf{Z}). Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations.
The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry.
It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics.
Graduate students interested in arithmetic groups and their applications to number theory, geometry and topology.
Contemporary Mathematics, Volume: 460
2008; 401 pp; softcover
ISBN-13: 978-0-8218-4486-1
Not yet published.Expected publication date is July 11, 2008.
Toric topology is the study of algebraic, differential, symplectic-geometric, combinatorial, and homotopy-theoretic aspects of a particular class of torus actions whose quotients are highly structured. The combinatorial properties of this quotient and the equivariant topology of the original manifold interact in a rich variety of ways, thus illuminating subtle aspects of both the combinatorics and the equivariant topology. Many of the motivations and guiding principles of the field are provided by (though not limited to) the theory of toric varieties in algebraic geometry as well as that of symplectic toric manifolds in symplectic geometry.
This volume is the proceedings of the International Conference on Toric Topology held in Osaka in May-June 2006. It contains about 25 research and survey articles written by conference speakers, covering many different aspects of, and approaches to, torus actions, such as those mentioned above. Some of the manuscripts are survey articles, intended to give a broad overview of an aspect of the subject; all manuscripts consciously aim to be accessible to a broad reading audience of students and researchers interested in the interaction of the subjects involved. We hope that this volume serves as an enticing invitation to this emerging field.
Graduate students and research mathematicians interested in different aspects of torus actions, such as topological, combinatorial, and symplectic or algebra-geometric.
Proceedings of Symposia in Pure Mathematics, Volume: 77
2008; approx. 689 pp; hardcover
ISBN-13: 978-0-8218-4471-7
Expected publication date is August 30, 2008.
This book addresses a new interdisciplinary area emerging on the border between various areas of mathematics, physics, chemistry, nanotechnology, and computer science. The focus here is on problems and techniques related to graphs, quantum graphs, and fractals that parallel those from differential equations, differential geometry, or geometric analysis. Also included are such diverse topics as number theory, geometric group theory, waveguide theory, quantum chaos, quantum wire systems, carbon nano-structures, metal-insulator transition, computer vision, and communication networks.
This volume contains a unique collection of expert reviews on the main directions in analysis on graphs (e.g., on discrete geometric analysis, zeta-functions on graphs, recently emerging connections between the geometric group theory and fractals, quantum graphs, quantum chaos on graphs, modeling waveguide systems and modeling quantum graph systems with waveguides, control theory on graphs), as well as research articles.
Graduate students and research mathematicians interested in various areas of analysis, dynamical systems, groups and their actions on graphs, mathematical physics.
Analysis on combinatorial graphs
Review articles
R. Band, I. Oren, and U. Smilansky -- Nodal domains on graphs--How to count them and why?
M. D. Horton, H. M. Stark, and A. A. Terras -- Zeta functions of weighted graphs and covering graphs
T. Sunada -- Discrete geometric analysis
Research articles
M. J. Gruber, D. H. Lenz, and I. Veselic -- Uniform existence of the integrated
density of states for combinatorial and metric graphs over Z^d
D. Guido, T. Isola, and M. L. Lapidus -- Bartholdi zeta functions for periodic simple graphs
M. Kelbert and Y. Suhov -- Asymptotic properties of Markov processes on Cayley trees
Analysis on fractals
Review articles
V. Nekrashevych and A. Teplyaev -- Groups and analysis on fractals
Research articles
R. Grigorchuk and Z. Sunic -- Schreier spectrum of the Hanoi Towers group on three pegs
A. Grigor'yan and T. Kumagai -- On the dichotomy in the heat kernel two sided estimates
M. L. Lapidus and E. P. J. Pearse -- Tube formulas for self-similar fractals
R. Peirone -- Existence of Eigenforms on nicely separated fractals
Analysis on quantum graphs
Review articles
J. Bolte and S. Endres -- Trace formulae for quantum graphs
J. Harrison -- Quantum graphs with spin Hamiltonians
J. P. Keating -- Quantum graphs and quantum chaos
P. Kuchment -- Quantum graphs: An introduction and a brief survey
Research articles
G. Berkolaiko -- Two constructions of quantum graphs and two types of spectral statistics
B. M. Brown, M. S. P. Eastham, and I. G. Wood -- An example on the discrete spectrum of a star graph
B. M. Brown, M. Langer, and K. M. Schmidt -- The HELP inequality on trees
R. Carlson -- Boundary value problems for infinite metric graphs
T. Ekholm, R. L. Frank, and H. Kovarik -- Remarks about Hardy inequalities on metric trees
H. Flechsig and S. Gnutzmann -- On the spectral gap in Andreev graphs
G. Freiling, M. Ignatiev, and V. Yurko -- An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graphs
M. J. Gruber, M. Helm, and I. Veselic -- Optimal Wegner estimates for random Schrodinger operators on metric graphs
V. Kostrykin, J. Potthoff, and R. Schrader -- Contraction semigroups on metric graphs
K. Pankrashkin -- Localization in a quasiperiodic model on quantum graphs
O. Post -- Equilateral quantum graphs and boundary triples
B. Winn -- A conditionally convergent trace formula for quantum graphs
Applications
Review articles
S. Avdonin -- Control problems on quantum graphs
P. Exner -- Leaky quantum graphs: A review
D. Grieser -- Thin tubes in mathematical physics, global analysis and spectral geometry
O. Hul, M. Lawniczak, S. Bauch, and L. Sirko -- Simulation of quantum graphs by microwave networks
D. Krejcirik -- Twisting versus bending in quantum waveguides
Research articles
B. Bellazzini, M. Burrello, M. Mintchev, and P. Sorba -- Quantum field theory on star graphs
H. D. Cornean, P. Duclos, and B. Ricaud -- On the skeleton method and an application to a quantum scissor
S. A. Fulling and J. H. Wilson -- Vacuum energy and closed orbits in quantum graphs
P. Schapotschnikow and S. Gnutzmann -- Spectra of graphs and semi-conducting polymers
Graduate Studies in Mathematics, Volume: 93
2008; approx. 493 pp; hardcover
ISBN-13: 978-0-8218-2003-2
Expected publication date is August 28, 2008.
This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann manifolds, isometric actions, and symplectic and Poisson geometry.
The layout of the material stresses naturality and functoriality from the beginning and is as coordinate-free as possible. Coordinate formulas are always derived as extra information. Some attractive unusual aspects of this book are as follows:
Initial submanifolds and the Frobenius theorem for distributions of nonconstant rank (the Stefan-Sussman theory) are discussed.
Lie groups and their actions are treated early on, including the slice theorem and invariant theory.
De Rham cohomology includes that of compact Lie groups, leading to the study of (nonabelian) extensions of Lie algebras and Lie groups.
The Frolicher-Nijenhuis bracket for tangent bundle valued differential forms is used to express any kind of curvature and second Bianchi identity, even for fiber bundles (without structure groups). Riemann geometry starts with a careful treatment of connections to geodesic structures to sprays to connectors and back to connections, going via the second and third tangent bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point of view.
Symplectic and Poisson geometry emphasizes group actions, momentum mappings, and reductions.
This book gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good knowledge of undergraduate analysis and linear algebra.
Graduate students, research mathematicians and physicists interested in differential geometry, mechanics, and relativity.
Manifolds and vector fields
Lie groups and group actions
Differential forms and de Rham cohomology
Bundles and connections
Riemann manifolds
Isometric group actions or Riemann G-manifolds
Symplectic and Poisson geometry
List of symbols
Bibliography
Index