Graduate Studies in Mathematics, Volume: 153
2014; 338 pp; hardcover
ISBN-13: 978-1-4704-1564-8
Expected publication date is August 8, 2014.
In the fifth of his famous list of 23 problems, Hilbert asked if every topological group which was locally Euclidean was in fact a Lie group. Through the work of Gleason, Montgomery-Zippin, Yamabe, and others, this question was solved affirmatively; more generally, a satisfactory description of the (mesoscopic) structure of locally compact groups was established. Subsequently, this structure theory was used to prove Gromov's theorem on groups of polynomial growth, and more recently in the work of Hrushovski, Breuillard, Green, and the author on the structure of approximate groups.
In this graduate text, all of this material is presented in a unified manner, starting with the analytic structural theory of real Lie groups and Lie algebras (emphasising the role of one-parameter groups and the Baker-Campbell-Hausdorff formula), then presenting a proof of the Gleason-Yamabe structure theorem for locally compact groups (emphasising the role of Gleason metrics), from which the solution to Hilbert's fifth problem follows as a corollary. After reviewing some model-theoretic preliminaries (most notably the theory of ultraproducts), the combinatorial applications of the Gleason-Yamabe theorem to approximate groups and groups of polynomial growth are then given. A large number of relevant exercises and other supplementary material are also provided.
Graduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups.
Contemporary Mathematics, Volume: 620
2014; 328 pp; softcover
ISBN-13: 978-0-8218-9474-3
Expected publication date is August 4, 2014.
This volume contains the proceedings of the Stanford Symposium on Algebraic Topology: Applications and New Directions, held from July 23-27, 2012, at Stanford University, Stanford, California. The symposium was held in honor of Gunnar Carlsson, Ralph Cohen and Ib Madsen, who celebrated their 60th and 70th birthdays that year. It showcased current research in Algebraic Topology reflecting the celebrants' broad interests and profound influence on the subject.
The topics varied broadly from stable equivariant homotopy theory to persistent homology and application in data analysis, covering topological aspects of quantum physics such as string topology and geometric quantization, examining homology stability in algebraic and geometric contexts, including algebraic K-theory and the theory of operads.
Graduate students and research mathematicians interested in algebraic topology and its applications.}
T. Church, J. S. Ellenberg, and B. Farb -- Representation stability in cohomology and asymptotics for families of varieties over finite fields
T. Church, B. Farb, and A. Putman -- A stability conjecture for the unstable cohomology of SLnZ, mapping class groups, and Aut(Fn)
W. Dwyer and K. Hess -- The Boardman-Vogt tensor product of operadic bimodules
S. Galatius and O. Randal-Williams -- Detecting and realising characteristic classes of manifold bundles
B. Goldfarb and T. K. Lance -- Controlled algebraic G-theory, II
I. Hambleton and E. K. Pedersen -- More examples of discrete co-compact group actions
L. Hesselholt -- On the K-theory of planar cuspical curves and a new family of polytopes
M. A. Hill and M. J. Hopkins -- Equivariant multiplicative closure
M. Kahle -- Topology of random simplicial complexes: A survey
L. Katzarkov, E. Lupercio, L. Meersseman, and A. Verjovsky -- The definition of a non-commutative toric variety
N. Kitchloo -- The stable symplectic category and quantization
G. Segal -- A geometric perspective on quantum field theory
M. Vejdemo-Johansson -- Sketches of a platypus: A survey of persistent homology and its algebraic foundations
K. Wickelgren -- Cartier's first theorem for Witt vectors on Zn?0?0
CBMS Regional Conference Series in Mathematics, Number: 120
2014; 69 pp; softcover
ISBN-13: 978-1-4704-1034-6
Expected publication date is September 1, 2014.
Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables.
The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics.
Graduate students and research mathematicians interested in fractal geometry and ergodic theory.
Introduction to fractals
Dimension
Trees and fractals
Invariant sets
Probability trees
Galleries
Probability trees revisited
Elements of ergodic theory
Galleries of trees
General remarks on Markov systems
Markov operator T and measure preserving transformation T
Probability trees and galleries
Ergodic theorem and the proof of the main theorem
An application: The k-lane property
Dimension and energy
Dimension conservation
Ergodic theorem for sequences of functions
Dimension conservation for homogeneous fractals: The main steps in the proof
Verifying the conditions of the ergodic theorem for sequences of functions
Bibliography
Index
Graduate Studies in Mathematics, Volume: 155
2014; approx. 437 pp; hardcover
ISBN-13: 978-1-4704-0966-1
Expected publication date is September 22, 2014.
Representation theory is an important part of modern mathematics, not only as a subject in its own right but also as a tool for many applications. It provides a means for exploiting symmetry, making it particularly useful in number theory, algebraic geometry, and differential geometry, as well as classical and modern physics.
The goal of this book is to present, in a motivated manner, the basic formalism of representation theory as well as some important applications. The style is intended to allow the reader to gain access to the insights and ideas of representation theory--not only to verify that a certain result is true, but also to explain why it is important and why the proof is natural.
The presentation emphasizes the fact that the ideas of representation theory appear, sometimes in slightly different ways, in many contexts. Thus the book discusses in some detail the fundamental notions of representation theory for arbitrary groups. It then considers the special case of complex representations of finite groups and discusses the representations of compact groups, in both cases with some important applications. There is a short introduction to algebraic groups as well as an introduction to unitary representations of some noncompact groups.
The text includes many exercises and examples.
Graduate students and research mathematicians interested in representation theory and its applications throughout mathematics.
Introduction and motivation
The language of representation theory
Variants
Linear representations of finite groups
Abstract representation theory of compact groups
Applications of representations of compact groups
Other groups: a few examples
Some useful facts
Bibliography
Index
American Mathematical Society Translations--Series 2, Volume: 233
Advances in the Mathematical Sciences
2014; approx. 252 pp; hardcover
ISBN-13: 978-1-4704-1683-6
Expected publication date is September 29, 2014.
This volume is dedicated to V. A. Marchenko on the occasion of his 90th birthday. It contains refereed original papers and survey articles written by his colleagues and former students of international stature and focuses on the areas to which he made important contributions: spectral theory of differential and difference operators and related topics of mathematical physics, including inverse problems of spectral theory, homogenization theory, and the theory of integrable systems. The papers in the volume provide a comprehensive account of many of the most significant recent developments in that broad spectrum of areas.
Research mathematicians interested in spectral theory and differential operators and their applications.
V. M. Adamyan, G. J. Martin, and B. S. Pavlov -- Local inverse scattering problem as a tool of perturbation analysis for resonance systems
H.-D. Alber -- The continuous theory of dislocations for a material containing dislocations to one Burgers vector only
A. I. Aptekarev -- Spectral problems of high order recurrences
A. Boutet de Monvel, I. Loutsenko, and O. Yermolayeva -- On the multifractal spectrum of the whole-plane Levy-Loewner evolution
A. Boutet de Monvel and L. Zielinski -- Asymptotic behavior of large eigenvalues of a modified Jaynes-Cummings model
S. Yu. Dobrokhotov, G. Makrakis, and V. E. Nazaikinskii -- Fourier integrals and a new representation of Maslov's canonical operator near caustics
J. Eckhardt, A. Kostenko, and G. Teschl -- Inverse uniqueness results for one-dimensional weighted Dirac operators
A. Eremenko and A. Gabrielov -- Spectral loci of Sturm-Liouville operators with polynomial potentials
O. Guedon, A. Lytova, A. Pajor, and L. Pastur -- The central limit theorem for linear eigenvalue statistics of the sum of independent matrices of rank one
T. Kappeler, P. Lohrmann, and P. Topalov -- On the spectrum of nonself-adjoint Zakharov-Shabat operators on R
E. Y. Khruslov, M. V. Goncharenko, and N. K. Radyakin -- Homogenized model of oscillations of elastic medium with small caverns filled with viscous incompressible fluid
V. Y. Novokshenov and A. A. Shchelkonogov -- Double scaling limit in Painleve IV equation and asymptotics of the Okamoto polynomials
A. M. Savchuk and A. A. Shkalikov -- Recovering of a potential of the Sturm-Liouville problem from finite sets of spectral data
R. Weder -- High-velocity estimates, inverse scattering and topological effects