Graduate Studies in Mathematics, Volume: 168
2015; approx. 320 pp; hardcover
ISBN-13: 978-1-4704-1913-4
Expected publication date is January 25, 2016.
This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of self-adjoint operators through Anderson localization--presented here via the fractional moment method, up to recent results on resonant delocalization.
The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of Herglotz-Pick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results.
The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.
Graduate students and researchers interested in random operator theory.
Introduction
General relations between spectra and dynamics
Ergodic operators and their self-averaging properties
Density of states bounds: Wegner estimate and Lifshitz tails
The relation of Green functions to eigenfunctions
Anderson localization through path expansions
Dynamical localization and fractional moment criteria
Fractional moments from an analytical perspective
Strategies for mapping exponential decay
Localization at high disorder and at extreme energies
Constructive criteria for Anderson localization
Complete localization in one dimension
Diffusion hypothesis and the Green-Kubo-Streda formula
Integer quantum Hall effect
Resonant delocalization
Phase diagrams for regular tree graphs
The eigenvalue point process and a conjectured dichotomy
Elements of spectral theory
Herglotz-Pick functions and their spectra
Bibliography
Index
Graduate Studies in Mathematics, Volume: 169
2015; 281 pp; hardcover
ISBN-13: 978-1-4704-1881-6
Expected publication date is January 3, 2016.
This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses.
The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the features one wants without the overhead of measure theory.
There is additional material provided for readers who would like to learn more than the core material, and there are numerous exercises to help solidify one's understanding. The text should be suitable for advanced undergraduates or for beginning graduate students including those in engineering or the sciences.
Professors, graduate students, and others interested in teaching and learning partial differential equations.
Introduction
Where do PDE come from
First order scalar semilinear equations
First order scalar quasilinear equations
Distributions and weak derivatives
Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation
Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle
The Fourier transform: Basic properties, the inversion formula and the heat equation
The Fourier transform: Tempered distributions, the wave equation and Laplace's equation
PDE and boundaries
Duhamel's principle
Separation of variables
Inner product spaces, symmetric operators, orthogonality
Convergence of the Fourier series and the Poisson formula on disks
Bessel functions
The method of stationary phase
Solvability via duality
Variational problems
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 207
2015; approx. 489 pp; hardcover
ISBN-13: 978-1-4704-2558-6
Expected publication date is January 9, 2016.
This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter.
The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.
Graduate students and researchers interested in partial differential equations and stochastic processes.
Stationary Fokker-Planck-Kolmogorov equations
Existence of solutions
Global properties of densities
Uniqueness problems
Associated semigroups
Parabolic Fokker-Planck-Kolmogorov equations
Global parabolic regularity and upper bounds
Parabolic Harnack inequalities and lower bounds
Uniquess of solutions to Fokker-Planck-Kolmogorov equations
The infinite-dimensional case
Bibliography
Subject index
Mathematical Surveys and Monographs, Volume: 209
2015; 218 pp; hardcover
ISBN-13: 978-1-4704-2545-6
Expected publication date is January 2, 2016.
Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work.
The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. The book can be used as a text for a course on applied topology or data analysis.
Graduate students and researchers interested in algebraic topology and applications to data analysis.