Graduate Studies in Mathematics, Volume: 172; 2016; 461 pp; Hardcover
Print ISBN: 978-0-8218-4841-8
Over the last fifteen years a variety of problems in combinatorics have been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a “stochastic special function theory” for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail.
Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.
After an elementary introduction, the authors give a new presentation of classical results due to Beurling, Kahane, and Landau. The main part of the book focuses on recent progress in the area, such as construction of universal sampling sets, high-dimensional and non-analytic phenomena.
The reader will see how methods of harmonic and complex analysis interplay with various important concepts in different areas, such as Minkowski's lattice, Kolmogorov's width, and Meyer's quasicrystals.
The book is addressed to graduate students and researchers interested in analysis and its applications. Due to its many exercises, mostly given with hints, the book could be useful for undergraduates.
Graduate students and research mathematicians interested in harmonic analysis and signal theory.
Graduate Studies in Mathematics, Volume: 173
2016; 192 pp; Hardcover
Print ISBN: 978-1-4704-2799-3
This is a graduate text in differentiable dynamical systems. It focuses on structural stability and hyperbolicity, a topic that is central to the field. Starting with the basic concepts of dynamical systems, analyzing the historic systems of the Smale horseshoe, Anosov toral automorphisms, and the solenoid attractor, the book develops the hyperbolic theory first for hyperbolic fixed points and then for general hyperbolic sets. The problems of stable manifolds, structural stability, and shadowing property are investigated, which lead to a highlight of the book, the Ω-stability theorem of Smale.
While the content is rather standard, a key objective of the book is to present a thorough treatment for some tough material that has remained an obstacle to teaching and learning the subject matter. The treatment is straightforward and hence could be particularly suitable for self-study.
Graduate students and research mathematicians interested in the hyperbolic theory of dynamical systems.
Cours Specialises, Volume: 23
2016; 284 pp; Hardcover
Print ISBN: 978-2-85629-822-0
For more than two centuries, integral geometry, also called the theory of geometric probabilities, has followed the development of probability, measure theory and geometry.
The first chapter recalls historical results: Buffon's needle and Bertand's paradoxes.
Chapters 2 to 7 and 10 present the basic notions and methods: slice using all affine lines or planes, project on all affine lines and planes, and average. The method is applied to curves in R2 or R3 and surfaces in R3 n fact results in dimensions 2 and 3 extend easily to higher dimensions and to space-forms. The last section of Chapters 7, 8, and 9 are devoted to statements of the form “topology implies some lower bound on the total curvature”. Again the objects are curves or surfaces contained in R3
Ph.D. students, graduate students, and research mathematicians interested in integral geometry.
Memoires de la Societe Mathematique de France
Volume: 144; 2016; 164 pp; Softcover
MSC: Primary 35; 42; 47;
Print ISBN: 978-2-85629-829-9
It was shown recently that solutions of boundary value problems for some second order elliptic equations (or systems) in divergence form with measurable coefficients can be constructed from solutions of generalised Cauchy-Riemann systems, in the spirit of what can be done for the Laplace equation. This involves a first order bisectorial operator of Dirac type on the boundary whose bounded holomorphic functional calculus on L2 is proved by techniques from the solution of the Kato problem, and the system can henceforth be solved by a semigroup for L2 data in a spectral space.
This memoir investigates the properties of this semigroup and, more generally, of the functional calculus on other spaces: Lp near 2 and adapted Hardy spaces otherwise. This yields non-tangential maximal functions estimates and Lusin area estimates for solutions of the boundary value problems.
Graduate students and research mathematicians interested in first order elliptic systems.
Memoires de la Societe Mathematique de France, Volume: 145
2016; 138 pp; Softcover
Print ISBN: 978-2-85629-830-5
The asymptotic behavior of the first eigenvalue of a magnetic Laplacian in the strong field limit and with the Neumann realization in a smooth domain is characterized for dimensions 2 and 3 by model problems inside the domain or on its boundary. In dimension 2, for polygonal domains, a new set of model problems on sectors has to be taken into account.
In this work, the authors consider the class of general corner domains. In dimension 3, they include as particular cases polyhedra and axisymmetric cones. The authors attach model problems not only to each point of the closure of the domain but also to a hierarchy of “tangent substructures” associated with singular chains. They investigate spectral properties of these model problems, namely semicontinuity and the existence of bounded generalized eigenfunctions, and prove estimates for the remainders of their asymptotic formula.
Lower bounds are obtained with the help of an IMS type partition based on adequate two-scale coverings of the corner domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems. A part of the authors' analysis extends to any dimension.
Graduate students and research mathematicians interested in the magnetic Laplacian.