CRM Monograph Series, Volume: 28
2009; 127 pp; hardcover
ISBN-13: 978-0-8218-4878-4
Expected publication date is October 31, 2009.
Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward.
In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD.
The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the use of the GCD promises to be efficient.
Clay Mathematics Proceedings, Volume: 8
2009; 562 pp; softcover
ISBN-13: 978-0-8218-4476-2
Expected publication date is November 14, 2009.
This book is based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen. Intended for graduate students and recent Ph.D.'s, this volume will introduce readers to modern techniques and outstanding conjectures at the interface of number theory and algebraic geometry.
The main focus is rational points on algebraic varieties over non-algebraically closed fields. Do they exist? If not, can this be proven efficiently and algorithmically? When rational points do exist, are they finite in number and can they be found effectively? When there are infinitely many rational points, how are they distributed?
For curves, a cohesive theory addressing these questions has emerged in the last few decades. Highlights include Faltings' finiteness theorem and Wiles's proof of Fermat's Last Theorem. Key techniques are drawn from the theory of elliptic curves, including modular curves and parametrizations, Heegner points, and heights.
The arithmetic of higher-dimensional varieties is equally rich, offering a complex interplay of techniques including Shimura varieties, the minimal model program, moduli spaces of curves and maps, deformation theory, Galois cohomology, harmonic analysis, and automorphic functions. However, many foundational questions about the structure of rational points remain open, and research tends to focus on properties of specific classes of varieties.
N. Elkies -- About the cover: Rational curves on a K3 surface
Curves
H. Darmon -- Rational points on curves
H. Chapdelaine -- Non-abelian descent and the generalized Fermat equation
M. Rebolledo -- Merel's theorem on the boundedness of the torsion of elliptic curves
P. Charollois -- Generalized Fermat equations (d'apres Halberstadt-Kraus)
S. Dasgupta and J. Voight -- Heegner points and Sylvester's conjecture
J. Voight -- Shimura curve computations
M. Greenberg -- Computing Heegner points arising from Shimura curve parametrizations
M. Greenberg -- The arithmetic of elliptic curves over imaginary quadratic fields and Stark-Heegner points
Y. I. Manin -- Lectures on modular symbols
Surfaces
B. Hassett -- Rational surfaces over nonclosed fields
D. Harari -- Non-abelian descent
B. G. Vioreanu -- Mordell-Weil problem for cubic surfaces, numerical evidence
Higher-dimensional varieties
Y. Tschinkel -- Algebraic varieties with many rational points
D. Abramovich -- Birational geometry for number theorists
J. M. Starr -- Arithmetic over function fields
N. Ratazzi and E. Ullmo -- Galois + Equidistribution = Manin-Mumford
E. Ullmo and A. Yafaev -- The Andre-Oort conjecture for products of modular curves
C.-L. Chai and F. Oort -- Moduli of abelian varieties and $p$-divisible groups
D. Kaledin -- Cartier isomorphism and Hodge Theory in the non-commutative case
A publication of the Theta Foundation.
2009; 246 pp; hardcover
ISBN-13: 978-973-87899-5-1
Expected publication date is November 13, 2009.
This volume contains the proceedings of the Potential Theory and Stochastics Conference, which was held in Albac, Romania, from September 4 to 8, 2007. It is also intended as a memorial volume for Aurel Cornea. Besides a presentation of the life and work of Aurel Cornea, it includes twenty refereed papers of the participants, covering the main topics of the conference: geometric aspects in potential theory, Dirichlet structures and stochastic analysis, potential theoretical methods for the analysis of infinite dimensional processes, stochastic partial differential equations, non-linear partial differential equations and potential theory.
D. Bakry, F. Baudoin, M. Bonnefont, and B. Qian -- Subelliptic Li-Yau estimates on three dimensional spaces
V. Barbu, P. Blanchard, G. Da Prato, and M. Rockner -- Self-organized criticality via stochastic partial differential equations
K. Ben Ali and M. Bezzarga -- On a nonhomogenous quasilinear problem in Sobolev spaces with variable exponent
M. Biroli -- $\Gamma$-convergence for strongly local Dirichlet forms in open sets with holes
H.-P. Blatt -- Divergence of rational approximants to non-analytic functions
N. Boboc and G. Bucur -- Non-symmtetric resistance forms
S.-L. Eriksson and H. Leutwiler -- Hyperbolic harmonic functions and their function theory
M. Fukushima -- On extended Dirichlet spaces and the space of BL functions
S. J. Gardiner and T. Sjodin -- Partial balayage and the exterior inverse problem of potential theory
M. Ghergu and V. R?dulescu -- The influence of the distance function in some singular elliptic problems
K. GowriSankaran -- A Fatou type theorem for multiply superharmonic functions
Z.-C. Hu, Z.-M. Ma, and W. Sun -- Some remarks on representations of non-symmetric local Dirichlet forms
N. Jacob, A. Potrykus, and M. Schicks -- Operators associated with multi-parameter families of probability measures
K. Kuwae -- Jensen's inequality over $\mathrm{CAT}(\kappa)$-space with small diameter
P. A. Loeb -- Rich measure spaces
Y. Mizuta, T. Ohno, and T. Shimomura -- Integrability of maximal functions for generalized Lebesque spaces $L{^p}^{(\cdot)}(\mathrm{log}\thinspace L){^q}^{(\cdot)}$
M. N. Pascu -- Probabilistic approaches to monotonicity and maximum principles
E. Popa -- Polyinvariant elements in a semi-dynamical system
W. Stannat -- Lipschitz continuity of the pseudo resolvent of the stochastic Burgers equation
C. Udrea -- Excessive functions with respect to Monge-Ampere resolvents and concave functions
History of Mathematics, Volume: 35
2009; approx. 196 pp; hardcover
ISBN-13: 978-0-8218-4899-9
Expected publication date is December 17, 2009.
The name Central Limit Theorem covers a wide variety of results involving the determination of necessary and sufficient conditions under which sums of independent random variables, suitably standardized, have cumulative distribution functions close to the Gaussian distribution. As the name Central Limit Theorem suggests, it is a centerpiece of probability theory which also carries over to statistics.
Part One of The Life and Times of the Central Limit Theorem, Second Edition traces its fascinating history from seeds sown by Jacob Bernoulli to use of integrals of $\exp (x^2)$ as an approximation tool, the development of the theory of errors of observation, problems in mathematical astronomy, the emergence of the hypothesis of elementary errors, the fundamental work of Laplace, and the emergence of an abstract Central Limit Theorem through the work of Chebyshev, Markov and Lyapunov. This closes the classical period of the life of the Central Limit Theorem, 1713-1901.
The second part of the book includes papers by Feller and Le Cam, as well as comments by Doob, Trotter, and Pollard, describing the modern history of the Central Limit Theorem (1920-1937), in particular through contributions of Lindeberg, Cramer, Levy, and Feller.
The Appendix to the book contains four fundamental papers by Lyapunov on the Central Limit Theorem, made available in English for the first time.
Early life and middle years
A seed is sown
Approximation by integrals of $e^{-x^{2}}$
Impetus provided by the theory of errors of observation
Impetus provided by mathematical astronomy
The flowering of the central limit theorem begins
The development of the hypothesis of elementary errors
The emergence of an abstract central limit theorem
Chebyshev's pupils: A. A. Markov and A. M. Lyapunov
Bibliography
The modern era
W. Feller, The fundamental limit theorems in probability
L. Le Cam, The central limit theorem around 1935
H. F. Trotter, J. L. Doob, David Pollard, and L. Le Cam, Comments and rejoinder
Appendix
A. M. Lyapunov, On a theorem in probability theory
A. M. Lyapunov, On a theorem in probability theory
A. M. Lyapunov, A general proposition in probability theory
A. M. Lyapunov, A new form of a theorem on the limit of a probability
Index
Contemporary Mathematics, Volume: 498
2009; 232 pp; softcover
ISBN-13: 978-0-8218-4628-5
Expected publication date is November 1, 2009.
This volume represents the proceedings of the conference on Foliations, Geometry, and Topology, held August 6-10, 2007, in Rio de Janeiro, Brazil, in honor of the 70th birthday of Paul Schweitzer. The papers concentrate on the theory of foliations and related areas such as dynamical systems, group actions on low dimensional manifolds, and geometry of hypersurfaces.
There are survey papers on classification of foliations and their dynamical properties, including codimension one foliations with Bott-Morse singularities. Other papers involve the relationship of foliations with characteristic classes, contact structures, and Eliashberg-Mishachev wrinkled mappings.
S. Hurder -- Classifying foliations
T. Tsuboi -- Classifying spaces for groupoid structures
B. Scardua and J. Seade -- Compact foliations with Bott-Morse singularities
G. Hector and M. A. Chaouch -- Dynamiques Source-Puits et Flots transversalement affines
J. L. Arraut and C. Maquera -- Structurally stable singular actions of $\mathbb{R}^{2}$ having a first integral
S. Matsumoto -- The parameter rigid flows on orientable 3-manifolds
D. Calegari -- The Euler class of planar groups
Y. Matsuda -- Global fixed points for groups of homeomorphisms of the circle
P. G. Walczak -- Orthogonal total foliations: Godbillon-Vey forms via local conformal invariants
F. J. Andrade, J. L. M. Barbosa, and J. H. S. de Lira -- Prescribed mean curvature hypersurfaces in warped products
H. Kodama, Y. Mitsumatsu, S. Miyoshi, and A. Mori -- On Thurston's inequality for spinnable foliations
S. Miyoshi and A. Mori -- Reeb components and Thurston's inequality
Y. M. Eliashberg and N. M. Mishachev -- Wrinkled embeddings