The book's structure reflects the fact that its content is based on a set of lectures delivered by one of the authors at the first Orthonet Summer School in Seville, Spain in 2016. The presentation of the material is self-contained and will be valuable to students and researchers interested in a novel approach to the study of orthogonal polynomials, focusing on their analytic properties.

]]>The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto’s calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given.

The book is suitable for a course at the graduate level, and also includes an appendix with a series of exercises along with their solutions.

]]>The essays cover the large spectrum of topics in which Turaev has been interested, including knot and link invariants, quantum representations, TQFTs, state sum constructions, geometric structures on knot complements, Kleinian groups, geometric group theory and its relationship with 3-manifolds, mapping class groups, operads, mathematical physics, Grothendieck’s program, the philosophy of mathematics, and several other topics.

At the same time, this volume will give an overview of topics that are at the forefront of current research in topology and geometry. Some of the essays are research articles and contain new results, sometimes answering questions that were raised by Turaev. The rest of the essays are surveys that will introduce the reader to some key ideas in the field.

]]>In brief, the topics covered include Friedrichs, Hardy, and Lieb–Thirring inequalities, eigenvalue bounds and asymptotics, Feshbach–Schur maps and perturbation theory, scattering theory and orthogonal polynomials, stability of matter, electron density estimates, Bose–Einstein condensation, Wehrl-type entropy inequalities, Bogoliubov theory, wave packet evolution, heat kernel estimates, homogenization, d-bar problems, Brezis–Nirenberg problems, the nonlinear Schrödinger equation in magnetic fields, classical discriminants, and the two-dimensional Euler–Bardina equations. In addition, Ari’s multifaceted service to the mathematical community is also touched upon.

Altogether the volume presents a collection of research articles which will be of interest to any active scientist working in one of the above mentioned fields.

]]>Evolution is a complex phenomenon driven by various processes, such as mutation and recombination of genetic material, reproduction of individuals, and selection of favourable types. These processes all have intrinsically random elements, which give rise to a wealth of phenomena that cannot be explained by deterministic models. Examples of such effects are the loss of genetic variability due to random reproduction and the emergence of random genealogies.

The collection is centred around the stochastic processes in population genetics and population dynamics. On the one hand, these are individual-based models of predator-prey and of coevolution type, of adaptive dynamics, or of experimental evolution, considered in the usual forward direction of time. They lead to processes describing the evolution of type frequencies, which may then be analysed via suitable limit theorems. On the other hand, one traces the ancestral lines of individuals back into the past; this leads to random genealogies. Beyond the classical concept of Kingman's coalescent, emphasis is on genealogies with multiple mergers and on ancestral structures that take into account selection, recombination, or migration.

The contributions in this volume will be valuable to researchers interested in stochastic processes and their biological applications, or in mathematical population biology.

]]>These lecture notes aim to present a fast-track study of some important topics in classical parts of von Neumann algebra theory that were developed in the 1970s. Starting with Tomita–Takesaki theory, this book covers topics such as the standard form, Connes’ cocycle derivatives, operator-valued weights, type III structure theory and non-commutative integration theory.

The self-contained presentation of the material makes this book useful not only to graduate students and researchers who want to know the fundamentals of von Neumann algebras, but also to interested undergraduates who have a basic knowledge of functional analysis and measure theory.

]]>
This book explains the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition. The discussion is conducted in the language of probability, enabling the theory to be developed under rather general and explicit conditions; for the finer conclusions, Stein's method emerges as the key ingredient. The book is thus of particular interest to graduate students and researchers in both combinatorics and probability theory.]]>

An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. This is a difficult small divisor problem due to complex resonance phenomena between the normal mode frequencies of oscillations. The proof requires various mathematical methods, ranging from Nash–Moser and KAM theory to reduction techniques in Hamiltonian dynamics and multiscale analysis for quasi-periodic linear operators, which are presented in a systematic and self-contained way. Some of the techniques introduced in this monograph have deep connections with those used in Anderson localization theory.

This book will be useful to researchers who are interested in small divisor problems, particularly in the setting of Hamiltonian PDEs, and who wish to get acquainted with recent developments in the field.]]>

This book presents a unified approach to the analysis of accuracy of deterministic mathematical models described by variational problems and partial differential equations of elliptic type. It is based on new mathematical methods developed to estimate the distance between a solution of a boundary value problem and any function in the admissible functional class associated with the problem in question. The theory is presented for a wide class of elliptic variational problems. It is applied to the investigation of modelling errors arising in dimension reduction, homogenization, simplification, and various conversion methods (penalization, linearization, regularization, etc.). A collection of examples illustrates the performance of error estimates.]]>