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Divergent Series, Summability and Resurgence I
Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view. The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations. The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra.
Divergent Series, Summability and Resurgence I-III
This three-volume work treats divergent series in one variable, especially those arising as solutions to complex ordinary differential or difference equations, and methods for extracting their analytic information. It provides a systematic construction, illustrated with examples, of the various theories of summability and the theory of resurgence developed since the 1980s. The Stokes phenomenon, for both linear and non-linear equations, plays an underlying and unifying role throughout the volumes. Applications presented include resurgent analyses of the First Painlevé equation and of the tangent-to-identity germs of diffeomorphisms of C, and links to differential Galois theory and the Riema...
Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation
These are the proceedings of a one-week international conference centered on asymptotic analysis and its applications. They contain major contributions dealing with - mathematical physics: PT symmetry, perturbative quantum field theory, WKB analysis, - local dynamics: parabolic systems, small denominator questions, - new aspects in mould calculus, with related combinatorial Hopf algebras and application to multizeta values, - a new family of resurgent functions related to knot theory.
Quasianalytic Monogenic Solutions of a Cohomological Equation
We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point.
Resurgence, Physics and Numbers
This book is issued from a conference around resurgent functions in Physics and multiple zetavalues, which was held at the Centro di Ricerca Matematica Ennio de Giorgi in Pisa, on May 18-22, 2015. This meeting originally stemmed from the impressive upsurge of interest for Jean Ecalle's alien calculus in Physics, in the last years – a trend that has considerably developed since then. The volume contains both original research papers and surveys, by leading experts in the field, reflecting the themes that were tackled at this event: Stokes phenomenon and resurgence, in various mathematical and physical contexts but also related constructions in algebraic combinatorics and results concerning numbers, specifically multiple zetavalues.
On the Splitting of Invariant Manifolds in Multidimensional Near-Integrable Hamiltonian Systems
Presents the problem of the splitting of invariant manifolds in multidimensional Hamiltonian systems, stressing the canonical features of the problem. This book offers introduction of a canonically invariant scheme for the computation of the splitting matrix.
Positive Definite Functions on Infinite-Dimensional Convex Cones
A memoir that studies positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces. It studies representations of convex cones by positive operators on Hilbert spaces. It also studies the interplay between positive definite functions and representations of convex cones.
Moduli Spaces of Polynomials in Two Variables
Investigates the geometry of the orbit space. This book associates a graph with each polynomial in two variables that encodes part of its geometric properties at infinity. It also defines a partition of $\mathbb{C} x, y]$ imposing that the polynomials in the same stratum are the polynomials with a fixed associated graph
Dynamics of Topologically Generic Homeomorphisms
The goal of this work is to describe the dynamics of generic homeomorphisms of certain compact metric spaces $X$. Here ``generic'' is used in the topological sense -- a property of homeomorphisms on $X$ is generic if the set of homeomorphisms with the property contains a residual subset (in the sense of Baire category) of the space of all homeomorphisms on $X$. The spaces $X$ we consider are those with enough local homogeneity to allow certain localized perturbations of homeomorphisms; for example, any compact manifold is such a space. We show that the dynamics of a generic homeomorphism is quite complicated, with a number of distinct dynamical behaviors coexisting (some resemble subshifts o...
On Dynamical Poisson Groupoids I
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of (CDYBE) on simple Lie algebras as classified by the same authors.
