Theory and Applications of Exponential Asymptotics

摘要 Costin Costin将致力于构建可分析函数的理论。 通过取解析函数关于 函数运算（代数运算，包括除法， 组成、分化、整合等）它们保留了许多 解析函数属性，包括通过 关系的扩展（形式化）和持久性。除了对探索形式化极限的内在兴趣之外， 有一个主要的实际利益，在重建实际的 从形式展开的解决方案，为广泛的一类方程。 一个完整的可分析函数理论很可能有一个 影响了纯分析和应用分析的许多分支。 Costin严格执行了这一功能建设 作为线性或非线性微分系统的解出现 类属奇点附近Costin将使用新技术， 非线性非正则微分系统的分类 奇点，直到解析等价，在提供尖锐的 关于非线性系统奇异性的性质和位置的信息 系统，在封闭形式的方程的可解性的研究，在解决连接问题，并在证明和改善 超渐近数值方法 该技术还将 扩展到涵盖偏微分方程和差分方程。 渐近分析是研究复杂系统的一种重要方法。然而，在许多重要的情况下， 感兴趣的量受太小的参数的影响 被经典渐近控制。指数渐近领域已经被引入作为一个成功的理论和计算框架来处理这些情况，并应用于从数论到光学和量子力学的问题。本项目的目的是严格的基础领域，在应用其技术的研究微分或差分系统，以及在扩展和改进的计算方法，这一新的理论。

ABSTRACT Costin Costin will work on constructing a theory of analyzable functions. Obtained by taking a closure of analytic functions with respect to function operations (algebraic operations, including division, composition, differentiation, integration, etc.) they preserve many analytic function properties, including unique representation by expansions (formalizability) and permanence of relations. In addition to the intrinsic interest in exploring the limits of formalizability, there is a major practical interest in reconstructing actual solutions from formal expansions for a wide class of equations. A complete theory of analyzable functions is likely to have an impact in many branches of pure and applied analysis. Costin has carried out this construction rigorously for functions arising as solutions to linear or nonlinear differential systems near generic singularities. Costin will use new techniques in the classification of nonlinear differential systems near irregular singularities, up to analytic equivalence, in providing sharp information on the nature and location of singularities of nonlinear systems, in the study of solvability in closed form of equations, in solving connection problems and in justifying and improving hyper-asymptotic numerical methods. The technique will also be extended to cover PDE's and difference equations. Asymptotic analysis is an important method of investigation in complex systems. In many important cases however, the quantities of interest are influenced by parameters too small to be controlled by classical asymptotics. The field of exponential asymptotics has been introduced as a successful theoretical and computational framework to deal with these situations and applied in problems ranging from number theory to optics and quantum mechanics. The present project is aimed at a rigorous foundation of the field, at applying its techniques to the study of differential or difference systems, as well as at extending and improving the computational methods of this new theory.