Theory and Applications of Exponential Asymptotics

科斯廷·科斯廷将致力于构建一种可分析函数理论。通过取关于函数运算(包括除法、合成、微分、积分等代数运算)的解析函数的闭包得到的。它们保持了许多解析函数的性质，包括唯一的展开式表示(形式化)和关系的持久性。除了探索可形式化极限的内在兴趣外，还有一个主要的实际兴趣是从一大类方程的形式展开重建实际解。一个完整的可分析函数理论可能会对纯分析和应用分析的许多分支产生影响。Costin对一般奇点附近的线性或非线性微分系统的解所产生的函数严格地进行了这种构造。Costin将使用新的技术对非规则奇点附近的非线性微分系统进行分类，直到解析等价，在提供关于非线性系统奇点的性质和位置的尖锐信息，在研究封闭形式的方程的可解性，在解决连接问题以及在证明和改进超渐近数值方法方面。该技术还将扩展到包括偏微分方程组和差分方程组。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，感兴趣的量受到太小的参数的影响，而不是经典渐近性所能控制的。指数渐近作为处理这些情况的一个成功的理论和计算框架已经被引入，并被应用于从数论到光学和量子力学的各种问题。本项目旨在为这一领域奠定坚实的基础，将其技术应用于微分或差分系统的研究，并扩展和改进这一新理论的计算方法。

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.