Theory and Applications of Exponential Asymptotics

Costin将致力于构建可分析函数的理论。通过对函数运算（包括除法、复合、微分、积分等代数运算）取解析函数的闭包而得到，它们保留了解析函数的许多性质，包括通过展开式的唯一表示（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之外，还有一个主要的实际兴趣是从广泛的一类方程的形式展开中重建实际解。完备的可分析函数理论可能对纯分析和应用分析的许多分支产生影响。Costin对近似于一般奇点的线性或非线性微分系统的解严格地进行了这种构造。Costin将运用新技术对非线性微分系统在不规则奇点附近的分类，直至解析等价，提供关于非线性系统奇点的性质和位置的尖锐信息，研究方程的封闭形式的可解性，解决连接问题，证明和改进超渐近数值方法。该技术还将扩展到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，感兴趣的量受到参数的影响，这些参数太小，无法用经典渐近法控制。指数渐近已经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。本项目旨在为该领域建立一个严谨的基础，将其技术应用于微分或差分系统的研究，以及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.