On Numbers, Germs, and Series

这个项目应用代数和模型理论工具来研究函数的增长率。代数是数学最古老的分支之一，而模型论是数理逻辑的一个分支，一个相当新的学科，起源于世纪后期对数学基础的哲学研究。然而，近几十年来，逻辑在数学的其他部分，计算机科学，甚至工程学中也有许多应用。大约15年来，PI一直致力于开发渐近分析的模型理论处理方法。这些研究自然与数学中的其他领域（主要是逻辑和分析）有关，但也可能导致微分方程在科学和工程中的新应用。他们最近导致了一些决定性的积极成果，许多问题，似乎遥不可及以前现在可能是answered.More具体地说，这个项目的目标是统一三个看似非常不同的方法来丰富的真实的连续无穷小和无限数量：超真实的数，芽实值函数，和transseries。超实数有一种组合的味道，包括康托的序数，它们是由J。康威在1970年代与博弈论有关。实值函数的芽是分析中的中心对象;它们首先由P. du Bois-Reymond在19世纪70年代系统地研究。Transseries是一种形式化的对象，它模拟了这种细菌在无穷远处的增长率;它们在20世纪80年代出现在分析和逻辑中。它们的形式性质也使得transseries适合于计算机代数系统中的机器计算。本建议的目的是加深我们对这些结构的了解，并在它们之间建立联系。例如，我们想知道：是否存在与transseries具有相同逻辑特征的解析结构（哈代域）？超实数和跨级数之间是否存在自然同构？由于在过去十年中，我们对数、级数和函数的概念的理解取得了根本性的进展，这些问题的答案现在可能已经触手可及，并且将展示出它们之间迄今为止未知的关系。

This project applies algebraic and model-theoretic tools to study the growth rates of functions. Algebra is one of the oldest branches of mathematics, whereas model theory is a branch of mathematical logic, a fairly new subject that originated in the late 19th century with philosophical investigations into the foundations of mathematics. However, in recent decades logic has found many applications in other parts of mathematics, in computer science, and even in engineering. For about fifteen years, the PI has been involved in a collaborative effort to develop a model-theoretic treatment of asymptotic analysis. These investigations are naturally related to other fields within mathematics (mainly logic and analysis) but may also lead to novel applications of differential equations in science and engineering. They recently led to some decisive positive results, and many questions which seemed out of reach previously may now be answerable.More concretely, the goal of this project is to unify three seemingly very different approaches to enrich the real continuum by infinitesimal and infinite quantities: surreal numbers, germs of real-valued functions, and transseries. Surreal numbers have a combinatorial flavor and encompass Cantor's ordinal numbers; they were introduced by J. H. Conway in the 1970s in connection with game theory. Germs of real-valued functions are central objects in analysis; they were first systematically studied by P. du Bois-Reymond in the 1870s. Transseries are formal objects that model the growth rates of such germs at infinity; they arose in both analysis and logic during the 1980s. Their formal nature also makes transseries suitable for machine computations in computer algebra systems. The goal of this proposal is to deepen our understanding of these structures, and to establish links between them. For example, we would like to know: Are there analytic structures (Hardy fields) with the same logical features as transseries? Is there a natural isomorphism between surreals and transseries? Answers to questions such as these may now be within grasp due to fundamental advances in our understanding of the ideas of number, series, and function within the last decade, and would exhibit heretofore unknown relationships between them.