On Numbers, Germs, and Series

这个项目应用代数和模型论工具来研究函数的增长率。代数是最古老的数学分支之一，而模型理论是数理逻辑的一个分支，数理逻辑是一门相当新的学科，起源于19世纪末，对数学基础进行了哲学研究。然而，近几十年来，逻辑在数学的其他部分，在计算机科学，甚至在工程中都有许多应用。大约15年来，PI一直参与开发渐近分析的模型理论处理的合作努力。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结果，许多以前看起来遥不可及的问题现在可能得到回答。更具体地说，这个项目的目标是统一三种看起来非常不同的方法，通过无穷小和无限的量来丰富真实的连续体：超实数、实值函数芽和跨序列。超现实数有一种组合的味道，包含了康托的序数；它们是由J·H·康威在20世纪70年代与博弈论联系起来引入的。实值函数芽是分析的中心对象；19世纪70年代，P.Du Bois-Reymond首先系统地研究了实值函数芽。跨系列是形式化的对象，它模拟了这种细菌在无限大的生长速度；它们出现在20世纪80年代的分析和逻辑中。它们的形式化性质也使TransSeries适合于计算机代数系统中的机器计算。这项提议的目的是加深我们对这些结构的理解，并在它们之间建立联系。例如，我们想知道：有没有分析结构(哈代场)具有与跨系列相同的逻辑特征？超现实主义和跨系列之间有天然的同构吗？在过去的十年里，由于我们对数、级数和函数的概念的理解取得了根本性的进步，这些问题的答案现在可能是可以掌握的，并将展示它们之间迄今未知的关系。

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.