Model Theory of Valued Differential Fields

微分方程表达了一个函数和它的变化率之间的关系，用来理解科学和工程中的许多现象。通常，这些函数的最终行为是令人感兴趣的：它们是趋于稳定还是趋于无穷？它们的生长速度有多快？有值微分域是抽象形式化这些问题和可能答案的数学结构。该项目将从模型论的角度研究有值微分域，模型论是数学逻辑的一个分支，涉及研究可以用形式语言表达的数学结构的性质，从而揭示对这些结构及其复杂性的新见解。近年来，模型理论的思想和工具在其他数学领域取得了丰硕的应用，本项目将在有值微分领域的设置中继续这一趋势。该项目包括对本科生的培训和指导。有值微分域是将微分方程解的渐近比较形式化的代数结构。本项目旨在通过两种主要方式加深我们对这种结构及其模型理论的理解。首先，它将研究无序值微分域，其中大多数是极其复杂的跨列，它们比有序值微分域研究得少得多。后一种结构只描述非振荡行为，而振荡函数在数学和描述现实世界中起着重要作用，从水波到量子物理中的波函数。本计画将寻找允许振荡的渐近微分代数的一般定义域，并研究其复杂性。其次，本课题将研究由横列的转幂扩展产生的某些有序值微分域。接下来的工作将自然而然地进行，最重要的是研究与这些有序值微分场相关的某些商结构，称为虚数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Differential equations, which express a relationship between a function and its rates of change, are used to understand many phenomena in science and engineering. Often, the eventual behavior of these functions is of interest: Do they stabilize or tend to infinity? How fast do they grow? Valued differential fields are mathematical structures that abstractly formalize these questions and possible answers. This project will investigate valued differential fields from the perspective of model theory, a branch of mathematical logic which involves studying properties of mathematical structures that can be expressed in formal languages, in turn revealing new insights into these structures and their complexity. In recent years the ideas and tools of model theory have had fruitful applications in other areas of mathematics, a trend this project will continue in the setting of valued differential fields. The project includes the training and mentoring of undergraduate students. Valued differential fields are algebraic structures that formalize the asymptotic comparison of solutions to differential equations. This project aims to deepen our understanding of such structures and their model theory in two primary ways. First, it will investigate unordered valued differential fields, most ambitiously complex transseries, which have been much less studied than ordered valued differential fields. These latter structures describe only non-oscillating behavior, while oscillating functions play an important role in mathematics and in describing the real world, from water waves to wave functions in quantum physics. This project will search for universal domains for the kinds of asymptotic differential algebra in which oscillation is permitted and study their complexity. Second, the project will investigate certain ordered valued differential fields that can arise from transexponential extensions of transseries. The work will then address natural next steps, most significantly the study of certain quotient structures, called imaginaries, associated to such ordered valued differential fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.