Model Theory of Valued Differential Fields

微分方程表达了函数与其变化率之间的关系，用于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：它们是稳定还是趋于无穷大？它们长得有多快？值微分域是抽象形式化这些问题和可能答案的数学结构。本项目将从数学逻辑的分支模型论的角度研究有值微分场，该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，模型论的思想和工具在数学的其他领域有着卓有成效的应用，这一趋势将在值微分领域的设置中继续下去。该项目包括培训和指导本科生。值微分域是一种代数结构，它形式化了微分方程解的渐近比较。本项目旨在通过两种主要方式加深我们对此类结构及其模型理论的理解。首先，它将研究无序值微分场，最雄心勃勃的复杂transseries，这已经远远低于有序值微分场的研究。这些后一种结构只描述了非振荡行为，而振荡函数在数学和描述真实的世界中起着重要作用，从水波到量子物理学中的波函数。该项目将寻找允许振荡的各种渐进微分代数的通用域并研究它们的复杂性。其次，该项目将调查某些有序值微分场，可以从transseries的跨指数扩展。这项工作将解决自然的下一步，最重要的是研究某些商结构，称为矩阵，与这种有序值微分field.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.