Model Theory of Valued Differential Fields

微分方程式表示函数与其变化率之间的关系，被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令人感兴趣：它们是稳定的还是趋于无穷大？它们长得有多快？有值微分域是抽象地形式化这些问题和可能的答案的数学结构。这个项目将从模型理论的角度研究有价值的微分域，模型理论是数理逻辑的一个分支，它涉及研究可以用形式语言表达的数学结构的性质，进而揭示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数学的其他领域得到了卓有成效的应用，这一趋势将在有价值的微分领域的设置中继续下去。该项目包括对本科生的培训和指导。有值微分场是将微分方程解的渐近比较形式化的代数结构。这个项目旨在从两个主要方面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场，最雄心勃勃的复杂跨系列，它的研究一直比有序值微分场少得多。这些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发挥着重要作用，从量子物理中的水波到波函数。这个项目将寻找渐近微分代数中允许振荡的泛域，并研究它们的复杂性。其次，该项目将研究某些有序的值微分场，这些场可以由跨系列的跨指数扩展而产生。然后，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，称为想象，与这种有序的有价值的微分场相关。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.