Model theory, diophantine geometry and asymptotic analysis

该项目的中心目标是通过开发和利用数学逻辑的一个分支模型理论的基本工具和观点，在解决一系列不同的数学问题方面取得重大进展。模型理论的一个重要方面是从形式的观点研究数学对象的基本结构，然后利用这一点来获得新的方法来解决其他数学领域出现的问题，如代数、几何、算术、微分方程、图论，甚至更远的领域，如经济学、机器学习或工程学。这个项目的重点是开发某些模型理论工具在某些领域的最新应用，如丢番图几何(通过代数和几何方法求解整数方程的研究)和渐近分析(研究函数在其输入变大时的增长行为)，以便通过探索并从它们之间的相互作用中获益，在这些不同的领域获得新的见解，同时也为我们理解基本的模型理论框架本身做出基本贡献。该项目包括对本科生和研究生的培训，包括那些通过数学联盟来自代表性不足群体的学生。这个项目建议解决的问题的普遍特征是它们与模型理论中称为o-最小性的部分的联系，o-最小性是半代数和次解析几何的常见推广，其中所考虑的数学对象不表现出狂野的几何行为(例如，没有振荡，族中的有限界是一致的，以及函数的良好的渐近增长行为)。这个项目的一个主要焦点是Pila和Wilkie的计数定理，这是o-极小在丢番图几何中的开创性应用，以及这个结果的后续改进和应用。一个关键的目标是对这一领域中使用的主要解析几何工具之一--光滑参数化--有新的见解。另一种是通过开发有效的方法，进一步改进这些计数结果或使其得到新的应用。这个项目还将使用o-极小的技巧来解决丢番图和渐近分析中的问题，通过进一步了解o-极小环境下函数的渐近增长行为，并对这种分析的更广泛的适用性提出批判性的新见解。这个奖项反映了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。

The central goal of this project is to make significant progress on solving a diverse array of mathematical problems by developing and exploiting foundational tools and perspectives from model theory, a branch of mathematical logic. An important aspect of model theory is to study the underlying structure of mathematical objects from a formal viewpoint, and then to utilize this to obtain novel approaches to questions emerging from other areas of mathematics, such as algebra, geometry, arithmetic, differential equations, graph theory, or even beyond, such as from economics, machine learning or engineering. The particular focus of this project is on developing recent applications of certain model-theoretic tools to areas such as diophantine geometry (the study of solving equations in integers by means of algebra and geometry) and asymptotic analysis (the study of the growth behaviour of functions as their inputs become large), in order to obtain new insights in these different fields by exploring and profiting from the interplay between them, while also making fundamental contributions to our understanding of the foundational model-theoretic framework itself. The project includes the training of undergraduate and graduate students, including those from underrepresented groups through the Math Alliance. The prevalent feature of the problems that this project proposes to address is their connection to the part of model theory known as o-minimality, a common generalization of semi-algebraic and subanalytic geometries, where the mathematical objects under consideration do not exhibit wild geometric behaviour (such as no oscillation, finiteness bounds being uniform in families, and good asymptotic growth behaviour of functions). A principal focus of this project is on the counting theorem of Pila and Wilkie, a seminal application of o-minimality to diophantine geometry, as well as on subsequent refinements and applications of this result. One key aim is to gain new insights into one of the main analytic-geometric tools used in this area, namely smooth parameterization. Another is to obtain further improvements to or new applications of these counting results, in particular by exploiting effective methods. This project moreover will use techniques from o-minimality to address questions in diophantine and asymptotic analysis, by furthering our understanding of the asymptotic growth behaviour of functions in the o-minimal setting, and making critical new insights into the wider applicability of this analysis.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.