Algebraic Model Theory

代数模型理论

• 批准号: 1001550
• 负责人: Thomas Scanlon
• 金额: $ 36.81万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001550&HistoricalAwards=false
• 关键词: Algebraic; Model; Theory

Scanlon will study the theory of difference and differential equations through the vantage of mathematical logic and to apply the results of these investigations to fundamental problems in arithmetic and geometry.More specifically, he will develop a unified approach to the logical theory of difference and differential fields and to resolve deep structural questions about difference-differential equations through a general theory of linearization via jet and arc spaces giving a precise sense to which all geometric complexity of nonlinear equations actually reflects the complexity of associated linear equations. Scanlon will study Galois theory, or the theory of symmetries, of general systems of equations involving operators by employing this process of linearization, the logical theory of liaison groups, and a Tannakian formalism. In addition, he will address specific questions about algebraic relations on special points by using the fine structure theory for definable sets in difference fields as well as the model theory of valued fields and of real geometry via the theory of o-minimality. Finally, Scanlon will develop the model theory of the theory of the Witt vectors with analytic structure and the relative Frobenius. Scanlon will use this theory both as a proving ground for the general theory of metastability and as a logically tame theory in which arithmetic problems may be encoded believing that this theory may ground a new theory of motivic integration in which finite dimensional difference varieties play the role of algebraic varieties. Scanlon will develop a model theory of p-jets independent of the prime p from which uniformities in number theoretic problems may be deduced.Model theory, in the sense of mathematical logic, gives a unifying perspective for studying questions in disparate branches of mathematics as instances of a general theory. For example, it may ground fanciful, but suggestive, theories in which techniques appropriate to the study of geometry and differential equations are transposed to investigate numbers. In the opposite direction, its very general methods for understanding symmetries of structures and more importantly symmetries of one part of a structure relative to another part when specialized to concrete mathematical questions about differential and difference equations reveal otherwise unknown theories of symmetries. With this project, Scanlon will explore the underlying unity of mathematics as seen from mathematical logic from the theories of differential equations, to dynamical systems, to number theory.

Scanlon将通过数理逻辑的Vantage研究差分和微分方程的理论，并将这些研究的结果应用于算术和几何的基本问题。他将制定一个统一的方法，差异和微分场的逻辑理论，并解决有关差异的深层结构问题-微分方程通过一般理论的线性化，通过射流和弧空间给出了一个精确的意义，所有几何复杂性的非线性方程实际上反映了复杂性的相关线性方程。斯坎伦将研究伽罗瓦理论，或理论的对称性，一般系统的方程涉及运营商采用这一进程的线性化，逻辑理论的联络组，和一个Tannakian形式主义。 此外，他将解决具体问题的代数关系的特殊点，利用精细结构理论的可定义集在不同领域以及模型理论的价值领域和真实的几何通过理论的o-极小。 最后，斯坎伦将发展具有解析结构的维特向量理论和相关的弗罗贝纽斯理论的模型理论。 斯坎伦将使用这一理论既作为一个试验场的一般理论的亚稳定性和作为一个逻辑驯服的理论，其中算术问题可能会编码相信，这一理论可能会地面一个新的理论motivic整合，其中有限维差异品种发挥作用的代数品种。斯坎伦将制定一个模型理论的p-喷气机独立的素数p从中均匀性数论问题可能推导出。模型理论，在数理逻辑的意义上，提供了一个统一的角度来研究问题的不同分支的数学作为实例的一般理论。例如，它可能是幻想的基础，但暗示，理论中的技术适用于几何和微分方程的研究转来研究数字。 在相反的方向，它的非常一般的方法来理解对称性的结构，更重要的是对称性的一个部分的结构相对于另一个部分时，专门用于具体的数学问题，微分方程和差分方程揭示了其他未知的对称性理论。 通过这个项目，斯坎伦将探索数学的基本统一性，从微分方程理论的数学逻辑，到动力系统，再到数论。