CAREER: Geometric and Algebraic Model Theory

职业：几何和代数模型理论

• 批准号: 0450010
• 负责人: Thomas Scanlon
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0450010&HistoricalAwards=false
• 关键词: CAREER; Geometric; Algebraic; Model; Theory

Scanlon's work integrates arithmetic, geometry and logic via amodel-theoretic study of natural mathematical structures. Specifically,Scanlon investigates the theory of compact complex analytic manifolds froma model theoretic point of view and extends this research to relatedgeometric objects. In developing the theory of jets of differentialequations, Scanlon hopes to discern the fine structure of dependence onsolution sets to partial differential equations of infinite differentialdimension and to present a unified treatment of differential Galois theoryfor such equations. Scanlon plans to refine the study of enriched valuedfields and to draw out the Diophantine consequences of these results.Scanlon works, as well, on purely logical issues and will continuedeveloping the theory of thorn-independence with an eye towards eventualapplications.Scanlon's work integrates arithmetic, geometry and logic. Definable sets,namely those sets definable within a fixed formal language, are principalobjects of study in model theory and this special emphasis has beeninstrumental in the success of the model-theoretic approaches to numbertheory, algebra and analytic geometry. Scanlon works to tighten theconnection between logical and geometric methods and to find mathematicalinterpretations of theorems in logic.

斯坎伦的工作通过对自然数学结构的模型理论研究，整合了算术、几何和逻辑。具体来说，Scanlon从模型论的角度研究了紧复解析流形的理论，并将这一研究扩展到相关的几何对象。在发展微分方程喷流理论的过程中，斯坎伦希望能够发现无穷维偏微分方程的依赖解集的精细结构，并对这类方程提出微分伽罗瓦理论的统一处理方法。斯坎伦计划改进对富含价值油田的研究，并得出这些结果的丢番图后果。斯坎伦也在纯逻辑问题上工作，并将继续发展刺无关理论，着眼于最终的应用。斯坎伦的作品综合了算术、几何和逻辑。可定义集合，即那些在固定的形式语言中可定义的集合，是模型理论研究的主要对象，这种特殊的强调在数论、代数和解析几何的模型理论方法的成功中起了重要作用。斯坎伦致力于加强逻辑和几何方法之间的联系，并在逻辑中找到定理的数学解释。