Mathematical Sciences: Model Theory of Modules

数学科学：模的模型论

• 批准号: 9626708
• 负责人: Ivo Herzog
• 金额: $ 3.6万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626708&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Model; Theory; Modules

DMS-9626708 Ivo Herzog This project concerns the connection between (1) the model theory of modules and (2) the use of functorial methods in the representation theory of rings. In recent years this connection has been developed with an eye to algebraic applications of first order methods to modules, and in particular to artin algebras. The present project will expand this approach to include the model theory of modules with an endomorphism, or equivalently - the model theory of modules over the ring of polynomials. Also to be considered are (1) first-order variants of the Brauer-Thrall conjectures for an artin algebra, and (2) a theory of homology for positive-primitive formulas. This research involves the connection between logic and algebra. Logic provides the context in which one can study the expressive power of mathematical language; as such it is an essential tool in computing. Algebra provides a vehicle for understanding notions such as symmetry and mappings, and finds application in sciences and manufacturing. Professor Herzog's research takes place in a part of algebra called module theory, which is akin to, but broader than, linear algebra. The problems which arise are similar to solving systems of linear equations. By forging the relationships between logic and algebra, his research will provide new understanding of modules.

DMS-9626708 这个项目关注（1）模的模型理论和（2）环的表示理论中函子方法的使用之间的联系。 近年来，这种联系已经发展到着眼于代数应用的一阶方法的模块，特别是阿廷代数。 本项目将扩展这种方法，包括模的模型理论与自同态，或等价的-模的模型理论在环的多项式。 也要考虑的是（1）一阶变种的Brauer-Thrall代数的artin代数，和（2）理论的同调的积极的原始公式。 这项研究涉及到逻辑和代数之间的联系。 逻辑提供了一个可以研究数学语言表达能力的背景;因此，它是计算中的一个重要工具。 代数为理解对称性和映射等概念提供了工具，并在科学和制造业中得到应用。 赫尔佐格教授的研究发生在代数的一部分，称为模理论，这是类似于，但更广泛的，线性代数。 出现的问题类似于求解线性方程组。 通过建立逻辑和代数之间的关系，他的研究将提供对模块的新理解。