Mathematical Sciences: Model Theory of Modules

这个项目涉及(1)模块的模型理论和(2)环表示理论中泛函方法的使用之间的联系。近年来，随着一阶方法在模块的代数应用，特别是在代数中的应用，这种联系得到了发展。目前的项目将扩展这种方法，包括具有自同态的模块的模型理论，或者等效地-多项式环上的模块的模型理论。同样要考虑的是(1)一个artin代数的Brauer-Thrall猜想的一阶变异体，和(2)一个正原始公式的同调理论。这项研究涉及到逻辑和代数之间的联系。逻辑为研究数学语言的表达能力提供了环境；因此，它是计算中必不可少的工具。代数为理解对称和映射等概念提供了载体，并在科学和制造业中得到了应用。赫尔佐格教授的研究集中在代数的一个叫做模块理论的部分，它类似于线性代数，但比线性代数更广泛。出现的问题类似于解线性方程组。通过建立逻辑和代数之间的关系，他的研究将为模块提供新的理解。

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.