Applications of Model Theory to Additive Structures

该项目致力于与模模型理论密切相关的三个主题，模模型理论是数理逻辑的一个交叉学科分支，旨在利用线性代数的形式数学语言来理解环和代数的表示理论。第一个主题是半单李代数的有限维表示的整体理论。这种表示的分类是经典的，但只提供了一个局部理论，给出了两个有限维表示同构的准则。通过考虑有限维表示的理论，该项目试图研究被称为伪有限维表示的表示，因为它们在形式语言中满足有限维表示的所有定理。这些非标准表示将被用来研究各种有限维表示，从而导致有限维表示的整体理论。该项目的第二个活动是研究环上的模的模型理论和该环的K理论之间关系的精确性质。人们希望通过引入同调理论来建立这两个领域之间的精确关系，以研究形式表示语言中的公式。最后，不是通过直接使用模型理论的技术，而是通过在范畴环境中模仿模块模型理论中用于研究纯洁性的直觉来发展完全精确范畴的理论。该项目将致力于密切研究代数学家从事的活动之间的关系，这些代数学家对几何和物理的应用最感兴趣，但使用数理逻辑的形式方法。这些活动将使用语言来研究代数中出现的对称性，并反过来将这些对称性应用于形式语言。在该项目的其他活动中，PI将继续担任为俄亥俄州西北部和中部的小学和中学数学教师提供的专业发展计划的顾问。

The project is devoted to three topics closely related to the Model Theory of Modules, an interdisciplinary branch of Mathematical Logic that seeks to understand the Representation Theory of Rings and Algebras using the formal mathematical language of Linear Algebra. The first topic is a global theory of finite-dimensional representations of a semisimple Lie algebra. The classification of such representations is classical, but only offers a local theory, giving criteria for when two finite-dimensional representations are isomorphic. By considering the theory of finite-dimensional representations, the project seeks to study the representations called pseudo-finite dimensional representations, because they satisfy in the formal language all the theorems of finite-dimensional representations. These nonstandard representations will be used to study varieties of finite-dimensional representations, and thus lead to a global theory of finite-dimensional representations. The second activity of the project is to study the precise nature of the relationship between the Model Theory of Modules over a ring and the K-theory} of that ring. One hopes to establish a precise relationship between these two areas in terms of a theory of homology introduced to study formulae in the formal language of representations. Finally, a theory for complete exact categories is to be developed not by using the techniques of model theory directly, but rather by mimicking in a categorical setting the intuition used in the model theory of modules to study purity. The project will be devoted to a close examination of the relationship between activities carried on by algebraists most interested in applications to geometry and physics, but using the formal methodology of mathematical logic. These activities will use the language to study the symmetries that arise in algebra, and conversely, apply these symmetries to formal language. In other activities of the project, the PI will continue to act as consultant to professional development programs offered to elementary and middle school mathematics teachers in Northwest Central Ohio.