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.
