Mathematical Sciences: Model Theory of Modules

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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.