This is the title of a talk given at the CMU graduate student seminar on September 15, 2015. A video of the beginning of the talk is on Youtube. Due to a technical glitch, the rest of the talk has unfortunately not been recorded.
Universal classes are a general model-theoretic framework introduced in the seventies by Saharon Shelah to study certain classes of modules. It encompasses several natural algebraic objects, such as vector spaces and locally finite groups.
I will present generalizations to universal classes of several concepts and results of linear algebra. For example, a universal class which has a single model of a "high-enough" infinite size has a single model in every high-enough size. Moreover, such classes admit an independence notion generalizing linear independence in vector spaces. I will also discuss a more general framework (also due to Shelah), abstract elementary classes, and conjectured extensions of these results there.
John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 20, College Publications, 2009.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes, Preprint: pdf arXiv.