# Universal classes

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 (if you don't like Youtube, here is a direct link). Due to a technical glitch, the rest of the talk has unfortunately not been recorded.

## Abstract

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.

## References

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.