# Computably enumerable sets and vector spaces with thin complements

## Valentina Harizanov

### George Washington University

Simple, hypersimple, hyperhypersimple, and maximal sets play an important role in computability theory. Maximal sets are co-atoms in the lattice of computably enumerable sets modulo finite sets. Soare established that they are automorphic in this lattice. Similarly, maximal vector spaces play an important role in the study of the lattice of computably enumerable vector spaces modulo finite dimension. We will investigate principal filters determined by maximal spaces and how algebraic structure interacts with computability-theoretic properties. We will discuss the problem of finding orbits of maximal spaces under lattice automorphisms, the development of various notions of maximality for vector spaces, and recent progress that is joint work with R. Dimitrov.

Valentina Harizanov is a professor of mathematics at George Washington University, studying computable model theory. She did her graduate work at the University of Wisconsin-Madison, where her Ph.D. advisor was Terry Millar.