Georgia Institute of Technology College of Sciences

Ultrafilters formalize a generalized notion of convergence based on a prescribed idea of "largeness" for subsets of the natural numbers, and underlie constructions like ultraproducts. In the study of moduli spaces, they provide a clean way to encode degenerations and to establish uniformity results that are difficult to obtain using ordinary limits. This talk will discuss applications of ultrafilters to uniformity theorems in dynamics and arithmetic geometry. After introducing local results that arise from this approach, I will sketch some of the arithmetic consequences, including uniform bounds on rational torsion points on abelian varieties. This is joint work with Jit Wu Yap