Speaker: Reeve Garrett (OSU)
Abstract: In the 1920s and 1930s, mathematicians such as Cartan, Stone, and Tarski formulated the definition for and proved the existence of a new mathematical object that we refer to currently as an ultrafilter. At the time, ultrafilters were applied to topological and analytic problems, and as model theory developed, they have become a standard tool for that discipline as well. However, in recent years, ring theorists such as Loper, Olberding, and Schoutens have discovered their incredible utility in understanding and solving various problems in their field. In this talk, we will define what ultrafilters are and give a survey of some of the insight they provide in the context of commutative ring theory, introducing conepts and terminology from the field as needed. More particularly, we will discuss the ultrafilter topologies on the prime spectrum of a ring and on the Zariski-Riemann space of valuation overrings of a domain and the insight these topologies provide. We will also discuss the utility of ultraproducts as it pertains to a partial solution to a conjecture of Artin regarding the solution sets of homogeneous polynomials.