Mathematics Colloquium by Professor Jenna Zomback and Austyn Simpson, Williams College

Ultraproducts in Commutative Algebra by Professor Jenna Zomback and Austyn Simpson, Williams College, Wednesday July 12, 1:00 – 1:50 PM, North Science Building 113, Wachenheim.

Abstract: In this talk, we will define ultraproducts and investigate some of their applications to commutative algebra. An ultraproduct of mathematical structures (for example, commutative rings) is a quotient of an infinite direct product where we identify two sequences of elements if they agree “almost everywhere.” We can make this identification precise using the axiom of choice, and many so-called “non-standard” tools from logic and set theory can then be employed to answer questions which are seemingly purely algebraic. After surveying some well-known such applications, we will report on some of the ongoing projects involving ultraproducts in commutative algebra at SMALL 2023.