Mathematics Seminar - 10/23/26

Abstract: Infinitary operations, i.e. operations with infinitely many inputs, arise in various areas of mathematics. Perhaps most familiar are infinite sums and products from calculus. Even these simple operations have some algebraic subtleties  For instance, the existence of conditionally convergent series implies that the infinite sum operation (on real numbers) is only finitely commutative, i.e. permuting summands may change the value of the sum. But are there natural algebraic objects with non-commutative infinitary operations? In this talk, I'll give an introduction to the most well-known non-commutative infinitary structure: The infinite earring group. 

The infinite earring group is most easily defined as the fundamental group of the infinite earring space E but also has purely algebraic representations using infinite words and inverse limits. It is an infinitary analogue of a "free group." In the past two decades there has been a flurry of progress in understanding the algebraic topology of spaces like E. We'll discuss how infinitary operations and the infinite earring group lie at the heart of this work.