Algebra Seminar talk
2026-03-27
Nick Chapman
Studying the structure of the Borel hierarchy
Abstract:
The class of Borel sets is one of the most fundamental structures on a topological space. Its study lies at the intersection of several areas of mathematics; in this talk, we will investigate properties of the Borel algebra from the viewpoint of descriptive set theory and topology, focusing on the \textit{length} of such a hierarchy on a given subspace $X \subseteq {}^\omega \omega$ of the real numbers. The length $\operatorname{ord}(X)$ of the hierarchy is defined as the least ordinal $\alpha$ for which every Borel subset of $X$ is $\Sigma^0_\alpha$. The value of this ordinal turns out to be highly malleable, and a sophisticated forcing technique was developed by A. Miller to control this ordinal in forcing extensions. We will discuss its basic building block of $\alpha$-forcing as well as sketch the nature of rank arguments that yield consistency results about assignments of $\operatorname{ord}(X)$ to several spaces $X$ simultaneously. Time permitting, we will also delve into the speaker's recent contributions to this area, such as the development of analogous arguments in the field of generalized descriptive set theory of an uncountable cardinal $\kappa$.
This talk can be given in either German or English, and I will adjust the level of the exposition depending on the makeup of the audience.