Mini-course (25.04.2024 - 16.05.2024, 3 lectures) - 1^{th} lecture:

Given a cardinal \(\kappa\), a set of reals \(A \subseteq \mathbb R\) is \(\kappa\)-dense if its intersection with any open interval has size \(\kappa\). Baumgartner's axiom (BA) — proved consistent by Baumgartner in 1973 — states that all \(\aleph_1\)-dense sets of reals are order isomorphic with the induced linear order from \(\mathbb R\). This is the most straightforward generalization to the uncountable of Cantor's proof that all countable dense linear orders without endpoints are order isomorphic. BA has variations to other topological spaces — given a topological space \(X\), a subset \(A \subseteq X\) is \(\kappa\)-dense if its intersection with each non-empty open subset has size \(\kappa\). The axiom BA(\(X\)) states that given any two \(\aleph_1\)-dense subsets of \(X\), say \(A\) and \(B\), there is an autohomeomorphism of \(X\) mapping \(A\) onto \(B\). In this parlance BA is equivalent to BA(\(\mathbb R\)). Surprisingly BA is not equivalent to BA(\(\mathbb R^n\)) for any finite \(1< n < \omega\). In fact BA does not follow from Martin's Axiom (Abraham-Rubin-Shelah) though BA(\(\mathbb R^n\)) does (in fact from \(\mathfrak p > \aleph_1\)) for each \(n > 1\) (Steprāns-Watson).

In these three lectures I will discuss these ideas and some related ones including the question of when BA(\(X\)) implies BA(\(Y\)) for Polish spaces \(X\) and \(Y\). Central to these questions are the role of cardinal characteristics including the celebrated theorem of Todorčević that BA implies \(\mathfrak b > \aleph_1\) as well as a recent, higher dimensional analogue of this result that for any \(n < \omega\) BA(\(\mathbb R^n\)) implies \(\mathfrak b > \aleph_1\) (S.-Steprāns). There are many beautiful open problems in this area and I plan to make discussing them a focal point of the talks. The talks will start slowly and should be accessible to students. Time permitting, the final talk will include some new results. If and when these results are presented, they are joint work with Juris Steprāns.