Diamonds on trees, part 1

We explore a natural generalization of the diamond principle to nonspecial w1-trees based on the Todorcevic-Brodsky notion of stationarity. We propose $\diamondsuit_T$ as a structural invariant that naturally partitions nonspecial w1-trees $T$ according to their ``thickness'' into three categories: trees where $\diamondsuit_T$ holds in ZFC, ``thin'' trees where it is equivalent to the classical diamond, and certain nonspecial Aronszajn trees where it is strictly stronger than the classical diamond. The talks will also cover the consistency results for this last category, introducing new forcing properties---$\sigma(S)$-closed and strategically closed in models---that are preserved under countable support iterations.