Taut Foliations and Braid Positivity

The L-space conjecture has been in the news a lot lately: this conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology. In this talk, I'll discuss the motivation for this conjecture, as well as some forthcoming work that builds taut foliations in infinitely many 3-manifolds obtained by Dehn surgery along some special knots. Along the way, we will produce some novel obstructions to braid positivity. I will not assume any background knowledge in Floer or foliation theories; all are welcome!