Title: Floer homology and the fundamental group

Abstract: The most important invariant of a 3-manifold is its fundamental group. One of the most fruitful approaches to understanding the fundamental group is to study its homomorphisms into
simpler groups. SU(2) is an especially convenient choice of target because it is one of the simplest non-abelian Lie groups, and because Floer homology provides powerful tools for studying SU(2) representations of 3-manifold groups. I will highlight some results stemming from this connection between Floer homology and the fundamental group—from Kronheimer and Mrowka’s work on the Property P Conjecture in the early 2000s, to their 2010 proof that Khovanov homology detects the unknot, to some of my recent work with Steven Sivek.