Ciprian Manolescu
Stanford University, Palo Alto, USA

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in $\mathbb{R}^3$ of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincaré conjecture using Rasmussen's invariant from Khovanov homology. It is yet unclear whether their strategy can work. I will explain several recent results in this direction and some of the challenges that appear. A key problem is to certify when a knot is slice (bounds a disk in four-dimensional half-space), which can be tackled with machine learning. The talk is based on joint work with Sergei Gukov, Jim Halverson, Marco Marengon, Lisa Piccirillo, Fabian Ruehle, Mike Willis, and Sucharit Sarkar.