Recent progress in Mathematics and Statistics: Ciprian Manolescu -- Khovanov homology and four-dimensional topology

Title: Khovanov homology and four-dimensional topology Abstract: 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 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 Poincare 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.