Khovanov homology and four-dimensional topology

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 a new attempt to pursue it (joint work with Lisa Piccirillo) and some of the challenges we encountered. I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.

Biography

Ciprian Manolescu grew up in Romania. During high school he earned three gold medals with perfect scores at the International Mathematical Olympiad. He obtained his BA and PhD from Harvard University. He was a Clay Fellow at Princeton and IAS, and he had faculty appointments at Columbia and UCLA, before moving to Stanford in 2019.

Manolescu’s research is centered on constructing new versions of Floer homology and applying them to questions in topology. With collaborators, he showed that many Floer-theoretic invariants are algorithmically computable. He also developed a new variant of Seiberg-Witten Floer homology, which he used to disprove the triangulation conjecture in high dimensions. For his work, he received the Morgan prize in 2002, the EMS Prize in 2012, the AMS E. H. Moore Prize in 2019, and was awarded a Simons Investigator Award in 2020.