MATH: Colloquium: Khovanov homology and four-manifolds

Ciprian Manolescu, Professor, Stanford University

 

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, and I will explain some of its challenges. (This is based on joint work with Lisa Piccirillo.) I will also review other topological applications of Khovanov homology, including a recent detection result for some exotic compact 4-manifolds with boundary (work of Ren and Willis).

Thursday, April 4 at 1:30 pm to 2:30 pm

Palmer Engineering, 104