Khovanov Homology and Surfaces in Four-Manifolds

Explore the intricacies of Khovanov homology and its applications to surfaces in four-manifolds in this AMS Maryam Mirzakhani Lecture delivered by Stanford University's Ciprian Manolescu at the 2021 Joint Mathematics Meetings. Delve into advanced mathematical concepts such as exotic smooth structures, gauge theory applications, and the slice genus problem. Examine the differential on Khovanov homology, the Rasmussen invariant, and their implications for proving existing results and developing new applications. Investigate the potential approach to the smooth Poincaré conjecture in four dimensions (SPC4), Gluck twists, and the construction of homotopy 4-spheres. Analyze knots with identical 0-surgeries, special RBG links, and their significance in understanding four-dimensional topology. Conclude with an exploration of computer experiments and potentially slice knots, providing a comprehensive overview of cutting-edge research in four-dimensional topology and knot theory.

Intro
Outline
Four dimensions
Exotic smooth structures
Applications of gauge theory
Surfaces in 4-manifolds
Surfaces in B4
Slice genus
The differential on Khovanov homology
More on Khovanov homology
The Rasmussen invariant
New proofs of old results
A new application
The knot trace
New applications
A possible approach to SPC4
Gluck twists
A negative result
A more positive result
Another construction of homotopy 4-spheres
Knots with the same 0-surgeries
Special RBG links
Slides From a special RBG link we obtain a knot ko by sliding Gover Runtil
An example
Computer experiments
Possibly slice knots
More examples