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.