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Khovanov homology and 4-manifolds

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 ℝ³ 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, as well as a new attempt to pursue it (joint work with Lisa Piccirillo). I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.


About the speaker


Manolescu is currently a professor of mathematics at Stanford University.  He was on the faculty at Columbia and UCLA.  He earned both his undergraduate and doctorate at Harvard University, under the direction of Peter Kronheimer.
The 2019 E.H. Moore Research Article Prize was awarded to Manolescu for his paper that resolves the Triangulation Conjecture (published in the Journal of the American Mathematical Society, Vol. 29, No. 1).  Of his many other honors, Manolescu was an invited speaker at 2018 ICM in Rio de Janeiro, elected as a member of the 2017 class of Fellows of the American Mathematical Society, and awarded in 2012 one of the ten prizes of the European Mathematical Society.  He received the Clay Research Fellowship in 2004.
Professor Manolescu's research interests include gauge theory, low-dimensional topology, and symplectic geometry.  He has worked on the Seiberg-Witten equations, Heegaard Floer theory and Khovanov homology.