Khovanov-Rozansky homology counts solutions to PDE
by
B/2nd floor-B.203 - Seminar room
Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve)
Given a knot K in S^3, seen as the boundary of hyperbolic 4-space, the counts n_{g,d}(K) of minimal surfaces filling K of genus g and « twistor degree » d are knot invariants. J. Fine conjectured that they may be assembled to recover the HOMFLYPT polynomial.
Following ideas of Donaldson and Thomas, we conjecture a way to categorify this count. This is done via Floer homology techniques, where the complex is generated by those minimal surfaces and the boundary map is constructed out of a count of a special type of submanifolds called « associative ».
It is natural to believe that these homology knot invariants coincide with Khovanov-Rozansky homology. If so, knowledge of KR homology would give an existence result for solutions of non-linear PDEs!