categorification

Note from Arik: If you want to learn more about the Hilbert schemes and the GNR conjectures you might find this video interesting as a starting point: https://www.newton.ac.uk/seminar/20170628100011001. There seems to be some support in the department to have Negut speak in our algebra seminar in the fall (most likely via Zoom) and give an update on the status of these conjectures. As far as I am informed they are still open. But some progress has been made since 2017 (the year when this video was recorded).

One of the beautiful things about these link homology theories is that they have so many different (equivalent) constructions. For Khovanov homology (categorification of the Jones polynomial) I want to point out the following: Bar-Natan (TQFTs, cobordism pictures) https://arxiv.org/abs/math/0410495 Seidel–Smith (symplectic geometry) https://arxiv.org/abs/math/0405089 Stroppel (Lie theory, category O) http://ems.math.uni-bonn.de/people/stroppel/TemperleyLieb.pdf Cautis–Kamnitzer (derived categories of coherent sheaves, geometric Satake) https://arxiv.org/abs/math/0701194

You can pick the one closest to your field and learn some more by looking at these papers if you want. The triply graded link homology as discussed in the course can be found here https://arxiv.org/abs/math/0510265. An equivalent version using matrix factorizations can be found here https://arxiv.org/abs/math/0505056. Recall that the Jones polynomial is obtained as a certain specialization of the HOMFLY-PT polynomial. This process of “specialization” is categorified by a spectral sequence from the triply graded homology (HOMFLY-PT homology) to Khovanov homology https://arxiv.org/abs/math/0607544.