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date: 2022-03-21 15:34
modification date: Monday 21st March 2022 15:34:23
title: categorification
aliases: [categorification]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #todo/untagged
- Refs:
	- MSRI workshops on [higher algebra](Unsorted/higher%20category.md) and categorification: 
		- <https://www.msri.org/workshops/908> #resources/videos #projects/unstarted #resources/workshops 
		- <https://www.msri.org/workshops/907> #resources/videos #projects/unstarted #resources/workshops 
	- Course by Khovanov: <https://www.math.columbia.edu/~khovanov/cat2020/>
- Links:
	- #todo/create-links 

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# 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>.