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Based on joint work with Rozansky. In my talk I outline a construction that produces a $\mathbb{C}^*\times\mathbb{C}^*$-equivariant complex of
sheaves $S_b$ on $Hilb_n(\mathbb{C}^2)$ such that the space of global sections $H^*(S_b)$
of the complex are the Khovanov-Rozansky homology of the closure of the braid $b$.
The construction is functorial with respect to adding a full twist to the braid. Thus we prove a weak version of the conjecture by Gorsky-Negut-Rasmussen.
In the heart of our construction is a fully faithful functor from the category of Soergel bimodules to a particular category of matrix factorizations.
I will keep the matrix factorization part minimal and concentrate on the main idea of the construction as well as key properties of the categories that we use.

Notes: Notes from the talk