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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

1. Oblomkov-Rozansky link invariant


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      Problem 1.1.

      Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).


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          Problem 1.2.

          Prove that MF$((\mathscr{X}^\text{big})^\text{stable},w)$ is equivalent as a monoidal category to K$^b(\mathbb{S}\text{Bim}_n)$, where $\mathscr{X}^\text{big}=\mathfrak{g} \times G \times \mathfrak{n} \times G \times \mathfrak{n}$.


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              Problem 1.3.

              On HHH$^\text{geom}$ of a braid $\beta$ there is a mysterious 4th $\mathbb{Z}$-grading coming from the $\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\beta$.


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                  Problem 1.4.

                  To what extend can computers compute HHH$^\text{geom}$ directly from the definition?


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                      Problem 1.5.

                      The space $\mathscr{X}^{\text{big}}$ and category MF($\mathscr{X}^{\text{big}},w$) makes sense not just in type A. What is the analog of $(\mathscr{X}^{\text{big}})^{\text{stable}}$ in other types? What is the categorical analog of the map Br$_\text{ext} \to$ Br$_\text{fin}$.


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                          Problem 1.6.

                          What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?

                          In the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\mathscr{X}^\text{big}$?

                              Cite this as: AimPL: Categorified Hecke algebras, link homology, and Hilbert schemes, available at http://aimpl.org/catheckehilbert.