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4. Hilbert schemes


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      Problem 4.1.

      Fixing a slope, what corresponds to a stable basis in $K$-theory of Hilbert schemes in the cocenter of the Hecke algebra?


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          Problem 4.2.

          Compute endomorphism algebra (in the homotopy category) of projectors, and relate it to coordinate ring and of charts on Hilb$^n(\mathbb{C}^2)$


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              Problem 4.3.

              Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson’s conjecture. Describe $H^*(\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.

                •     We define Hilb$^n(x^2=0)=\{\text{codiminsional }n \text{ ideals in }\mathbb{C}[x,y]/(x^2=0)\}\subseteq$ Hilb$^n(\mathbb{C}^2)$

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                  Problem 4.4.

                  Relate Hilb$^n(x^2=0)$ to $HH$ of the arc algebra.


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                      Problem 4.5.

                      Let $\mathcal{F}(\beta)$ be the sheaf on Hilb$^n(\mathbb{C}^2)$ associated to $\beta$. How does $\mathcal{F}(\beta)|_{\text{Hilb}^n(x^2=0)}$ relate to Khovanov homology of $\beta$?


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                          Problem 4.6.

                          Construct link invariants for links in lens spaces using Hilb$([\mathbb{C}^2/(\mathbb{Z}/l\mathbb{Z}])$ and relate to wreath Macdonald polynomials.

                              Cite this as: AimPL: Algebra, geometry, and combinatorics of link homology, available at http://aimpl.org/agclinkhom.