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5. Spectral refinements (in the sense of algebraic topology)


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      Problem 5.1.

      [Mohammed Abouzaid] Are there spectral refinements of KLRW and nil Hecke algebra?
        1. Remark. One could attempt to study it a). combinatorially as in Lipshitz–Sarkar [MR3230817], b). symplectically as in Seidel–Smith [MR2254624] and Aganagic’s work (e.g., [arXiv:2207.14104]).
            • Remark. [Robert Lipshitz] There is some work trying to get a spectral $\mathfrak{sl}_2$-action on annular Khovanov spectrum via combinatoric, see Akhmechet–Krushkal–Willis [arXiv:2011.11234].
                • Remark. [Ko Honda] What about just Hecke algebras?


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                      Problem 5.2.

                      [Egor Shelukhin] Are there spectral refinements of link spectral invariants and applications?
                        1. Remark. [Ivan Smith] We don’t yet have spectral lifts of Lagrangian Floer theory, but we do for Hamiltonian Floer theory (e.g., Morava K-theories), so an initial goal could be to see if the quantitative spectral invariants in this case can be used to find applications.

                              Cite this as: AimPL: Floer theory of symmetric products and Hilbert schemes , available at http://aimpl.org/floerhilbert.