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4. Resolutions

    1. Compare $\Delta$-resolutions


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      Problem 4.1.

      Can we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?

      $\bullet$ Potential mechanism: (starting in degree 0)

      {vertices/points in stratification of $T^n$ arising in HHL} is in one to one correspondence with {additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}

        • Topology/Morse theory of cellular resolutions


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          Problem 4.2.

          Can we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through "cellular" operations)

            • Topology/Morse theory of cellular resolutions


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              Problem 4.3.

              The Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of "framed flow categories"). can this refined decomposition tell us something on the algebraic side?

                •     $T^n=K(\pi,1)$

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                  Problem 4.4.

                  Given a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\pi_1$?

                    • Topology/Morse theory of resolutions


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                      Problem 4.5.

                      Is there an analogue of "Morse theory for cellular resolutions" for virtual resolutions?

                        •     Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.

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                          Problem 4.6.

                          Is there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...? (c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)

                              Cite this as: AimPL: Syzygies and mirror symmetry, available at http://aimpl.org/syzygyms.