open problem list

  • Wall-crossing invariants of flat surfaces: Count saddle connections and cylinders on flat (half-translation) surfaces in a way which produces wall-crossing structures, i.e. so that wall-crossing formulas hold. For quadratic differentials with simple zeros and poles and satisfying a genericity condition, this was done in arXiv:2104.06018, but the case of non-generic surfaces (e.g. Veech surfaces) and higher order zeros remains.
  • Homological mirror symmetry for spaces with sheaves of categories: In some cases (e.g. punctured surfaces) one can define a Fukaya category of a symplectic manifold with coefficients in a perverse Schober. Is there a variant of homological mirror symmetry for such categories?
  • Hall algebras of Z/2-graded categories: Is there a notion of Hall algebra for (enhanced) triangulated categories which are Z/2-graded, i.e. Exti = Exti+2? A small hint came from arXiv:1908.10358. Toen. Bridgeland. Gorsky.
  • Derived categories over F1: Is there a useful notion of derived (stable) category over the “field with one element”? Potential examples come from quivers with monomial relations, toric varieties, and lagrangian skeleta.
  • Asymptotics of modified curve shortening flow: Taken from arxiv:1802.04123. Asymptotics of a certain PDE (non-linear heat flow) should be governed by combinatorics of a partially ordered set.
  • Dimension of triangulated categories: What is a good notion of dimension for triangulated categories, and what are its properties? For d-Calabi-Yau categories there is the obvious choice dim=d. Also fractional CY categories. More generally, we noticed in arxiv:1307.8418 that the entropy of Serre functor has something to do with dimension. Elagin-Lunts discuss the relation between different definitions and open problems.
  • Classification of objects in Fukaya categories of closed higher genus surfaces: PARTIALLY SOLVED by Auroux-Smith. SOLVED by Pascaleff-Sibilla
  • Thomas-Yau conjecture on Fukaya categories, Bridgeland stability, and special lagrangians. Joyce. This is a generalization of arxiv:1409.8611 to higher dimensions.