Stability conditions in symplectic topology

Abstract

We discuss potential (largely speculative) applications of Bridgeland's theory of stability conditions to symplectic mapping class groups.Date: November 2017. I.S. is partially funded by a Fellowship from EPSRC. Thanks to friends and colleagues at Cambridge for a decade and a half of moral support, intellectual stimulation and company lifting spirits.

An extension of the Siegel space of complex abelian varieties and conjectures on stability structures

“…Given these results it is natural to wonder how much of the theory of flat surfaces survives for spaces of stability conditions on more general categories, e.g. Fukaya categories, F (M), of higher-dimensional symplectic manifolds, M. See also the discussion by Smith [26]. The case when dim R M = 6 and M is compact is of particular interest in view of the theory of categorical Donaldson-Thomas invariants [17].…”

Relative stability conditions on Fukaya categories of surfaces

“…On the other side (A-side) of mirror symmetry there have been many indications that stability conditions can recover geometric data encoded by the Fukaya category, in particular regarding questions about special Lagrangian geometry [24,37]. In the main work [20] that this paper references, Haiden, Katzarkov and Kontsevich look at stability conditions on the (wrapped) Fukaya category of a marked surface Σ, and show that the spaces of stability conditions on F(Σ) are related to the geometry of quadratic differentials on Σ.…”