"Over the last decade, homological algebra has entered symplectic topology, largely thanks to the appearance of Fukaya categories in homological mirror symmetry. Applications of these new methods and ideas are still scarce. We propose a fundamentally new approach to studying symplectic dynamics, by studying the action of the symplectic mapping class group on the complex manifold of stability conditions on its Fukaya category. This can be seen as a first attempt to generalise classical Teichmueller theory to higher-dimensional symplectic manifolds. Many invariants arising in low-dimensional topology, including Khovanov cohomology for knots, are governed by the Fukaya categories of associated moduli spaces. We propose an ""uncertainty principle"" in topology, in which these invariants are intrinsically constrained by rigidity of this underlying categorical structure. Besides applications in topology, this suggests a framework for studying the sense in which topological complexity is the shadow of dynamical complexity."
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