Final Report Summary - SYMPTEICH (Towards symplectic Teichmueller theory)
Symplectic topology is a field of pure mathematics with close links to dynamical systems, to the geometry of high dimensional spaces, and to quantum mechanics and theoretical physics. It arises when studying periodic orbit problems for classical mechanical systems and dualities in superstring theory. The project brought to bear new techniques from non-commutative algebra and physics on classical problems of topology. Outcomes included growth phenomena for periodic orbits, rigidity theorems for Lagrangian boundary value submanifolds, and an unexpected link between billiard trajectories, BPS counts in string theory and abstract homological algebra. Formally, these results come from studying the behaviour of an infinite-dimensional gradient flow (akin to heat flow or fluid flow over an object), known as Floer cohomology, which gives a particular mathematical way of capturing persistent properties of critical points and bifurcations, persistent under perturbations of the system.