Floer theory beyond Floer (FloerPlus35)

The holomorphic curve theory of Gromov and Floer, introduced 35 years ago, revolutionised all aspects of symplectic topology. Floer cohomology now incorporates many of the algebraic structures present in ordinary cohomology: ring structures, exact triangles, equivariant cousins, Steenrod operations. The construction of these holomorphic curve invariants in general relies on difficult virtual perturbation methods. In the last years, new techniques from algebraic geometry, via mirror symmetry, and separately from stable homotopy theory, have entered the subject. This proposal is centred on developing one central idea in each of these new themes.

With Abouzaid and McLean, the PI is developing a theory of global Kuranishi charts, a new approach to genus zero curve theory bypassing many of the usual technicalities. Combining these with input from chromatic homotopy theory yields fundamentally new symplectic invariants, adapted to the orbifold nature of holomorphic curve moduli spaces. We will use these to build symplectic quantum (Morava) K-theory, with applications to Hamiltonian fibrations, products, blow-ups and exotic symplectic structures. Separately, with Sheridan, the PI initiated a program to study Lagrangian cobordism via the theory of rational equivalence of algebraic cycles on the mirror. This suggests constraints on the existence of unobstructed Lagrangians coming from Chow group computations, mirroring `counterexamples to the Hodge conjecture'. In both strands, the need to keep track of torsion information necessitates using frameworks going beyond classical holomorphic curve invariants.