Symplectic topologists are building a new mathematical toolkit to solve problems that have resisted 35 years of existing methods. The field studies the geometry of spaces where area can be measured but distance cannot—a setting that arises naturally in classical mechanics and string theory. Current techniques for constructing key invariants rely on notoriously difficult "virtual perturbation" methods that can stall progress for years. This project pursues two fresh approaches: one borrows ideas from algebraic geometry via mirror symmetry, the other from stable homotopy theory. The first strand aims to create global Kuranishi charts, a more streamlined way to handle the genus-zero curves that underpin many calculations. The second uses rational equivalence of algebraic cycles on mirror spaces to constrain which geometric objects—called Lagrangians—can exist without obstruction. If successful, the work would produce new symplectic invariants (quantum Morava K-theory) and potentially resolve long-standing questions about exotic symplectic structures. This is fundamental mathematics with no immediate practical application, but similar abstract geometric insights have historically underpinned developments in theoretical physics, particularly in string theory and quantum field theory.
View original technical description
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.
Plain English summaries and category classifications on this site are generated by AI and may not perfectly reflect the original research.
Is something wrong? Let us know