HMS SIGNED

Project "HMS" data sheet

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 Project objective

Mirror symmetry is a deep relationship between algebraic and symplectic geometry, with origins in string theory. It was originally envisioned as an isomorphism between the Hodge theory of an algebraic variety and the `quantum Hodge theory' of a `mirror' symplectic manifold, but it was subsequently realized by Kontsevich that the relationship went far deeper. His Homological Mirror Symmetry (HMS) conjecture predicts an equivalence between the derived category of the variety and the Fukaya category of its mirror, and has far-reaching implications for diverse areas of mathematics. In previous work I have proved the conjecture in fundamental cases, established its precise relationship with the Hodge-theoretic version of mirror symmetry, and used these results to solve questions in enumerative geometry and symplectic topology.

The proposed research centres on HMS, new aspects of its relationship with Hodge theory, and new applications to symplectic topology. It is split into four projects:

1. Prove HMS for Gross-Siebert mirrors (this covers the vast majority of mirror pairs proposed in the literature). As a preliminary step in this direction we will prove HMS for Batyrev mirrors.

2. Prove that HMS implies mirror symmetry for open Gromov-Witten invariants. The key step will be the construction of a mirror to the Abel-Jacobi map.

3. Enrich the Hodge-theoretic structures emerging from HMS with rational structures. The key step will be to show that the Gamma rational structure on quantum Hodge theory emerges from the topological K-theory of the Fukaya category.

4. The Lagrangian cobordism group is an important invariant of a symplectic manifold, which can be used to study some of the most fundamental questions in symplectic topology such as the classification of Lagrangian submanifolds. We will elucidate its structure by using its relationship, via HMS, with the Chow group of the mirror variety.