Descripción del proyecto
Estudiar las interacciones de la simetría especular homológica con la teoría de Hodge y la topología simpléctica
La simetría especular homológica es una conjetura matemática que intenta explicar un fenómeno denominado simetría especular y que fue observado, por primera vez, por físicos dedicados al estudio de la teoría de cuerdas. Esta conjetura matemática fue concebida originalmente como un isomorfismo entre la teoría de Hodge de una variedad algebraica y la teoría cuántica de Hodge de una topología simpléctica especular. La conjetura predice la equivalencia entre la categoría derivada de la variedad y la categoría de Fukaya de su simetría especular, lo tiene implicaciones relevantes para diversas áreas de las matemáticas. El proyecto HMS, financiado con fondos europeos, llevará a cabo una investigación para poner de relieve nuevos aspectos de la relación entre la simetría especular homológica y la teoría de Hodge, e identificará aplicaciones previamente desconocidas para la topología simpléctica.
Objetivo
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.
Ámbito científico
Palabras clave
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
EH8 9YL Edinburgh
Reino Unido