Lagrangian submanifolds: open Gromov-Witten theory and Mirror Symmetry

Abstract

The project's subject area is a subarea of mathematics named symplectic geometry. Geometrical objects are called manifolds. In symplectic geometry, Lagrangian submanifolds play a major role, as indicated by the famous Weinstein creed "Everything is a Lagrangian", meaning that one should express objects and constructions in symplectic geometry as Lagrangian submanifolds. In particular, by Strominger-Yau-Zaslow (SYZ), Lagrangian fibrations are in the core of the geometric explanation for the prolific and recent area of study, mirror symmetry, which relates symplectic geometric objects with algebraic geometric ones. Open Gromov-Witten theory, corresponding to the study of pseudo-holomorphic discs with boundary on a given Lagrangian, is the main tool in the recent approach to understand Lagrangians. This project contains 4 main subprojects involving classification of Lagrangians and the study of Lagrangian fibrations in several types of symplectic manifolds, with an ample range of applications in symplectic geometry and mirror symmetry. This projects are with international collaboration with renowned researchers, and in one case, in collaboration with a postdoctoral fellow. It also contains a list of future projects thatcan become works for PhD students, or a line of research for collaboration with postdocs. This project therefore contributes with USP's internationalisation endeavour and postgraduate students education. Explicitly, the main projects contemplate: classification of Lagrangian tori in CP^2; applications of the \Psi invariant for Lagrangian fluxes and unobstructedness of SYZ fibres; an open-string quantum Lefschetz formula; mirror symmetry for smoothing of algebraic cones. I have shown the existence of infinite many (up to symplectomorphism) monotone Lagrangian tori in CP^2 and later in all del Pezzo surfaces, which exhausts the monotone symplectic manifolds in dimension 4. This was the first example o infinite many non-equivalente Lagrangians in compact symplectic manifolds, which inspired future works, see citations. Nonetheless, classification of Lagrangian tori is extremely hard work, only develop in a few examples by Dimitroglou-Rizell and collaborators. Therefore, the first project has very high relevance in the field, though of long and hard execution. That being said, a result of Tonkonog implies that the potential, which is the invariant used to distinguished monotone Lagrangians up to symplectomorphism, of any monotone Lagrangian tori in CP^2 must agree with a potential of the toriconstructed by myself. Fixed an almost-complex structure J, we can determine the potential of a Lagrangian submanifold. But this in general vary with the choice of J for non-monotone Lagrangians. The \Psi invariant, formalised by Shelukhin-Tonkonog-V., extracts the invariant elements of the potential. Besides distinguishing between Lagrangians, it can be used to understand Lagrangian isotopies and, consequently, maximal shapes of Lagrangiantubular neighbourhoods. In the second project, we intend to show a deep application of how varying the \Psi invariant under Lagrangian isotopies implies the unobstructedness of SYZ fibres in Calabi-Yau manifolds. The third project describes a formula that relates the potentials of a Lagrangian in a symplectic hypersurface and its Biran lift to a Lagrangian in the initial symplectic manifold, under certain hypothesis. It provides the computation of potentials for Lagrangians in several dimensions (eg $\CP^n$), leading to another plethora of infinitely many Lagrangians. The fourth project intends to use the Lagrangian fibrations in smoothing of algebraic cones, the theme of my former ph.D. student Achig-Andrango, to provide mirror symmetry results for these manifolds. (AU)