Floer homology, low-dimensional topology, and mirror symmetry

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This project studies the applications of symplectic geometry, and in particular Lagrangian Floer homology, to mirror symmetry and to low-dimensional topology. Lagrangian submanifolds and Fukaya categories lie at the heart of Kontsevich's homological mirror symmetry conjecture and the Strominger-Yau-Zaslow approach to the construction of mirror pairs. The project will focus on geometric phenomena such as instanton corrections in order to gain a better understanding of mirror symmetry and broaden its scope. Lagrangian submanifolds also play a key role in low-dimensional topology, where they enter in the construction of various invariants of 3- and 4-manifolds (closed or with boundary). This motivates the study of bordered Heegaard-Floer homology of 3-manifolds and invariants of broken Lefschetz fibrations on 4-manifolds from the perspective of Fukaya categories of symmetric products, with the aim of providing a richer algebraic framework and revealing new connections.Broadly speaking, this project aims to reinforce the existing connections between various areas of geometry, topology and mathematical physics. Modern theoretical physics has had a tremendous impact on mathematics, and in particular on geometry, where equations arising from field theories have led to new invariants of topological spaces and new conjectures about their geometry. One goal of the project will be to clarify the mathematical validity and scope of predictions inspired by string theory, relating two different areas of mathematics to each other (algebraic geometry, which studies sets defined by polynomial equations, and symplectic geometry, which studies the phase spaces of classical mechanics). On the other hand, the same mathematical ideas have applications to the study of the topology of three and four-dimensional spaces. More specifically, the aim is to explore how slicing such spaces along two-dimensional surfaces can lead to new interpretations of various topological invariants.