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Topics in symplectic Gromov–Witten theory https://doi.org/10.17863/CAM.104156 Hirschi, Amanda Keating, Ailsa The main focus of this thesis is on the Gromov--Witten theory of general symplectic manifolds. Mohan Swaminathan and I construct a framework to define a virtual fundamental class for the moduli space of stable maps to a general closed symplectic manifold. Our construction, inspired by [AMS21], works for all genera and leads to a more straightfoward definition of symplectic Gromov--Witten invariants as was previously available. We prove a formula for the Gromov--Witten invariants of a product of two symplectic manifolds, conjectured in [KM94]. I generalise the product formula to a formula for the Gromov--Witten invariants of a suitable fibre product of symplectic manifolds. Our invariants satisfy the Kontsevich-Manin axioms and are extended to descendent Gromov--Witten invariants. I show that our definition of Gromov--Witten invariants agrees with the classical Gromov--Witten invariants defined by [RT97] for semipositive symplectic manifolds. Given a Hamiltonian group action on the target manifold, I construct equivariant Gromov--Witten invariants and prove a virtual Atiyah--Bott-type localisation formula, providing a tool for computations. Together with Soham Chanda and Luya Wang, I construct infinitely many exotic Lagrangian tori in complex projective spaces of complex dimension higher than $2$. We lift tori in $\mathbb{P}$², constructed by Vianna, and show that these lifts remain non-symplectomorphic, using an invariant derived from pseudoholomorphic disks. Noah Porcelli and I use Ljusternik-Schnirelmann theory, applied to moduli spaces of pseudoholomorphic curves, and homotopy theory to prove lower bounds on the number of intersection points of two (possibly non-transverse) Lagrangians in terms of the cuplength of the Lagrangian in generalised cohomology theories, improving previous lower bounds by Hofer. University of Cambridge 2023-08 Thesis Doctoral Doctor of Philosophy (PhD) eng Engineering and Physical Sciences Research Council (2434378) https://www.repository.cam.ac.uk/handle/1810/361000 https://www.repository.cam.ac.uk/bitstreams/411fb4a6-b075-496b-817c-f3943347b9aa/download 87eda9de84448d1f82354d60eee3eb5f https://www.repository.cam.ac.uk/bitstreams/9c3ce892-cf19-42d6-a87d-0827e1670f51/download 61863cedc413701caefa6fa51255e6c7 https://www.rioxx.net/licenses/all-rights-reserved/ Gromov-Witten theory symplectic