Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold

is derived equivalent to the category of coherent sheaves on the mirror complex manifold.

Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are

missing to prevent HMS from being true. One possible solution is to include coisotropic

branes into the Fukaya category. This thesis gives a construction for linear symplectic tori of

a version of Fukaya category including coisotropic branes by using a doubling procedure, and

discussing the relation between the Fukaya category of the doubling torus and the Fukaya

category of the original torus.

Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce a new interpretation of the Fukaya-Seidel category of a Laurent polynomial on (C*)^n, which has other potential applications, and give evidence of homological mirror symmetry for non-compact toric varieties by computing certain Floer-theoretic natural transformations.

We present two geometric constructions of Lagrangian surgeries between two Lagrangian submanifolds intersecting cleanly along a 1-dimensional submanifold. We show in a concrete example in 4-dimensions that the two constructions are isomorphic in the Fukaya Category and represent a mapping cone.

We produce for each tropical hypersurface $V(\phi)\subset Q=\RR^n$ a Lagrangian $L(\phi)\subset (\CC^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are mirror to hypersurfaces in a toric mirror. These constructions are extended to tropical varieties given by locally planar intersections, and to symplectic 4-manifolds with almost toric fibrations. We also explore relations to wall-crossing, dimer models, and Lefschetz fibrations. A complete example is worked out for the mirror pair $(\CP^2\setminus E, W), X_{9111}$.

We define a new class of symplectic spaces called ``pumpkin domains'', which roughly speaking comprise a Liouville domain and a Liouville hypersurface of its boundary. To such an object we assign an $A_\infty$-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a pumpkin domain, and the partially wrapped Fukaya category of this pumpkin domain is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates the partially wrapped Fukaya category of a pumpkin domain to the wrapped Fukaya category of its underlying Liouville domain. This operation is mirror to removing a divisor.

Given a punctured Riemann surface with a pair-of-pants decomposition, we compute its

wrapped Fukaya category in a suitable model by reconstructing it from those of various pairs

of pants. The pieces are glued together in the sense that the restrictions of the wrapped

Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree.

The A infinity-structures are given by those in the pairs of pants. The category of singularities

of the mirror Landau-Ginzburg model can also be constructed in the same way from local

ane pieces that are mirrors of the pairs of pants.

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs $X$ and $Y$ such that the complex geometry on $X$ mirrors the symplectic geometry on $Y$. It allows one to deduce information about $Y$ from known properties of $X$. Strominger-Yau-Zaslow (1996) described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich (1994) conjectured that a complex invariant on $X$ (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of $Y$ (the Fukaya category). This is known as homological mirror symmetry. In this project, we first use the construction of SYZ mirrors for hypersurfaces in abelian varieties following Abouzaid-Auroux-Katzarkov, in order to obtain $X$ and $Y$ as manifolds. The complex manifold comes from the genus 2 curve as a hypersurface in its Jacobian torus, and we equip the SYZ mirror manifold with a symplectic form. We then describe an embedding of the category on the complex side into a cohomological Fukaya-Seidel category of $Y$ as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations.

We construct an exotic monotone Lagrangian torus in CP2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian isotopic to the known Clifford

and Chekanov tori.