Given a closed oriented Riemann surface $\Sigma$ of genus greater than zero, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$. We show that $\mathcal{F}$ is an isomorphism of algebras. We also define a higher-dimensional analog of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p:W\to D\subset\mathbb{C}$ of a $2n$-dimensional Milnor fiber of the $A_{2\kappa-1}$ singularity. We represent a link by a $\kappa$-strand braid, which in turn is represented as an element $h$ of the symplectic mapping class group $\mathrm{Symp}(W,\partial W)$. We then apply the higher-dimensional Heegaard Floer homology machinery to the pair $(\boldsymbol{a},h(\boldsymbol{a}))$, where $\boldsymbol{a}$ is a collection of $\kappa$ unstable manifolds of $W$ which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant.

We generalize some properties of surface automorphisms of pseudo-Anosov type.

First, we generalize the Penner construction of a pseudo-Anosov homeomorphism

and show that a symplectic automorphism which is constructed by our generalized

Penner construction has an invariant Lagrangian branched submanifold and an invariant

Lagrangian lamination, which are higher-dimensional generalizations of a

train track and a geodesic lamination in the surface case. Moreover, if a pair consisting

of a symplectic automorphism and a Lagrangian branched surface B satisfies

some assumptions, we prove that there is an invariant Lagrangian lamination L

which is a higher-dimensional generalization of a geodesic lamination.

In this thesis we prove some classification results for symplectic and exact Lagrangian fillings in contact geometry. First we prove a classification result for symplectic fillings of certain contact manifolds. Let $(M,\xi)$ be a contact 3-manifold and $T^2 \subset (M,\xi)$ a mixed torus. We prove a JSJ-type decomposition theorem for strong and exact symplectic fillings of $(M,\xi)$ when $(M,\xi)$ is cut along $T^2$. As an application we prove the uniqueness of exact fillings when $(M,\xi)$ is obtained by Legendrian surgery on a knot in $(S^3,\xi_{std})$ which is stabilized both positively and negatively. Second we show a classification result for Lagrangian fillings of Legendrian representatives of positive braid closures in $S^3$. This second result follows from an injectivity result for augmentation categories of positive braids.

We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of κ-tuples of cotangent fibers and κ-tuples of conormal bundles of homotopically nontrivial simple closed curves in T ∗Σ with a certain braid skein group, where Σ is a closed oriented surface of genus > 0 and κ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to Σ. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where Σ = T 2 and the cotangent fibers and conormal bundles of curves are both parallel copies.

This dissertation contains results that contribute to and use the theory of convex hypersurfaces in contact manifolds. First, we generalize a $3$-dimensional convexity criterion result of Giroux \cite{giroux1991convexite}. Specifically, we show that the criterion holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori \cite{mori2011convex} is $C^{\infty}$-close to a convex hypersurface. Second, inspired by the techniques of Honda and Huang in \cite{honda2019convex}, we develop explicit local operations that may be applied to Liouville domains with the goal of simplifying the dynamics of the Liouville vector field. As an application, we show that Mitsumatsu's well-known Liouville-but-not-Weinstein domains are stably Weinstein, answering a question posed by Huang \cite{huang2019dynamical}.

We define splitting surfaces in contact manifolds, and develop a technique for decomposing the strong or exact symplectic fillings of a contact manifold which admits such a surface, using Eliashberg’s strategy of filling by holomorphic discs. We then apply this technique to the symplectic filling classification problem for several families of contact manifolds. In particular, we complete the classification of exact fillings for lens spaces, virtually overtwisted torus bundles, and virtually overtwisted circle bundles over Riemann surfaces. We also produce classification results for contact manifolds obtained by surgery on Legendrian negative cables, and for large families of contact structures on Seifert fibered spaces.

In this thesis, we give a proposed construction of cobordism maps in embedded contact homology via a count of $J$-holomorphic curves. We prove a new index formula in the $L$-supersimple setting of Bao-Honda and use it to classify degenerations of $1$-dimensional moduli spaces of curves in exact symplectic cobordisms. We correct these degenerations by using obstruction bundle techniques of Hutchings-Taubes to continue the moduli spaces by gluing in branched covers of trivial cylinders. We then use a new evaluation map to cut out a $1$-dimensional family in the resulting moduli space, modulo two conjectures on the behavior of this map. Finally, we analyze the new endpoints of the continuations and use them to complete our proposed definition of the cobordism maps.