For a closed oriented smooth 4-manifold X with $b^2_+(X)>0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This thesis describes well-defined counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2-forms, and it is shown that they recover the Seiberg-Witten invariants (modulo 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds.

The main results are the following. Given a suitable near-symplectic form w and tubular neighborhood N of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a map on the set of spin-c structures of X. They are furthermore equal to the Seiberg-Witten invariants with mod 2 coefficients, where w determines the "chamber" for defining the latter invariants when $b^2_+(X)=1$.

In the final chapter, as a non sequitur, a new proof of the Fredholm index formula for punctured pseudoholomorphic curves is sketched. This generalizes Taubes' proof of the Riemann-Roch theorem for compact Riemann surfaces.

This dissertation constructs a Morse-Bott version of embedded contact homology (ECH). The dissertation is comprised of two parts, corresponding to the two papers written by the author as a graduate student at UC Berkeley. The first part explains how to compute ECH in the Morse-Bott setting when certain transversality conditions are met and provided a certain correspondence theorem is true; and gives a large class of examples where the transversality conditions are satisfied. The second part provides the analytic foundations of the first part by giving a proof of the correspondence theorem.

Embedded contact homology (ECH) is a powerful three-manifold invariant that encodes information about the underlying contact manifold. It has been shown that embedded contact homology is isomorphic to certain versions of Seiberg-Witten Floer homology and Heegaard Floer homology. This thesis highlights the relations of ECH to other three-dimensional Floer theories. In Chapter 2, we use ideas from periodic Floer homology, a cousin of ECH, and techniques from pseudo-Anosov maps to answer a knot detection question in knot Floer homology. This is joint work with Ethan Farber and Braeden Reinoso. In Chapter 3, we show a connected sum formula for ECH by studying pseudo-holomorphic curves in the symplectization of the contact connected sum, without going through the tremendous isomorphisms to Seiberg-Witten Floer homology or Heegaard Floer homology. Our chain level description of the connected sum complex is useful for studying studying similar formulas for other contact homologies and ECH spectral invariants in the future. In Chapter 3, we give a relative version of ECH for contact three-manifolds with convex sutured boundaries. This generalizes both sutured ECH and Lipshitz's cylindrical reformulation of Heegaard Floer homology, and provides a potential framework for bordered ECH. This is joint work with Julian Chaidez, Oliver Edtmair, Yuan Yao and Ziwen Zhao.

Embedded contact homology (ECH) is an invariant of a contact three-manifold. In Part I of this thesis, we provide a combinatorial description of the ECH chain complex of certain ``toric'' contact manifolds. This is an extension of the combinatorial description appearing in Hutchings-Sullivan. ECH capacities are invariants of a symplectic four-manifold with boundary, which give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. In Part II of this thesis, we compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called ``concave toric domains''. Examples include the (nondisjoint) union of two ellipsoids in $\R^4$. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. This is a joint work with D. Cristofaro-Gardiner, D. Frenkel, M. Hutchings and V. G. B. Ramos.

This dissertation discusses the algebraic structures of fixed point Floer homologies. Thedissertation is divided into three chapters, and is adapted from two joint papers by the author. Chapter 1 gives a brief review of the fixed point Floer homology. Chapter 2 gives a detailed computation of the product and the coproduct structures on the fixed point Floer homology of iterations of a single Dehn twist on a surface. Chapter 3 gives a direct verification of the closed-string mirror symmetry for nodal curves based on the computations from Chapter 2.

This thesis explores quantitative invariants in symplectic geometry. The first chapter is based on joint work with Julian Chaidez [Invent. Math. 229 (2022), 243–301], where we construct the first examples of dynamically convex domains that are not symplectomorphic to any convex domain. A key component of this work is a quantitative convexity criterion involving the Ruelle invariant.In the second chapter, based on the paper [arXiv:2206.07847], we establish a sharp dynamical characterization of convex domains in four-dimensional Euclidean space that admit a symplectic embedding into a given cylinder. More specifically, we demonstrate that the cylindrical capacity of a convex domain matches the minimum symplectic area of a disk-like global surface of section for the natural Reeb flow on the boundary of the domain. This contributes to progress toward the strong Viterbo conjecture regarding the equivalence of symplectic capacities on convex domains. The third chapter comprises a joint paper with Michael Hutchings [arXiv:2110.02463]. We show a quantitative smooth closing lemma for area preserving surface diffeomorphisms. Our proof relies on spectral invariants arising from periodic Floer homology. We also establish a new Weyl law for these spectral invariants.

This dissertation is comprised of two papers studying the topology of certain spaces of symplectic embeddings.

The first paper shows how given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result which claims that given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, one can prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible.

The second paper proves how given two 2n-dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, a certain map from the n-torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map at the level of homology with mod 2 coefficients.

This dissertation is a collection of four papers involving embedded contact homology (ECH). ECH is a three-manifold invariant recently defined by Michael Hutchings. The dissertation covers both technical results about ECH (for example a proof that the absolute grading in ECH is a topological invariant) and applications of ECH to Reeb dynamics and symplectic embedding problems. All four papers were written while the author was a graduate student at UC Berkeley.

The embedded contact homology (ECH) of a 3-manifold Y is a topological invariant defined using a contact form on Y which counts certain pseudoholomorphic curves in its symplectization. We compute the ECH of nontrivial circle bundles over Riemann surfaces.

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.