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ItemConstructions and Invariants of High-Dimensional Legendrian Submanifolds(Georgia Institute of Technology, 2023-07-17) Roy, AgnivaThis thesis explores the question of understanding Legendrian submanifolds in contact manifolds of dimension greater than 3. There are two primary contributions. First, we explore two natural constructions of Legendrian spheres from supporting open book decompositions and show that these always yield the standard Legendrian unknot. Second, in joint work with Hughes, we explore the Legendrians obtained from the doubling construction in dimension 5, and represent them as Legendrian weaves. We show that a large family of pairwise non-isotopic Legendrians can be obtained by looking at doubles associated to torus links T(2, n). Further we also mention theorems in progress regarding the fillability of these doubled Legendrians.
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ItemContact geometric theory of Anosov flows in dimension three and related topics(Georgia Institute of Technology, 2022-07-20) Hozoori, SurenaThis thesis consists of the author's work on the contact and symplectic geometric theory of Anosov flows in low dimensions, as well as the related topics from Riemannian geometry. This includes the study of the interplay between various geometric, topological and dynamical features of such flows. After reviewing some basic elements from the theory of contact and symplectic structures in low dimensions, we discuss a characterization of Anosov flows on three dimensional manifolds, purely in terms of those geometric structure. This is based on the previous observations of Mitsumatsu and Eliashberg-Thurston in the mid 90s, and in the context of a larger class of dynamics, namely projectively Anosov flows. Our improvement of those observations, which have been left unexplored to a great extent in our view, facilitates employing new geometric tools to the study of questions about (projectively) Anosov flows and vice versa. We then discuss another characterization of Anosov three flows, in terms of the associated underlying Reeb dynamics. Beside the contact topological consequences of this result, it sheds light on contact geometric interpretation of the existence of an invariant volume form for these flows, a condition which is well known to have deep consequences in the dynamics of the flow from the viewpoint of the long term behavior of the flow (transitivity) and measure theory (ergodicity). The implications of these results on various related theories, namely, Liouville geometry, the theory of contact hyperbolas and bi-contact surgery, are discussed as well. As contact Anosov flows are an important and well studied special case of volume preserving Anosov flows, we also make new observation regarding these flows, utilizing the associated Conley-Zehnder indices of their periodic orbits, a classical tool from the field of contact dynamics. We finally discuss some Riemannian geometric motivations in the study of contact Anosov flows in dimension three. In particular, this bridges our study to the curvature properties of Riemannian structures, which are compatible with a given contact manifold. Our study of the curvature in this context goes beyond the study of Anosov dynamics, although has implications on the topic. In particular, we investigate a natural curvature realization for compatible Riemannian structures, namely Ricci-Reeb realization problem. The majority of the results in this manuscript, with the exception of some parts of Chapter 5, can be found in the author's previous papers.
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ItemOn embeddings of 3-manifolds in symplectic 4-manifolds(Georgia Institute of Technology, 2022-07-13) Mukherjee, AnubhavWe proposed a conjecture that every 3-manifolds smoothly embedded in some closed symplectic 4-manifolds. This work shows that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary.
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ItemClassification fo tight contact structures on the Weeks manifold(Georgia Institute of Technology, 2021-08-18) Min, Hyun KiOne of important problems in 3-dimensional contact geometry is to figure out which 3-manifolds admit tight contact structures and classify tight contact structures on the manifolds which admit tight contact structures. It turned out that this problem is closely related to the topology of 3-manifolds. For Seifert fibered spaces and toroidal manifolds, there has been a lot of results. On the other hand, there has been relatively few results for hyperbolic manifolds. In this thesis, we classify tight contact structures on the Weeks manifold, which is a hyperbolic L-space, having the smallest volume among all closed orientable hyperbolic 3-manifolds.
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ItemBranched Covers and Braided Embeddings(Georgia Institute of Technology, 2021-06-25) Kolay, SudiptaWe study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.