We apply branched covering techniques to construct minimal simply-connected symplectic 4-manifolds with small χ_h values. We also use these constructions to provide an alternate proof that for each s ≥ 0, there exists a positive integer λ(s) such that each pair (j,8j+s) with j ≥ λ(s) is realized as (χ_h(M),c_1^2(M)) for some minimal simply-connected symplectic M. The smallest values of λ(s) currently known to the author are also explicitly computed for 0 ≤ s ≤ 99. Our computations in these cases populate 19 952 points in the (χ,c)-plane not previously realized in the existing literature.

In data analysis, data points are usually analyzed based on their relations to other points (e.g., distance or inner product). This kind of relation can be analyzed on the manifold of the data set. Manifold learning is an approach to understand such relations. Various manifold learning methods have been developed and their effectiveness has been demonstrated in many real-world problems in pattern recognition and signal processing. However, most existing manifold learning algorithms only consider one manifold based on one dissimilarity matrix. In practice, multiple measurements may be available, and could be utilized. In pattern recognition systems, data integration has been an important consideration for improved accuracy given multiple measurements. Some data integration algorithms have been proposed to address this issue. These integration algorithms mostly use statistical information from the data set such as uncertainty of each data source, but they do not use the structural information (i.e., the geometric relations between data points). Such a structure is naturally described by a manifold. Even though manifold learning and data integration have been successfully used for data analysis, they have not been considered in a single integrated framework. When we have multiple measurements generated from the same data set and mapped onto different manifolds, those measurements can be integrated using the structural information on these multiple manifolds. Furthermore, we can better understand the structure of the data set by combining multiple measurements in each manifold using data integration techniques. In this dissertation, I present a new concept, manifold integration, a data integration method using the structure of data expressed in multiple manifolds. In order to achieve manifold integration, I formulated the manifold integration concept, and derived three manifold integration algorithms. Experimental results showed the algorithms' effectiveness in classification and dimension reduction. Moreover, for manifold integration, I showed that there are good theoretical and neuroscientific applications. I expect the manifold integration approach to serve as an effective framework for analyzing multimodal data sets on multiple manifolds. Also, I expect that my research on manifold integration will catalyze both manifold learning and data integration research.

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On a bounded domain of $IR^N$, we are interested in the nonlinear Schrödinger problem $-Delta u + V(x)u = vert uvert^{p-2}u$ submitted to the Dirichlet boundary conditions or Neumann boundary conditions. This equation has many interests in astrophysics and quantum mechanics. Depending on the domain and the potential $V$, we are studying numerically (by making and computing algorithms) and theoretically the structure of ground state (resp. least energy nodal) solution, i.e. one-signed (resp. sign-changing) solutions with minimal energy. We prove some symmetry and symmetry breaking results and make a lot of conjectures. We also pay attention to the $p$-Laplacian case and we change the nonlinearity $vert uvert^{p-2}u$.

The thesis is based on the use of mathematical theories and techniques to gain qualitative and quantitative insight into the transmission dynamics of hepatitis C virus (HCV) in an IDU (injecting drug user) population. A deterministic model, which stratifies the IDU population into eight mutually-exclusive compartments (based on epidemiological status), is considered. Rigorous qualitative analysis of the model establishes, for the first time, the presence of the phenomenon of backward bifurcation in HCV transmission dynamics. Three routes (or causes) to such a dynamic phenomenon have been established. Furthermore, five main parameters that play a dominant role on the transmission dynamics of the disease have been identified. Numerical simulations of the model show that the re-infection of recovered individuals has marginal effect on the HCV burden (as measured in terms of the cumulative incidence and prevalence of the disease) in the IDU community.

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Thesis advisor: Robert Meyerhoff / Thurston showed that for all but a finite number of Dehn Surgeries on a cusped hyperbolic 3-manifold, the resulting manifold admits a hyperbolic structure. Global bounds on this number have been set, and gradually improved upon, by a number of Mathematicians until Lackenby and Meyerhoff proved the sharp bound of 10, which is realized by the figure-eight knot exterior. We improve this result by proving a stronger version of Gordon’s conjecture: that excluding the figure-eight knot exterior, cusped hyperbolic 3-manifolds have at most 8 non-hyperbolic Dehn Surgeries. To do so we make use of the work of Gabai et. al. from a forthcoming paper which parameterizes measurements of the cusp, then uses a rigorous computer aided search of the space to classify all hyperbolic 3-manifolds up to a specified cusp size. Their approach hinges on the discreteness of manifold points in the parameter space, an assumption which cannot be made if the manifolds have infinite volume. In this paper we also show that infinite-volume manifolds, which must be Free Bicuspid, can have cusp volume as low as 3.159. As such, these manifolds are a concern for any future expansion of the approach of Gabai et. al. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Peer-to-Peer (P2P) refers to a class of systems and/or applications that use distributed resources in a decentralized and autonomous manner to achieve a goal. A number of successful applications, like BitTorrent (for file and content sharing) and SETI@Home (for distributed computing) have demonstrated the feasibility of this approach. <p> As a new form of distributed computing, P2P computing has the same coordination problems as other forms of distributed computing. Coordination has been considered an important issue in distributed computing for a long time and many coordination models and languages have been developed. <p> This research focuses on how to solve coordination problems in P2P computing. In particular, it is to provide a seamless P2P computing environment so that the migration of computation components is transparent. This research extends Manifold, an event-driven coordination model, to meet P2P computing requirements and integrates the P2P-Manifold model into an existing platform. The integration hides the complexity of the coordination model and makes the model easy to use.

Peer-to-Peer (P2P) refers to a class of systems and/or applications that use distributed resources in a decentralized and autonomous manner to achieve a goal. A number of successful applications, like BitTorrent (for file and content sharing) and SETI@Home (for distributed computing) have demonstrated the feasibility of this approach. <p> As a new form of distributed computing, P2P computing has the same coordination problems as other forms of distributed computing. Coordination has been considered an important issue in distributed computing for a long time and many coordination models and languages have been developed. <p> This research focuses on how to solve coordination problems in P2P computing. In particular, it is to provide a seamless P2P computing environment so that the migration of computation components is transparent. This research extends Manifold, an event-driven coordination model, to meet P2P computing requirements and integrates the P2P-Manifold model into an existing platform. The integration hides the complexity of the coordination model and makes the model easy to use.