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Modularity of some elliptic curves over totally real fields
(2014-06-06)
In this thesis, we investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular j-invariants. By analyzing quadratic points on some modular ...
Regeneration of Elliptic Chains with Exceptional Linear Series
(2014-06-06)
We study two dimension estimates regarding linear series on algebraic curves. First, we generalize the classical Brill-Noether theorem to many cases where the Brill-Noether number is negative. Second, we extend results of ...
The complex geometry of Teichmüller space
(2014-06-06)
We study isometric maps between Teichmüller spaces and bounded symmetric domains in their Kobayashi metric. We prove that every totally geodesic isometry from a disk to Teichmüller space is either holomorphic or anti-holomorphic; ...
The Eigencurve is Proper
(2014-06-06)
Coleman and Mazur constructed a rigid analytic curve Cp,N, called the eigencurve, whose points correspond to all finite slope overconvergent p-adic eigenforms. We prove the conjecture that the eigencurve Cp,N is proper ...
Arithmetic Properties of Moduli Spaces and Topological String Partition Functions of Some Calabi-Yau Threefolds
(2014-06-06)
This thesis studies certain aspects of the global properties, including geometric and arithmetic, of the moduli spaces of complex structures of some special Calabi-Yau threefolds (B-model), and of the corresponding topological ...
Poverty, Disease, and the Ecology of Complex Systems
(Public Library of Science, 2014)
Understanding why some human populations remain persistently poor remains a significant challenge for both the social and natural sciences. The extremely poor are generally reliant on their immediate natural resource base ...
Open Gromov-Witten Invariants on Elliptic K3 Surfaces and Wall-Crossing
(2013-10-08)
We defined a new type of open Gromov-Witten invariants on hyperK\"aher manifolds with holomorphic
Dynamics of HIV treatment and social contagion
(2013-10-08)
Modern-day management of infectious diseases is critically linked to the use of mathematical models to understand and predict dynamics at many levels, from the mechanisms of pathogenesis to the patterns of population-wide ...
Holomorphically parametrized L2 Cramer's rule and its algebraic geometric applications
(2013-10-08)
Suppose $f,g_1,\cdots,g_p$ are holomorphic functions over $\Omega\subset\cxC^n$. Then there raises a natural question: when can we find holomorphic functions $h_1,\cdots,h_p$ such that $f=\sum g_jh_j$? The celebrated Skoda ...
Statistical Analysis of Molecular Signal Recording
(Public Library of Science, 2013)
A molecular device that records time-varying signals would enable new approaches in neuroscience. We have recently proposed such a device, termed a “molecular ticker tape”, in which an engineered DNA polymerase (DNAP) ...