Now showing items 1-10 of 156
Modularity of some elliptic curves over totally real fields
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
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
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
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
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 ...
Open Gromov-Witten Invariants on Elliptic K3 Surfaces and Wall-Crossing
We defined a new type of open Gromov-Witten invariants on hyperK\"aher manifolds with holomorphic
Dynamics of HIV treatment and social contagion
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
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 ...
Symmetric Spaces and Knot Invariants from Gauge Theory
In this thesis, we set up a framework to define knot invariants for each choice of a symmetric space. In order to address this task, we start by defining appropriate notions of singular bundles and singular connections for ...
Chiral Principal Series Categories
This thesis begins a study of principal series categories in geometric representation theory using the Beilinson-Drinfeld theory of chiral algebras. We study Whittaker objects in the unramified principal series category. ...