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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 ...
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
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 ...
Symmetric Spaces and Knot Invariants from Gauge Theory
(2014-06-06)
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
(2014-06-06)
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. ...
Relative Jacobians of Linear Systems
(2014-06-06)
Let X be a smooth projective variety. Given any basepoint-free linear system, |D|, there is a dense open subset parametrizing smooth divisors, and over that subset, we can consider the relative Picard variety of the universal ...