Now showing items 1-20 of 81

• #### 2-Selmer groups and Heegner points on elliptic curves ﻿

(2015-05-07)
This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and Swinnerton-Dyer conjecture when the 2-Selmer group has rank one. For certain elliptic ...
• #### A Model 2-Category of Enriched Combinatorial Premodel Categories ﻿

(2019-09-10)
Quillen equivalences induce equivalences of homotopy theories and therefore form a natural choice for the "weak equivalences" between model categories. In [21], Hovey asked whether the 2-category Mod of model categories ...
• #### A p-adic Jacquet-Langlands Correspondence ﻿

(2017-05-16)
In this paper, we construct a candidate p-adic Jacquet-Langlands correspondence. This is a correspondence between unitary continuous admissible representations of GL2(Qp) valued in p-adic Banach spaces, and unitary continuous ...
• #### Algebraicity Criteria and Their Applications ﻿

(2016-05-04)
We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz p-curvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields. The Grothendieck–Katz ...
• #### Algorithms and Models for Genome Biology ﻿

(2014-02-25)
New advances in genomic technology make it possible to address some of the most fundamental questions in biology for the first time. They also highlight a need for new approaches to analyze and model massive amounts of ...
• #### Alternate Compactifications of Hurwitz Spaces ﻿

(2012-12-19)
We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus g curves expressed as d-sheeted, simply branched covers of genus h curves. They are obtained by allowing the branch points of the ...
• #### Anabelian Intersection Theory ﻿

(2012-12-19)
Let F be a field finitely generated and of transcendence degree 2 over $\bar{\mathbb{Q}}$. We describe a correspondence between the smooth algebraic surfaces X defined over $\bar{\mathbb{Q}}$ with field of rational ...
• #### Analysis of Some PDEs over Manifolds ﻿

(2013-02-14)
In this dissertation I discuss and investigate the analytic aspect of several elliptic and parabolic partial differential equations arising from Rimannian and complex geometry, including the generalized Ricci flow, Gaussian ...
• #### The Arithmetic of Simple Singularities ﻿

(2012-08-10)
We investigate some arithmetic orbit problems in representations of linear algebraic groups arising from Vinberg theory. We aim to give a description of the orbits in these representations using methods with an emphasis ...
• #### 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 ...
• #### 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. ...
• #### Complete Homogeneous Varieties via Representation Theory ﻿

(2016-05-02)
Given an algebraic variety $X\subset\PP^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a \textit{homogeneous variety} if ...
• #### 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; ...
• #### Covers of an Elliptic Curve E and Curves in ExP1 ﻿

(2015-04-30)
We describe the hyperplane sections of the Severi variety of curves in ExP1 in a similar fashion to Caporaso—Harris’ seminal work. From this description we almost get a recursive formula for the Severi degrees—we get the ...
• #### D-Modules on Spaces of Rational Maps and on Other Generic Data ﻿

(2012-12-13)
Fix an algebraic curve X. We study the problem of parametrizing geometric data over X, which is only generically deﬁned. E.g., parametrizing generically deﬁned maps from X to a fixed target scheme Y. There are three methods ...
• #### Degenerations, Log K3 Pairs and Low Genus Curves on Algebraic Varieties ﻿

(2016-04-28)
We investigate several questions pertaining to the enumerative and deformation-theoretic behavior of low-genus curves on algebraic varieties, using specialization techniques.
• #### Derived categories and birational geometry of Gushel-Mukai varieties ﻿

(2016-05-17)
We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether ...
• #### 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 ...
• #### Entire Surfaces of Prescribed Curvature in Minkowski 3-Space ﻿

(2018-05-16)
This thesis concerns the global theory of properly embedded spacelike surfaces in 3 dimensional Minkowski space with prescribed Gaussian curvature. We prove that every regular domain which is not a wedge contains a unique ...
• #### Entropy, Dimension and Combinatorial Moduli for One-Dimensional Dynamical Systems ﻿

(2013-09-30)
The goal of this thesis is to provide a unified framework in which to analyze the dynamics of two seemingly unrelated families of one-dimensional dynamical systems, namely the family of quadratic polynomials and continued ...