Browsing FAS Theses and Dissertations by FAS Department "Mathematics"
Now showing items 120 of 65

2Selmer groups and Heegner points on elliptic curves
(20150507)This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and SwinnertonDyer conjecture when the 2Selmer group has rank one. For certain elliptic ... 
Algebraicity Criteria and Their Applications
(20160504)We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz pcurvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields. The Grothendieck–Katz ... 
Algorithms and Models for Genome Biology
(20140225)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
(20121219)We construct several modular compactifications of the Hurwitz space \(H^d_{g/h}\) of genus g curves expressed as dsheeted, simply branched covers of genus h curves. They are obtained by allowing the branch points of the ... 
Analysis of Some PDEs over Manifolds
(20130214)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
(20120810)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 CalabiYau Threefolds
(20140606)This thesis studies certain aspects of the global properties, including geometric and arithmetic, of the moduli spaces of complex structures of some special CalabiYau threefolds (Bmodel), and of the corresponding topological ... 
Chiral Principal Series Categories
(20140606)This thesis begins a study of principal series categories in geometric representation theory using the BeilinsonDrinfeld theory of chiral algebras. We study Whittaker objects in the unramified principal series category. ... 
Complete Homogeneous Varieties via Representation Theory
(20160502)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
(20140606)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 antiholomorphic; ... 
Covers of an Elliptic Curve E and Curves in ExP1
(20150430)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 ... 
DModules on Spaces of Rational Maps and on Other Generic Data
(20121213)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
(20160428)We investigate several questions pertaining to the enumerative and deformationtheoretic behavior of lowgenus curves on algebraic varieties, using specialization techniques. 
Derived categories and birational geometry of GushelMukai varieties
(20160517)We study the derived categories of coherent sheaves on GushelMukai 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
(20140606)Coleman and Mazur constructed a rigid analytic curve Cp,N, called the eigencurve, whose points correspond to all finite slope overconvergent padic eigenforms. We prove the conjecture that the eigencurve Cp,N is proper ... 
Entire Surfaces of Prescribed Curvature in Minkowski 3Space
(20180516)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 OneDimensional Dynamical Systems
(20130930)The goal of this thesis is to provide a unified framework in which to analyze the dynamics of two seemingly unrelated families of onedimensional dynamical systems, namely the family of quadratic polynomials and continued ... 
Equivariant Weiss Calculus and Loops of Stiefel Manifolds
(20160518)In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration of the Stiefel manifolds O(V ;W) and U(V ;W) of orthogonal and unitary, respectively, maps V > V ⊕W stably split as a wedge sum of Thom spaces defined ... 
A Formula for Some Shalika Germs
(20150517)In this article, for nilpotent orbits in (the Lie algebras of) ramified quasisplit unitary groups with two Jordan blocks, we give the values of their Shalika germs at certain equivalued elements with halfintegral depth ... 
Galois Deformation Ring and BarsottiTate Representations in the Relative Case
(20160513)In this thesis, we study finite locally free group schemes, Galois deformation rings, and BarsottiTate representations in the relative case. We show three independent but related results, assuming p > 2. First, we give ...