Browsing FAS Theses and Dissertations by FAS Department "Mathematics"
Now showing items 1-20 of 65
-
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 ... -
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 ... -
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 ... -
Equivariant Weiss Calculus and Loops of Stiefel Manifolds
(2016-05-18)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
(2015-05-17)In this article, for nilpotent orbits in (the Lie algebras of) ramified quasi-split unitary groups with two Jordan blocks, we give the values of their Shalika germs at certain equi-valued elements with half-integral depth ... -
Galois Deformation Ring and Barsotti-Tate Representations in the Relative Case
(2016-05-13)In this thesis, we study finite locally free group schemes, Galois deformation rings, and Barsotti-Tate representations in the relative case. We show three independent but related results, assuming p > 2. First, we give ...