Browsing FAS Theses and Dissertations by FAS Department "Mathematics"
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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 ... 
$\ell^{\infty}$Selmer Groups in Degree $\ell$ Twist Families
(20200515)Suppose $E$ is an elliptic curve over $\QQ$ with no nontrivial rational $2$torsion point. Given a nonzero integer $d$, take $E^d$ to be the quadratic twist of $E$ coming from the field $\QQ(\sqrt{d})$. For every nonnegative ... 
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 ... 
Geometric Properties of Families of Galois Representations
(20180510)This thesis concerns families of Galois representations arising as etale local systems on a variety over a number field or a padic field. The first part of the thesis studies families of Galois representations of number ... 
Geometric Variational Problems for Mean Curvature
(20180501)This thesis investigates variational problems related to the concept of mean curvature on submanifolds. Our primary focus is on the area functional, whose critical points are the minimal submanifolds and whose gradient ... 
The Geometry of Hurwitz Space
(20130930)We explore the geometry of certain special subvarieties of spaces of branched covers which we call the Maroni and CasnatiEkedahl loci. Our goal is to understand the divisor theory on compactifications of Hurwitz space, ... 
The Geometry of the WeilPetersson Metric in Complex Dynamics
(20140606)In this work, we study an analogue of the WeilPetersson metric on the space of Blaschke products of degree 2 proposed by McMullen. We show that the WeilPetersson metric is incomplete and study its metric completion. Our ... 
Goodwillie Approximations to Higher Categories
(20150514)Goodwillie calculus involves the approximation of functors between higher categories by socalled polynomial functors. We show (under mild hypotheses) how to associate to a higher category C a Goodwillie tower, consisting ... 
High Dimensional Normality of Noisy Eigenvectors
(20200515)We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional ... 
Holomorphically parametrized L2 Cramer's rule and its algebraic geometric applications
(20131008)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 ... 
Indirect Reciprocity With Optional Interactions and Private Information and Stochastic Evolution of Staying Together
(20190417)We explore indirect reciprocity with optional interactions and private information and stochastic evolution of staying together. Indirect reciprocity is cooperation based on reputation in a group of players; my behavior ... 
Integral canonical models for Gbundles on Shimura varieties of abelian type
(20170502)This thesis builds on Kisin's theories of Smodules and integral models for Shimura varieties of abelian type to further our understanding of the arithmetic of Shimura varieties in several directions. First, we show that ... 
Interpolation and Vector Bundles on Curves
(20150505)Interpolation is a property of vector bundles on curves closely related to slope stability. The notion is motivated by the deformation theory of curves in projective space incident to given fixed subvarieties. If the normal ... 
Local Statistics of Dyson Brownian Motion
(20180426)The time to equilibrium of the local statistics of Dyson Brownian motion with general initial data is investigated. In the bulk of the spectrum it is established that the local statistics coincide with the invariant GOE/GUE ... 
LocalGlobal Compatibility and the Action of Monodromy on nearby Cycles
(20121219)In this thesis, we study the compatibility between local and global Langlands correspondences for \(GL_n\). This generalizes the compatibility between local and global class field theory and is related to deep conjectures ... 
Mapping Class Groups, Homology and Finite Covers of Surfaces
(20121220)Let S be an orientable surface of genus g with n punctures, such that \(\chi(S) = 2 − 2g − n < 0\). Let \(\psi \epsilon Mod(S)\) denote an element in its mapping class group. In this thesis, we study the action of \(\psi\) ...