Browsing FAS Theses and Dissertations by FAS Department "Mathematics"
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Moduli of Galois Representations
(20130925)The theme of this thesis is the study of moduli stacks of representations of an associative algebra, with an eye toward continuous representations of profinite groups such as Galois groups. The central object of study is ... 
The Moduli Space of S1Type Zero Loci for Z/2 Harmonic Spinors in Dimension 3
(20150517)Let M be a compact oriented 3dimensional smooth manifold. In this paper, we will construct a moduli space consisting of the following date {(Σ,ψ)} where Σ is a C1embedding S1 curve in M, ψ is a Z/2harmonic spinor vanishing ... 
Nearby Cycles of Whittaker Sheaves
(20180513)In this thesis we study the nearby cycles of a Whittaker sheaf as it degenerates to an object of the principal series category. In the case of a finitedimensional flag variety, we prove that the nearby cycles sheaf ... 
Nilpotence and Descent in Stable Homotopy Theory
(20170418)We study various applications of the ideas of descent and nilpotence to stable homotopy theory. In particular, we give a descenttheoretic calculation of the Picard group of topological modular forms and we prove a partial ... 
On Newforms for Split Special Odd Orthogonal Groups
(20130918)The theory of local newforms has been studied for the group of \(PGL_n\) and recently \(PGSp_4\) and some other groups of small ranks. In this dissertation, we develop a newform theory for generic supercuspidal representations ... 
On the Arithmetic of Hyperelliptic Curves
(20160506)My research involves answering various numbertheoretic questions involving hyperelliptic curves. A hyperelliptic curve is a generalization of elliptic curves to curves of higher genus but which still have explicit ... 
On the Framed Singular Instanton Floer Homology From Higher Rank Bundles
(20160511)In this thesis we study the framed singular instanton Floer homology defined by by Kronheimer and Mrowka in \cite{KM3}. Given a 3manifold $Y$ with a link $K$ and $\delta \in H^2(Y,\mathbb{Z})$ satisfying a nonintegral ... 
On the MoyPrasad Filtration and Stable Vectors
(20160519)Let K be a maximal unramified extension of a nonarchimedean local field of residual characteristic p > 0. Let G be a reductive group over K which splits over a tamely ramified extension of K. To a point x in the Bruhat–Tits ... 
Open GromovWitten Invariants on Elliptic K3 Surfaces and WallCrossing
(20131008)We defined a new type of open GromovWitten invariants on hyperK\"aher manifolds with holomorphic 
Pencils of quadrics and Jacobians of hyperelliptic curves
(20131008)Using pencils of quadrics, we study a construction of torsors of Jacobians of hyperelliptic curves twice of which is Pic^1. We then use this construction to study the arithmetic invariant theory of the actions of SO2n+1 ... 
PicardFuchs Systems Arising From Toric and Flag Varieties
(20180512)This thesis studies the PicardFuchs systems for families arising as vector bundles zero loci in toric or partial flag varieties, including RiemannHilbert type theorems and arith metic properties of the differential ... 
PicardLefschetz Oscillators for the DrinfeldLafforgueVinberg Compactification
(20150501)We study the singularities of the DrinfeldLafforgueVinberg compactification of the moduli stack of Gbundles on a smooth projective curve for a reductive group G. The study of these compactifications was initiated by V. ... 
qdeformed Interacting Particle Systems, RSKs and Random Polymers
(20160428)We introduce and study four $q$randomized RobinsonSchenstedKnuth (RSK) insertion tableau dynamics. Each of them is a discrete time Markov dynamics on twodimensional interlacing particle arrays (these arrays are in ... 
Ramification of the Hilbert Eigenvariety
(20190514)Andreatta–Iovita–Pilloni constructed eigenvarieties for cuspidal Hilbert modular forms. The eigenvariety has a natural map to the weight space, called the weight map. We compute the dimension of the tangent space of the ... 
Rational Curves on Hypersurfaces
(20150518)We investigate the dimensions of the spaces of rational curves on hypersurfaces, and various related questions. 
Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory
(20130930)The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed ... 
Real Orientations of LubinTate Spectra and the Slice Spectral Sequence of a C4Equivariant Height4 Theory
(20190516)In this thesis, we show that LubinTate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the GoerssHopkinsMiller theorem to algebras with involution. For each height ... 
Regeneration of Elliptic Chains with Exceptional Linear Series
(20140606)We study two dimension estimates regarding linear series on algebraic curves. First, we generalize the classical BrillNoether theorem to many cases where the BrillNoether number is negative. Second, we extend results of ... 
Relative Jacobians of Linear Systems
(20140606)Let X be a smooth projective variety. Given any basepointfree 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 ... 
Restrictions of Steiner Bundles and Divisors on the Hilbert Scheme of Points in the Plane
(20120918)The Hilbert scheme of \(n\) points in the projective plane parameterizes degree \(n\) zerodimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert ...