Browsing by Subject "Mathematics"
Now showing items 120 of 86

A Study of Edge Toric Ideals using Associated Graphs
(20120426)This thesis studies properties of edge toric ideals and resolutions by analyzing the associated graphs of algebraic structures. It mainly focused on proving that the repeated edges in a graph wouldn't change some properties ... 
Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves
(2018)To study periods of fundamental groups of algebraic varieties, one requires an explicit algebraic de Rham theory for completions of fundamental groups. This thesis develops such a theory in two cases. In the first case, ... 
Algorithms for the Reeb Graph and Related Concepts
(2014)This thesis is concerned with a structure called the Reeb graph. There are three main problems considered. The first is devising an efficient algorithm for comnstructing the Reeb graph of a simplicial complex with respect ... 
Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics
(2016)The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian ... 
Analyzing Stratified Spaces Using Persistent Versions of Intersection and Local Homology
(20080805)This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun ... 
Applications of Persistent Homology to Time Varying Systems
(2013)This dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In ... 
Applications of Spatial Models to Ecology and Social Systems
(2015)Interacting particle systems have been applied to model the spread of infectious diseases and opinions, interactions between competing species, and evolution of forest landscapes. In this thesis, we study three spatial ... 
Applications of Topological Data Analysis and Sliding Window Embeddings for Learning on Novel Features of TimeVarying Dynamical Systems
(2017)This work introduces geometric and topological data analysis (TDA) tools that can be used in conjunction with sliding window transformations, also known as delayembeddings, for discovering structure in time series and ... 
Approximately Counting Perfect and General Matchings in Bipartite and General Graphs
(2009)We develop algorithms to approximately count perfect matchings in bipartite graphs (or permanents of the corresponding adjacency matrices), perfect matchings in nonbipartite graphs (or hafnians), and general matchings in ... 
Aspects of Motives: Finitedimensionality, ChowKunneth Decompositions and Intersections of Cycles
(2016)This thesis analyzes the Chow motives of 3 types of smooth projective varieties: the desingularized elliptic self fiber product, the Fano surface of lines on a cubic threefold and an ample hypersurface of an Abelian variety. ... 
Aspects of the (0,2)McKay Correspondence
(2015)We study first order deformations of the tangent sheaf of resolutions of CalabiYau threefolds that are of the form $\CC^3/\ZZ_r$, focusing on the cases where the orbifold has an isolated singularity. We prove a lower ... 
Atlas Simulation: A Numerical Scheme for Approximating Multiscale Diffusions Embedded in High Dimensions
(2014)When simulating multiscale stochastic differential equations (SDEs) in highdimensions, separation of timescales and highdimensionality can make simulations expensive. The computational cost is dictated by microscale ... 
Augmentations and exact Lagrangian cobordisms
(2017)To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category.An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots.We ... 
Augmentations and Rulings of Legendrian Links
(2016)For any Legendrian knot in R^3 with the standard contact structure, we show that the existence of an augmentation to any field of the ChekanovEliashberg differential graded algebra over Z[t,t^{1}] is equivalent to the ... 
Autonomous Modeling, Statistical Complexity and Semiannealed Treatment of Boolean Networks
(2012)This dissertation presents three studies on Boolean networks. Boolean networks are a class of mathematical systems consisting of interacting elements with binary state variables. Each element is a node with a Boolean logic ... 
Axelrod's Model in Two Dimensions
(2014)In 1997 R. Axelrod introduced a model in which individuals have one of $Q$ possible opinions about each of $F$ issues and neighbors interact at a rate proportional to the fraction of opinions they share. Thanks to work by ... 
Bifurcations in the EchebarriaKarma Modulation Equation for Cardiac Alternans in One Dimension
(2009)While alternans in a single cardiac cell appears through a simpleperioddoubling bifurcation, in extended tissue the exact natureof the bifurcation is unclear. In particular, the phase ofalternans can exhibit wavelike ... 
Categorification of quantum sl_3 projectors and the sl_3 ReshetikhinTuraev invariant of framed tangles
(2012)Quantum sl_3 projectors are morphisms in Kuperberg's sl_3 spider, a diagrammatically defined category equivalent to the full pivotal subcategory of the category of (type 1) finitedimensional representations of the quantum ... 
Coarsening of Thin Fluid Films
(20080415)Observed in many physical systems, coarsening is an orderly decrease in the number of localized structures, such as particles, drops, shear bands, solitons, or point defects. Coarsening is a type of pattern formation in ... 
Cohomology Jumping Loci and the Relative Malcev Completion
(20071212)Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev completion of its fundamental group G and the cohomology groups of X with coefficients in rank one ...