Mathematics
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Novel uses of the Mallows model in coloring and matching
A natural model of a highly ordered random ranking is the Mallows model. Disorder is measured by the number of inversions; these are pairs of elements whose order is reversed. The Mallows model assigns to each ranking of ... 
Bin packing, number balancing, and rescaling linear programs
This thesis deals with several important algorithmic questions using techniques from diverse areas including discrepancy theory, machine learning and lattice theory. In Chapter 2, we construct an improved approximation ... 
Nonlocal operators, jump diffusions and FeynmanKac tranforms
Nonlocal operators are analytically defined by integrals over the whole space, hence hard to study certain properties. This thesis studies inverse local times at $0$ of onedimensional reflected diffusions on $[0, ... 
Bispectral Operator Algebras
This dissertation is an amalgamation of various results on the structure of bispectral differential operator algebras, ie. algebras of differential operators with possibly noncommutative coefficients in a variable $x$ ... 
Algorithms in Discrepancy Theory and Lattices
This thesis deals with algorithmic problems in discrepancy theory and lattices, and is based on two projects I worked on while at the University of Washington in Seattle. A brief overview is provided in Chapter 1 (Introduction). ... 
A Survey of Tverberg Type Problems
Tverberg's theorem, which celebrates its fiftieth anniversary this year, is a central result in the fields of discrete geometry and topological combinatorics. Proved in 1966, it was a major step in solving questions whether, ... 
On fvectors of polytopes and matroids
The fvector of a simplicial complex is a fundamental invariant that counts the number of faces in each dimension. A natural question in the theory of simplicial complexes is to understand the relationship between the ... 
Problems in Algebraic Vision
This thesis studies several fundamental mathematical problems that arise from computer vision using techniques in algebraic geometry and optimization. Chapters 2 and 3 consider the fundamental question of the existence of ... 
Cornered Asymptotically Hyperbolic Metrics
This thesis considers asymptotically hyperbolic manifolds that have a finite boundary in addition to the usual infinite boundary – cornered asymptotically hyperbolic manifolds. A theorem of CartanHadamard type near infinity ... 
On TSemisimplicity of Iwasawa Modules and Some Computations with Z3Extensions
For certain Zpextensions of abelian number fields, we study the Iwasawa module associated to the ideal class groups. We show that generic Zpextensions of abelian number fields are Tsemisimple. We also construct the ... 
Some Inverse Problems in Analysis and Geometry
The aim of a typical inverse problem is to recover the interior properties of a medium by making measurements only on the boundary. These types of problems are motivated by geophysics, medical imaging and quantum mechanics ... 
Formal group laws and hypergraph colorings
This thesis demonstrates a connection between formal group laws and chromatic symmetric functions of hypergraphs, two seemingly unrelated topics in the theory of symmetric functions. A formal group law is a symmetric ... 
On the Geometry of Rectifiable Sets with Carleson and Poincaretype inequlaities
A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$dimensional subset $M$ of $\mathbb{R}^{n+d}$ is well ... 
Random recursion
We study the limiting behavior of three stochastic processes. Two are interacting particle systems, the frog model and coalescing random walk. We work out transience and recurrence properties on various graphs. The last ... 
Boundary Harnack Principle for StableLike Processes
We establish the boundary Harnack principle for certain classes of symmetric stablelike processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stablelike processes on $\mathcal{C}^{1,1}$domains. Using ... 
Geometry and Optimization of Relative Arbitrage
This thesis is devoted to the mathematics of volatility harvesting, the idea that extra portfolio growth may be created by systematic rebalancing. First developed by E. R. Fernholz in the late 90s and the early 2000s, ... 
Computational aspects of modular parametrizations of elliptic curves
\abstract{ We investigate computational problems related to modular parametrizations of elliptic curves defined over $\mathbb{Q}$. We develop algorithms to compute the Mazur SwinnertonDyer critical subgroup of elliptic ... 
On Selmer groups and factoring padic Lfunctions
Samit Dasgupta has proved a formula factoring a certain restriction of a 3variable RankinSelberg padic Lfunction as a product of a 2variable padic Lfunction related to the adjoint representation of a Hida family and ... 
Results on singularities of pairs
Singularities of algebraic varieties have been studied extensively, and recently also the properties of singularities of pairs have been investigated. This thesis presents several results on singularities of different kinds ... 
Aspects of Markov Chains and Particle Systems
The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure ...