Mathematics
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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 ... 
Path algebras and monomial algebras of finite GKdimension as noncommutative homogeneous coordinate rings
This thesis sets out to understand the categories QGr A where A is either a monomial algebra or a path algebra of finite GelfandKirillov dimension. The principle questions are: \begin{enumerate} \item What is the structure ... 
Grothendieck Duality on Diagrams of Schemes
The Du Bois complex and Du Bois singularities, which extend results of Hodge theory to singular complex varieties, can be defined in terms of a cubical hyperresolution. In this dissertation I further develop the language ... 
Heat Kernel Estimates for Markov Processes Associated with TimeDependent Dirichlet Forms
In this paper, timeinhomogeneous stablelike processes are investigated. We establish the relation between the transition operators and timedependent parabolic equations, as well as upper heat kernel estimates. 
Matrix free methods for large scale optimization
Sequential quadratic optimization (SQP) methods are widely used to solve largescale nonlinear optimization problems. We build two matrixfree methods for approximately solving exact penalty subproblems that arise when ... 
Competing Brownian Particles
Consider a finite system of N Brownian particles on the real line. Rank them from bottom to top: the (currently) lowest particle has rank 1, the second lowest has rank 2, etc., up to the top particle, which has rank N. The ... 
Inverse Problems for Scalar Elliptic Equations and Systems
In this thesis, we discuss inverse boundary value problems for scalar equations and for systems. First we introduce the famous Calder\'on problem and its recent developments. We focus on deriving the stability estimate of ...