### Recent Submissions

• #### Path algebras and monomial algebras of finite GK-dimension 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 Gelfand-Kirillov 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 Time-Dependent Dirichlet Forms ﻿

In this paper, time-inhomogeneous stable-like processes are investigated. We establish the relation between the transition operators and time-dependent 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 large-scale nonlinear optimization problems. We build two matrix-free 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 ...
• #### Sheaves on support varieties and varieties of elementary subalgebras ﻿

We present several results about two closely related types of objects: the projectivized scheme $\PG$ of one parameter subgroups of an infinitesimal group scheme $G$ and the variety $\bE(\fg)$ of maximal elementary subalgebras ...
• #### The C*-algebra of a finite T_0 topological space ﻿

We are concerned with the following motivating question: how can one extend the classical Gelfand-Naimark theorem to the simplest non-Hausdorff topological spaces? Our model space is a finite $T_0$ topological space, or ...
• #### Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology ﻿

The Hochschild cohomology $HH^\bullet(A)$ of an algebra $A$ is a derived invariant of the algebra which admits both a graded ring structure (called the cup product) and a compatible graded Lie algebra structure (called the ...
• #### Toward the compactification of the stack of Lie(G)-forms using perfect complexes ﻿

To establish geometric properties of an algebraic stack, one can find a compactification. This method has been successfully employed to find irreducible components for example of the moduli stack of curves [DM69], vector ...
• #### On numerics and inverse problems ﻿

In this thesis, two projects in inverse problems are described. The first concerns a simple mathematical model of synthetic aperture radar with undirected beam, modeled as a 2D circular Radon transform with centers restricted ...
• #### Some Linear and Nonlinear Geometric Inverse Problems ﻿

Inverse problems is an area at the interface of several disciplines and has become a prominent research topic due to its potential applications. A wide range of these problems can be formulated under various geometric ...
• #### Convex Optimization over Probability Measures ﻿

The thesis studies convex optimization over the Banach space of regular Borel measures on a compact set. The focus is on problems where the variables are constrained to be probability measures. Applications include ...
• #### Spectral Theory of Z^d Substitutions ﻿

In this paper, we generalize and develop results of Queffelec allowing us to characterize the spectrum of an aperiodic substitution in Z^d by describing the Fourier coefficients of mutually singular measures of pure type ...
• #### Finite-Difference Methods for Second-Order Wave Equations with Reduced Dispersion Errors ﻿

Finite Difference (FD) schemes have been used widely in computing approximations for partial differential equations for wave propagation, as they are simple, flexible and robust. However, even for stable and accurate ...
• #### The Zeros of Elliptic Curve L-functions: Analytic Algorithms with Explicit Time Complexity ﻿

Elliptic curves are central objects of study in modern-day algebraic number theory. The problem of how to determine the rank of a rational elliptic curve is a difficult one, and at the time of the writing of this thesis ...
• #### Some cohomology of finite general linear groups ﻿

We prove that the degree r(2p − 3) cohomology of any (untwisted) finite group of Lie type over F_(p^r), with coefficients in characteristic p, is nonzero as long as its Coxeter number is at most p. We do this by providing ...
• #### Inverse Boundary-Value Problems on an Infinite Slab ﻿

In this work, we study the stability aspect of two inverse boundary-value problems (IBVPs) on an infinite slab with partial data. The uniqueness aspects of these IBVPs were considered and studied by Li and Uhlmann for the ...
• #### Time-like graphical models ﻿

We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical ...
• #### The Positive Semidefinite Rank of Matrices and Polytopes ﻿

The positive semidefinite (psd) rank of a nonnegative <italic>p</italic> × <italic>q</italic> matrix <italic>M</italic> is defined to be the smallest integer <italic>k</italic> such that there exist <italic>k</italic> × ...