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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 ... 
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 GelfandNaimark theorem to the simplest nonHausdorff topological spaces? Our model space is a finite $T_0$ topological space, or ... 
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 ... 
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 ... 
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 ... 
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 ... 
FiniteDifference Methods for SecondOrder 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 ... 
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 ... 
The Zeros of Elliptic Curve Lfunctions: Analytic Algorithms with Explicit Time Complexity
Elliptic curves are central objects of study in modernday 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 BoundaryValue Problems on an Infinite Slab
In this work, we study the stability aspect of two inverse boundaryvalue 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 ... 
Timelike 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> × ... 
The Grothendieck Groups of Module Categories over Coherent Algebras
Let <italic>k</italic> be a field and <italic>B</italic> either a finitely generated free <italic>k</italic>algebra, or a regular <italic>k</italic>algebra of global dimension two with at least three generators, generated ... 
Interacting particle systems with partial annihilation through membranes
This thesis studies the <italic>hydrodynamic limit</italic> and the <italic>fluctuation limit</italic> for a class of interacting particle systems in domains. These systems are introduced to model the transport of positive ... 
Eigenvalue fluctuations for random regular graphs
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are <italic>universal</italic>. We probe the edges of universality by studying ...