Browsing Mathematics by Title
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The Radiative Transfer Equation in Photoacoustic Imaging
(20130725)Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, ... 
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 ... 
The regularity of Loewner curves
The Loewner differential equation, a classical tool that has attracted recent attention due to SchrammLoewner evolution (SLE), provides a unique way of encoding a simple 2dimensional curve into a continuous 1dimensional ... 
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 ... 
Selfshrinking Solutions to Mean Curvature Flow
We construct new examples of selfshrinking solutions to mean curvature flow. We first construct an immersed and nonembedded sphere selfshrinker. This result verifies numerical evidence dating back to the 1980's and shows ... 
Selmer groups for elliptic curves with isogenies of prime degree
(2003)The MordellWeil theorem states that the points of an elliptic curve defined over a number field form a finitely generated, abelian group. The rank of this group, generally referred to as the rank of the elliptic curve, ... 
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 ... 
Shimura Degrees for Elliptic Curves over Number Fields
A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve E over <bold>Q</bold> of conductor N, there is a nonconstant map from the modular curve of level N ... 
Smoothness of Loewner Slits
(20130225)In this dissertation, we show that the chordal Loewner differential equation with C^{beta} driving function generates a C^{beta + 1/2} slit for 1/2 < beta <= 2, except when beta = 3/2 the slit is only proved to be weakly C^{1,1}. 
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 ... 
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 ... 
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 ... 
Stable processes with opposing drifts
(1996)A strong Markov process, $W\sp0$, is constructed by a natural linking together of two independent stable processes of type ($\alpha,\ \beta\sb1$) and ($\alpha,\ \beta\sb2$). The drift for a stable process X of type ($\alpha,\ ... 
Stationary distribution for spinning reflecting diffusions
(20120913)This dissertation studies two different types of interaction of diffusion processes with the boundary of a domain $DsubRR^n$, which is assumed to be bounded, and of class $C^2(RR^n)$. The first process that is studied is ... 
Strichartz estimates for wave equations with coefficients of Sobolev regularity
(2005)Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. In this work, such methods are used to show Strichartz inequalities for ... 
A TensorTriangulated Approach to Derived Categories of NonNoetherian Rings
(20120913)We investigate the subcategories and Bousfield lattices of derived categories of general commutative rings, extending previous work done under a Noetherian hypothesis. Maps between rings R → S induce adjoint functors between ... 
Thermoacoustic Tomography in Elastic Media
(20131114)We investigate the problem of recovering the initial displacement f for a solution u of a linear, isotropic, nonhomogeneous elastic wave equation, given measurements of u on [0, T ] × boundary of Omega, where Omega in R3 ... 
Three Problems in Discrete Probability
In this thesis we present three problems. The first problem is to find a good description of the number of fixed points of a 231avoiding permutation. We use a bijection from Dyck paths to 231avoiding permutations that ... 
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 topology of graph homomorphisms
(2007)In this thesis we consider topological aspects of graph homomorphisms. Our main object of study is the Hom complex of graphs, a space first introduced by Lovasz to obtain lower bounds on chromatic number, and more recently ...