Browsing Mathematics by Title
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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 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 ... 
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 ... 
Some Theorems on the Resolution Property and the Brauer map
Using formallocal methods, we prove that a separated and normal DeligneMumford surface must satisfy the resolution property, this includes the first class of separated algebraic spaces which are not schemes. Our analysis ... 
Spectral analysis in bipartite biregular graphs and community detection
This thesis concerns to spectral gap of random regular graphs and consists of two main con tributions. First, we prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for ... 
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 the wave equation on Riemannian manifolds of bounded curvature
Wave packet methods have proven to be a useful tool for the study of dispersive effects of the wave equation with coefficients of limited differentiability. In this thesis, we use scaled wave packet methods to prove ... 
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 ... 
Structure and complexity in nonconvex and nonsmooth optimization
Complexity theory drives much of modern optimization, allowing a fair comparison between competing numerical methods. The subject broadly seeks to both develop efficient algorithms and establish limitations on efficiencies ... 
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 ... 
The geometry of uniform measures
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. They were first studied in the groundbreaking work of Preiss where he proved that a Radon measure ... 
The Inverse Problem of Thermoacoustic Tomography in Attenuating Media
Thermoacoustic tomography is a developing medical imaging technique that combines the propagation of electromagnetic and ultrasound waves with the purpose of producing a high contrast and high resolution internal image of ... 
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 ...