Browsing Mathematics by Subject "Mathematics"
Now showing items 120 of 63

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 ... 
Arithmetic Properties of the Derived Category for CalabiYau Varieties
This thesis develops a theory of arithmetic FourierMukai transforms in order to obtain results about equivalences between the derived category of CalabiYau varieties over nonalgebraically closed fields. We obtain answers ... 
Brownian Motion on Spaces with Varying Dimension
In this thesis we introduce and study Brownian motion with or without drift on state spaces with varying dimension. Starting with a concrete such state space that is the plane with an infinite pole on it, we construct a ... 
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 ... 
Classification of connected Hopf algebras up to primecube dimension
We classify all connected Hopf algebras up to p^3 dimension over an algebraically closed field of characteristic p>0 under the mild restriction such that in dimension p^3, we only work over odd primes p when the primitive ... 
Combinatorial Laguerre Series
(20140224)We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials. We describe how such a series can be computed by ... 
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 ... 
Conformal welding of uniform random trees
A conformally balanced tree is an embedding of a given planar map into the plane with constraints on the harmonic measure of its edges such that the resulting set is unique up to scale and rotation. Bishop (2013) showed ... 
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 ... 
Deformation invariance of rational pairs
Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs (X,D). Here X is a normal variety and D is a reduced divisor on X. Integral to the definition of a rational pair is the ... 
Deformations of Categories of Coherent Sheaves and FourierMukai Transforms
(20130725)In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent ... 
Dual Equivalence Graphs and their Applications
In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's ... 
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 ... 
Eigenvalue Fluctuations of Random Matrices beyond the Gaussian Universality Class
(20131114)The goal of this thesis is to develop one of the threads of what is known in random matrix theory as universality, which essentially is that a large class of matrices generalizing the Gaussian matrices (certain Wigner ... 
Elliptic Inverse Problems
Inverse problems arise in various areas of science and engineering including medical imaging, computer vision, geophysics, solid mechanics, astronomy, and so forth. A wide range of these problems involve elliptic operators. ... 
Estimating Norms of Matrix Functions using Numerical Ranges
(20131114)We study Crouzeix's conjecture: for any polynomial p and any square matrix A, the spectral norm of the matrix p(A) is at most double of the supremum norm of the polynomial p on the numerical range of the matrix A. 
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 ... 
FiniteDifference Methods for the Wave Equation with Reduced Dispersion Errors
(20130417)A new methodology was proposed in Finkelstein and Kastner (2007,2008) to derive finitedifference (FD) schemes in the joint timespace domain to reduce dispersion error. The key idea is that the true dispersion relation ... 
Four Problems in Probability and Optimization
(20140224)This thesis studies bootstrap percolation, a problem in probability, as well as several topics in the application of sums of squares to combinatorial optimization. In the chapter on percolation, we bound the critical ... 
A Geometric Perspective on FirstPassage Competition
(20130417)We study the macroscopic geometry of firstpassage competition on the integer lattice <bold>Z</bold><super>d</super>, with a particular interest in describing the behavior when one species initially occupies the exterior ...