Now showing items 1-20 of 77

    • Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology 

      Negron, Cris
      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 Calabi-Yau Varieties 

      Ward, Matthew J
      This thesis develops a theory of arithmetic Fourier-Mukai transforms in order to obtain results about equivalences between the derived category of Calabi-Yau varieties over non-algebraically closed fields. We obtain answers ...
    • Aspects of Markov Chains and Particle Systems 

      Ganguly, Shirshendu
      The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure ...
    • Boundary Harnack Principle for Stable-Like Processes 

      Rudnick, Christian
      We establish the boundary Harnack principle for certain classes of symmetric stable-like processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stable-like processes on $\mathcal{C}^{1,1}$-domains. Using ...
    • Brownian Motion on Spaces with Varying Dimension 

      Lou, Shuwen
      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 

      McMurdie, Christopher Robert
      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 ...
    • Classification of connected Hopf algebras up to prime-cube dimension 

      Wang, Xingting
      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 

      Taylor, Jair Patrick (2014-02-24)
      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 

      Sarantsev, Andrey
      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 ...
    • Computational aspects of modular parametrizations of elliptic curves 

      Chen, Hao
      \abstract{ We investigate computational problems related to modular parametrizations of elliptic curves defined over $\mathbb{Q}$. We develop algorithms to compute the Mazur Swinnerton-Dyer critical subgroup of elliptic ...
    • Conformal welding of uniform random trees 

      Barnes, Joel
      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 

      Jordan-Squire, Christopher
      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 ...
    • Cornered Asymptotically Hyperbolic Metrics 

      McKeown, Stephen Edward
      This thesis considers asymptotically hyperbolic manifolds that have a finite boundary in addition to the usual infinite boundary – cornered asymptotically hyperbolic manifolds. A theorem of Cartan-Hadamard type near infinity ...
    • Deformation invariance of rational pairs 

      Erickson, Lindsay
      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 Fourier-Mukai Transforms 

      Grigg, Nathan (2013-07-25)
      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 

      Roberts, Austin
      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 

      Johnson, Tobias Lee
      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 

      Paquette, Elliot (2013-11-14)
      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 

      Yang, Yang
      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 

      Choi, Daeshik (2013-11-14)
      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.