Recent Submissions

  • On the g2-number of various classes of spheres and manifolds 

    Zheng, Hailun
    For a $(d-1)$-dimensional simplicial complex $\Delta$, we let $f_i=f_i(\Delta)$ be the number of $i$-dimensional faces of $\Delta$ for $-1\leq i\leq d-1$. One classic problem in geometric combinatorics is the following: ...
  • Structure and complexity in non-convex and non-smooth optimization 

    Paquette, Courtney
    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 ...
  • The geometry of uniform measures 

    Nimer, Abdalla Dali
    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 ...
  • Spectral analysis in bipartite biregular graphs and community detection 

    Brito, Gerandy
    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 ...
  • Strichartz estimates for the wave equation on Riemannian manifolds of bounded curvature 

    Chen, Yuanlong
    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 ...
  • Bin packing, number balancing, and rescaling linear programs 

    Hoberg, Rebecca Anne
    This thesis deals with several important algorithmic questions using techniques from diverse areas including discrepancy theory, machine learning and lattice theory. In Chapter 2, we construct an improved approximation ...
  • Non-local operators, jump diffusions and Feynman-Kac tranforms 

    Wang, Lidan
    Non-local operators are analytically defined by integrals over the whole space, hence hard to study certain properties. This thesis studies inverse local times at $0$ of one-dimensional reflected diffusions on $[0, ...
  • Novel uses of the Mallows model in coloring and matching 

    Levy, Avi William
    A natural model of a highly ordered random ranking is the Mallows model. Disorder is measured by the number of inversions; these are pairs of elements whose order is reversed. The Mallows model assigns to each ranking of ...
  • Algorithms in Discrepancy Theory and Lattices 

    Ramadas, Harishchandra
    This thesis deals with algorithmic problems in discrepancy theory and lattices, and is based on two projects I worked on while at the University of Washington in Seattle. A brief overview is provided in Chapter 1 (Introduction). ...
  • Bispectral Operator Algebras 

    Casper, William Riley
    This dissertation is an amalgamation of various results on the structure of bispectral differential operator algebras, ie. algebras of differential operators with possibly noncommutative coefficients in a variable $x$ ...
  • A Survey of Tverberg Type Problems 

    Thng, Ivana
    Tverberg's theorem, which celebrates its fiftieth anniversary this year, is a central result in the fields of discrete geometry and topological combinatorics. Proved in 1966, it was a major step in solving questions whether, ...
  • 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 ...
  • Problems in Algebraic Vision 

    Lee, Hon Leung
    This thesis studies several fundamental mathematical problems that arise from computer vision using techniques in algebraic geometry and optimization. Chapters 2 and 3 consider the fundamental question of the existence of ...
  • On f-vectors of polytopes and matroids 

    Samper Casas, Jose Alejandro
    The f-vector of a simplicial complex is a fundamental invariant that counts the number of faces in each dimension. A natural question in the theory of simplicial complexes is to understand the relationship between the ...
  • Some Inverse Problems in Analysis and Geometry 

    ASSYLBEKOV, YERNAT
    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 ...
  • On T-Semisimplicity of Iwasawa Modules and Some Computations with Z3-Extensions 

    Van Huele, Yannick
    For certain Zp-extensions of abelian number fields, we study the Iwasawa module associated to the ideal class groups. We show that generic Zp-extensions of abelian number fields are T-semisimple. We also construct the ...
  • Formal group laws and hypergraph colorings 

    Taylor, Jair Patrick
    This thesis demonstrates a connection between formal group laws and chromatic symmetric functions of hypergraphs, two seemingly unrelated topics in the theory of symmetric functions. A formal group law is a symmetric ...
  • On the Geometry of Rectifiable Sets with Carleson and Poincare-type inequlaities 

    Merhej, Jessica
    A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+d}$ is well ...
  • 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 ...
  • Random recursion 

    Junge, Matthew S.
    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 ...

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