Mathematics Theses and Dissertations
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This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.
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Faithful tropicalization of hypertoric varieties
(University of Oregon, 20170906)The hypertoric variety M_A defined by an arrangement A of affine hyperplanes admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. In this thesis, we explicitly describe the polyhedral ... 
On the Subregular Jring of Coxeter Systems
(University of Oregon, 20170906)Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke algebra associated to (W, S) via KazhdanLusztig polynomials by Lusztig. We study a subalgebra J_C of J corresponding to the subregular cell C ... 
Motives of Log Schemes
(University of Oregon, 20170906)This thesis introduces two notions of motive associated to a log scheme. We introduce a category of log motives à la Voevodsky, and prove that the embedding of Voevodsky motives is an equivalence, in particular proving ... 
Gluing manifolds with boundary and bordisms of positive scalar curvature metrics
(University of Oregon, 20170906)This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalarflat manifolds with vanishing mean curvature on the boundary. Our methods involve tools ... 
Constructing a v2 Self Map at p=3
(University of Oregon, 20170906)Working at the prime p = 3, we construct a stably finite spectrum, Z, with a v_2^1 self map f. Further, both Ext_A(H*(Z),Z_3) and Ext_A(H*(Z),H*(Z)) have a vanishing line of slope 1/16 in (ts,s) coordinates, and the map ... 
Relations in the Witt Group of Nondegenerate Braided Fusion Categories Arising from the Representation Theory of Quantum Groups at Roots of Unity
(University of Oregon, 20170906)For each finite dimensional Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a modular tensor category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integrable $\hat{\mathfrak{g}}$modules of ... 
Equivariant Derived Categories Associated to a Sum of Potentials
(University of Oregon, 20170906)We construct a semiorthogonal decomposition for the equivariant derived category of a hypersurface associated to the sum of two potentials. More specifically, if $f,g$ are two homogeneous poynomials of degree $d$ defining ... 
Homological Properties of Standard KLR Modules
(University of Oregon, 20170501)KhovanovLaudaRouquier algebras, or KLR algebras, are a family of algebras known to categorify the upper half of the quantized enveloping algebra of a given Lie algebra. In finite type, these algebras come with a family ... 
Dancing in the Stars: Topology of Nonkequal Configuration Spaces of Graphs
(University of Oregon, 20161121)We prove that the nonkequal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit ... 
Categorical Actions on Supercategory O
(University of Oregon, 20161121)This dissertation uses techniques from the theory of categorical actions of KacMoody algebras to study the analog of the BGG category O for the queer Lie superalgebra. Chen recently reduced many questions about this ... 
The Geometry of quasiSasaki Manifolds
(University of Oregon, 20161027)Let (M,g) be a quasiSasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain ... 
GKZ Hypergeometric Systems and Projective Modules in Hypertoric Category O
(University of Oregon, 20161027)In this thesis I show that indecomposable projective and tilting modules in hypertoric category O are obtained by applying a variant of the geometric Jacquet functor of Emerton, Nadler, and Vilonen to certain Gel'fandKa ... 
Frames Generated by Actions of Locally Compact Groups
(University of Oregon, 20161027)Let $G$ be a second countable, locally compact group which is either compact or abelian, and let $\rho$ be a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_\rho$. We examine frames of the form $\{ ... 
Representations of KhovanovLaudaRouquier algebras of affine Lie type
(University of Oregon, 20161027)We study representations of KhovanovLaudaRouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced ... 
Zonotopes and Hypertoric Varieties
(University of Oregon, 20160223)Hypertoric varieties are a class of conical symplectic resolutions which are very computable. In the current literature, they are only defined constructively, using hyperplane arrangements. We provide an abstract definition ... 
Compact Group Actions on C*algebras: Classification, NonClassifiability and Crossed Products and Rigidity Results for Lpoperator Algebras
(University of Oregon, 20150818)This dissertation is concerned with representations of locally compact groups on different classes of Banach spaces. The first part of this work considers representations of compact groups by automorphisms of C*algebras, ... 
The Homotopy Calculus of Categories and Graphs
(University of Oregon, 20150818)We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the BissonTsemo model ... 
Abelian Arrangements
(University of Oregon, 20150818)An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. We are interested in the topology of the complement of an arrangement. If the arrangement ... 
Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A
(University of Oregon, 20150818)This thesis consists of two parts. In the first we prove that the KhovanovLaudaRouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger ... 
Linking Forms, Singularities, and Homological Stability for Diffeomorphism Groups of Odd Dimensional Manifolds
(University of Oregon, 20150818)Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)dimensional manifolds, with respect to forming the connected sum with (2n1)connected, (4n+1)dimensional manifolds that are ...