Browsing Mathematics Theses and Dissertations by Title
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On Algebras Associated to Finite Ranked Posets and Combinatorial Topology: The Koszul, Numerically Koszul and CohenMacaulay Properties
(University of Oregon, 20140929)This dissertation studies new connections between combinatorial topology and homological algebra. To a finite ranked poset Γ we associate a finitedimensional quadratic graded algebra RΓ. Assuming Γ satisfies a combinatorial ... 
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 ... 
Plumbers' knots and unstable Vassiliev theory
(University of Oregon, 201006)We introduce a new finitecomplexity knot theory, the theory of plumbers' knots, as a model for classical knot theory. The spaces of plumbers' curves admit a combinatorial cell structure, which we exploit to algorithmically ... 
Primitive and Poisson spectra of nonsemisimple twists of polynomial algebras
(University of Oregon, 2001)We examine a family of twists of the complex polynomial ring on n generators by a nonsemisimple automorphism. In particular, we consider the case where the automorphism is represented by a single Jordan block. The ... 
Probability on graphs: A comparison of sampling via random walks and a result for the reconstruction problem
(University of Oregon, 201009)We compare the relaxation times of two random walks  the simple random walk and the metropolis walk  on an arbitrary finite multigraph G. We apply this result to the random graph with n vertices, where each edge is ... 
Quantum Cluster Characters
(University of Oregon, 2012)We de ne the quantum cluster character assigning an element of a quantum torus to each representation of a valued quiver (Q; d) and investigate its relationship to external and internal mutations of a quantum cluster ... 
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 ... 
Representations of Hecke algebras and the Alexander polynomial
(University of Oregon, 201006)We study a certain quotient of the IwahoriHecke algebra of the symmetric group Sd , called the super TemperleyLieb algebra STLd. The Alexander polynomial of a braid can be computed via a certain specialization of the ... 
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 ... 
Representations of the Oriented Brauer Category
(University of Oregon, 20150818)We study the representations of a certain specialization $\mathcal{OB}(\delta)$ of the oriented Brauer category in arbitrary characteristic $p$. We exhibit a triangular decomposition of $\mathcal{OB}(\delta)$, which we use ... 
The RO(G)graded Serre Spectral Sequence
(University of Oregon, 200806)The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable ... 
Semisimplicity of Certain Representation Categories
(University of Oregon, 20131003)We exhibit a correspondence between subcategories of modules over an algebra and subbimodules of the dual of that algebra. We then prove that the semisimplicity of certain such categories is equivalent to the existence ... 
Solvable Particle Models Related to the BetaEnsemble
(University of Oregon, 20131003)For beta > 0, the betaensemble corresponds to the joint probability density on the real line proportional to prod_{n > m}^N abs{x_n  x_m}^beta prod_{n = 1}^N w(x_n) where w is the weight of the system. It has the ... 
Stabilization of chromatic functors
(University of Oregon, 200906)We study the Bousfield localization functors known as [Special characters omitted], as described in [MahS]. In particular we would like to understand how they interact with suspension and how they stabilize. We prove ... 
Summability of Fourier orthogonal expansions and a discretized Fourier orthogonal expansion involving radon projections for functions on the cylinder
(University of Oregon, 200906)We investigate Cesàro summability of the Fourier orthogonal expansion of functions on B d × I m , where B d is the closed unit ball in [Special characters omitted] and I m is the m fold Cartesian product of the ... 
A Super Version of Zhu's Theorem
(University of Oregon, 200806)We generalize a theorem of Zhu relating the trace of certain vertex algebra representations and modular invariants to the arena of vertex super algebras. The theorem explains why the space of simple characters for the ... 
The Einstein Constraint Equations on Asymptotically Euclidean Manifolds
(University of Oregon, 20150818)In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz ... 
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 ... 
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 ... 
Topics in Random Walks
(University of Oregon, 20131003)We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to ...