Browsing Mathematics Theses and Dissertations by Title
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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 ... 
Infinite dimensional versions of the SchurHorn theorem
(University of Oregon, 201106)We characterize the diagonals of four classes of selfadjoint operators on infinite dimensional Hilbert spaces. These results are motivated by the classical SchurHorn theorem, which characterizes the diagonals of selfadjoint ... 
Koszul and generalized Koszul properties for noncommutative graded algebras
(University of Oregon, 200906)We investigate some homological properties of graded algebras. If A is an R algebra, then E (A) := Ext A ( R, R ) is an Ralgebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R ... 
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 ... 
Manifolds with indefinite metrics whose skewsymmetric curvature operator has constant eigenvalues
(University of Oregon, 2000)Relative to a nondegenerate metric of signature (p, q), an algebraic curvature tensor is said to be IP if the associated skewsymmetric curvature operator R(π) has constant eigenvalues and if the kernel of R(π) has constant ... 
Metrics of positive scalar curvature and generalised Morse functions
(University of Oregon, 200906)We study the topology of the space of metrics of positive scalar curvature on a compact manifold. The main tool we use for constructing such metrics is the surgery technique of Gromov and Lawson. We extend this technique ... 
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 ... 
Motivic Integral of K3 Surfaces over a NonArchimedean Field
(University of Oregon, 20140929)We prove a formula expressing the motivic integral of a K3 surface over C((t)) with semistable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete ... 
Multinets in P^2 and P^3
(University of Oregon, 20131003)In this dissertation, a method for producing multinets from a net in P^3 is presented. Multinets play an important role in the study of resonance varieties of the complement of a complex hyperplane arrangement and very few ... 
Multiplier Theorems on Anisotropic Hardy Spaces
(University of Oregon, 2012)We extend the theory of singular integral operators and multiplier theorems to the setting of anisotropic Hardy spaces. We first develop the theory of singular integral operators of convolution type in the anisotropic ... 
Nonexistence of a stable homotopy category for pcomplete abelian groups
(University of Oregon, 200906)We investigate the existence of a stable homotopy category (SHC) associated to the category of p complete abelian groups [Special characters omitted]. First we examine [Special characters omitted] and prove [Special ... 
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 ...