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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.

### Recent Submissions

• #### Kazhdan-Lusztig Polynomials of Matroids and Their Roots ﻿

(University of Oregon, 2018-10-31)
The Kazhdan-Lusztig polynomial of a matroid M, denoted P_M( t ), was recently defined by Elias, Proudfoot, and Wakefield. These polynomials are analogous to the classical Kazhdan-Lusztig polynomials associated with Coxeter ...
• #### Stability Within T<sup>2</sup>-Symmetric Expanding Spacetimes ﻿

(University of Oregon, 2018-09-06)
We prove a nonpolarized analogue of the asymptotic characterization of T<sup>2</sup>-symmetric Einstein flow solutions completed recently by LeFloch and Smulevici. We impose a far weaker condition, but obtain identical ...
• #### Factorizable Module Algebras, Canonical Bases, and Clusters ﻿

(University of Oregon, 2018-09-06)
The present dissertation consists of four interconnected projects. In the first, we introduce and study what we call factorizable module algebras. These are $U_q(\mathfrak{g})$-module algebras $A$ which factor, potentially ...
• #### A Structure Theorem for RO(C2)-graded Cohomology ﻿

(University of Oregon, 2018-09-06)
Let C2 be the cyclic group of order two. We present a structure theorem for the RO(C2)-graded Bredon cohomology of C2-spaces using coefficients in the constant Mackey functor F2. We show that, as a module over the cohomology ...
• #### Higher Congruences Between Modular Forms ﻿

(University of Oregon, 2018-09-06)
In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms ...
• #### RO(C2)-graded Cohomology of Equivariant Grassmannian Manifolds ﻿

(University of Oregon, 2018-09-06)
We compute the RO(C2)-graded Bredon cohomology of certain families of real and complex C2-equivariant Grassmannians.
• #### Unimodal Levy Processes on Bounded Lipschitz Sets ﻿

(University of Oregon, 2018-09-06)
We give asymptotics near the boundary for the distribution of the first exit time of the isotropic alpha-stable Levy process on bounded Lipschitz sets in real euclidean space. These asymptotics bear some relation to the ...
• #### NC-algebroid thickenings of moduli spaces and bimodule extensions of vector bundles over NC-smooth schemes ﻿

(University of Oregon, 2018-04-10)
We begin by reviewing the theory of NC-schemes and NC-smoothness, as introduced by Kapranov in \cite{Kapranov} and developed further by Polishchuk and Tu in \cite{PT}. For a smooth algebraic variety $X$ with a ...
• #### Periodic Margolis Self Maps at p=2 ﻿

(University of Oregon, 2018-04-10)
The Periodicity theorem of Hopkins and Smith tells us that any finite spectrum supports a $v_n$-map for some $n$. We are interested in finding finite $2$-local spectra that both support a $v_2$-map with a low power of ...
• #### Faithful tropicalization of hypertoric varieties ﻿

(University of Oregon, 2017-09-06)
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 J-ring of Coxeter Systems ﻿

(University of Oregon, 2017-09-06)
Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke algebra associated to (W, S) via Kazhdan-Lusztig polynomials by Lusztig. We study a subalgebra J_C of J corresponding to the subregular cell C ...
• #### Motives of Log Schemes ﻿

(University of Oregon, 2017-09-06)
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, 2017-09-06)
This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools ...
• #### Constructing a v2 Self Map at p=3 ﻿

(University of Oregon, 2017-09-06)
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 (t-s,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, 2017-09-06)
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, 2017-09-06)
We construct a semi-orthogonal 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, 2017-05-01)
Khovanov-Lauda-Rouquier 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 Non-k-equal Configuration Spaces of Graphs ﻿

(University of Oregon, 2016-11-21)
We prove that the non-k-equal 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, 2016-11-21)
This dissertation uses techniques from the theory of categorical actions of Kac-Moody 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 quasi-Sasaki Manifolds ﻿

(University of Oregon, 2016-10-27)
Let (M,g) be a quasi-Sasaki 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 ...