• Homological Algebra for Polynomial Mackey Rings over Prime Cyclic Groups 

  Casebolt, Ross (University of Oregon, 2022-10-26)

  Let $C_l$ denote the cyclic group of prime order $l$ and let $k$ be a field. We define a Mackey $\underline{k}$-algebra $\underline{k}[x_\theta]$ which is constructed by adjoining a free commutative variable to the free ...
• The Distribution of the Cusped Hypocycloidal Mahler Measure 

  Hunter, Nathan (University of Oregon, 2022-10-26)

  We explore generalized Mahler measures associated to regions in the complex plane. These generalized Mahler measures describe the complexity of polynomials in C[x] by comparing the geometry of their roots to subsets of C. ...
• Families of Differential Operators Acting on Overconvergent Hilbert Modular Forms 

  Aycock, Jon (University of Oregon, 2022-10-26)

  We construct differential operators acting on overconvergent Hilbert modular forms. This extends work of Katz in the case of p-adic Hilbert modular forms, and of Harron--Xiao and Liu for overconvergent Siegel modular forms. ...
• Fixed-Point-Free Involution Word Diagrams 

  Potter, Jake (University of Oregon, 2022-10-04)

  We define a useful diagram for studying fixed-point-free involution words. Following the example of Little, we define a specific and a general bumping algorithm on these diagrams. These algorithms serve as the basis for ...
• The Curtis-Wellington Spectral Sequence Through Cohomology 

  Hunter, Dana (University of Oregon, 2022-10-04)

  We study stable homotopy through unstable methods applied to its representing infinite loop space, as pioneered by Curtis and Wellington. Using cohomology instead of homology, we find a width filtration whose subquotients ...
• Moduli Space of A-Infinity Structures and Nonreduced Curves of Genus 0 

  Zhang, Wei (University of Oregon, 2022-10-04)

  In this thesis, we study $A_\infty$-structures arising from derived categories of certain algebraic curves. More precisely, we consider pairs $(\mathcal{O}_C,\mathcal{O}_D)$, where $C$ is an irreducible projective curve ...
• Linear Recurrence Sequences in Diophantine Analysis 

  Bellah, Elisa (University of Oregon, 2022-10-04)

  Diophantine analysis is an area of number theory concerned with finding integral solutions to polynomial equations defined over the rationals, or more generally over a number field. In some cases, it is possible to associate ...
• Diagrammatic Representation Theory of the Rank Two Symplectic Group 

  Bodish, Elijah (University of Oregon, 2022-10-04)

  We study the diagrammatic representation theory of the group $Sp_4$ and the quantum group $U_q(\mathfrak{sp}_4)$, expanding on the previous results of Kuperberg about type $B_2= C_2$ webs. In particular, we construct a ...
• $RO(C_3)$-graded Bredon Cohomology and $C_p$-surfaces 

  Pohland, Kelly (University of Oregon, 2022-10-04)

  Let $p$ be an odd prime, and let $C_p$ denote the cyclic group of order $p$. We use equivariant surgery methods to classify all closed, connected $2$-manifolds with an action of $C_p$. We then use this classification in ...
• Ensemble Averages of Assorted Log-Gas Models 

  Wolff, Elisha (University of Oregon, 2022-10-04)

  We use techniques in the shuffle and exterior algebras to present the partition functions for several log-gas models in terms of either the Hyperpfaffian or the Berezin integral of an appropriate alternating tensor. Our ...
• Modules with Good Filtrations over Generalized Schur Algebras 

  Weinschelbaum, Ilan (University of Oregon, 2022-10-04)

  In this dissertation we examine generalized Schur algebras, as defined by Kleshchev and Muth. Given a quasi-hereditary superalgebra $A$, Kleshchev and Muth proved that for $n \geq d$, the generalized Schur algebra $T^A ...
• The Combinatorial PT-DT Correspondence 

  Webb, Gautam (University of Oregon, 2021-11-23)

  We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas ...
• The Combinatorics of log-Coulomb Gases in p-Fields 

  Webster, Joe (University of Oregon, 2021-09-13)

  This thesis is based on the article [16], which studies the integral? ? ?a? ?b? s ρ(x1,...,xN) max|xi −xj| min|xi −xj| |xi −xj| ij dx1 ...dxN KN i<j i<j i<j where K is an arbitrary p-field, ρ is a well-behaved function ...
• Extending Higher Bruhat Orders to Non-longest Words in S_n 

  Hothem, Daniel (University of Oregon, 2021-09-13)

  In this paper, we extend Manin and Schechtman's higher Bruhat orders for the symmetric group to higher Bruhat orders for non-longest words w in the symmetric group. We prove that the higher Bruhat orders of non-longest ...
• An Odd Analog of Plamenevskaya's Invariant of Transverse Knots 

  Montes de Oca, Gabriel (University of Oregon, 2020-12-08)

  Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. We define an analog of Plamenevskaya’s invariant in the odd Khovanov homology of Ozsváth, Rasmussen, ...
• A Special Family of Binary Forms, Their Invariant Theory, and Related Computations 

  Dethier, Christophe (University of Oregon, 2020-12-08)

  In this manuscript we study the family of diagonalizable forms, a special family of integral binary forms. We begin with a summary of definitions and known results relevant to binary forms, diagonalizable forms, Thue ...
• AT-algebras from zero-dimensional dynamical systems 

  Herstedt, Paul (University of Oregon, 2020-09-24)

  We outline a particular type of zero-dimensional system (which we call "fiberwise essentially minimal"), which, together with the condition of all points being aperiodic, guarantee that the associated crossed product ...
• Some Extension Algebras of Standard Modules over Khovanov-Lauda-Rouquier Algebras of Type A, Including A-Infinity Structure 

  Buursma, Doeke (University of Oregon, 2020-09-24)

  We give an explicit description of the category of Yoneda extensions of standard modules over KLR algebras for two special cases in Lie type A. In these two special cases, the A-infinity category structure of the Yoneda ...
• Combinatorics of the Double-Dimer Model 

  Jenne, Helen (University of Oregon, 2020-09-24)

  We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the ...
• A Categorical sl_2 Action on Some Moduli Spaces of Sheaves 

  Takahashi, Ryan (University of Oregon, 2020-09-24)

  We study a certain sequence of moduli spaces of stable sheaves on a K3 surface of Picard rank 1 over $\mathbb{C}$. We prove that this sequence can be given the structure of a geometric categorical $\mathfrak{sl}_2$ action, ...

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