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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Item Open Access RO(C₂)-graded Stable Stems and Equivariant Framed Bordism(University of Oregon, 2024-12-19) McGinnis, Stewart; Dugger, DanielThe purpose of this dissertation is to prove fundamental relations in the $RO(C_2)$-graded stable equivariant homotopy groups of spheres $\pi_{*,*}$ using geometric methods. The main tool we use is a singular version of the Pontryagin-Thom isomorphism which holds in the equivariant setting. Our work then consists of writing down explicit bordisms between manifold representatives of homotopy classes. Selected relations include $\epsilon \eta = \eta$, $\rho \eta = 1 + \epsilon$, and $24 \nu = 0$ where $\eta$ and $\nu$ are equivariant Hopf maps, $\epsilon$ is a unit in $\pi_{0,0}$, and $\rho$ is the generator of $\pi_{-1,-1}$. We also completely characterize the periodic portion of the topological zero-stem $\pi_{0,*}$ using singular manifold representatives which are the products $C_2 \times D^k$ equipped with various $C_2$-actions. While we focus on $C_2$, most of the theory we develop applies to $RO(G)$-graded homotopy groups for arbitrary finite groups $G$.Item Open Access PSEUDO SYMMETRIC MULTIFUNCTORS: COHERENCE AND EXAMPLES(University of Oregon, 2024-08-07) Manco, Diego; Dugger, DanielDonald Yau introduced pseudo symmetric Cat-multifunctors and proved that Mandell's inverse K-theory multifunctor is stably equivalent to a pseudo symmetric one. We prove a coherence result for pseudo symmetric Cat-multifunctors in the form of a 2-adjunction. As a consequence, we obtain that pseudo symmetric Cat-multifunctors preserve E_n-algebras parameterized by free Cat-operads at the cost of changing the parameterizing Cat-operad by its product with the categorical Barrat-Eccles operad. Since Mandell's inverse K-theory is pseudo symmetric, we derive that E_n-algebras parameterized by free E_n Cat-operads in the symmetric monoidal category of $\Gamma$-categories can be realized, up to stable equivalence, as the K-theory of some E_n-algebra in the multicategory of permutative categories. This result can be regarded as a multiplicative version of a theorem by Thomason that says that any connective spectrum can be realized as the K-theory of a suitable symmetric monoidal category up to stable equivalence. Our coherence theorem also allows for a simple description of a 2-category defined by Yau which has Cat-multicategories as 0-cells and pseudo symmetric Cat-multifunctors as 1-cells. We also provide new examples of pseudo symmetric Cat-multifunctors by proving that the free algebra functor of a symmetric, pseudo commutative, strong 2-monad, as defined by Hyland and Power, can be seen as a pseudo symmetric Cat-multifunctor. This result can be interpreted as a coherence result, and it implies a coherence result for pseudo commutative, strong 2-monads conjectured by Hyland and Power.Item Open Access Two Problems in Symmetric Tensor Categories(University of Oregon, 2024-08-07) Czenky, Agustina; Ostrik, VictorFix an algebraically closed field k of characteristic $p \geq 0$. A symmetric fusion category $\mathcal C$ over k is a fusion category endowed with a braiding $c_{X,Y}: X\otimes Y \to Y \otimes X$ such that $c_{Y,X}c_{X,Y}=\id_{X\otimes Y} $ for all $X, Y \in \mathcal{C}$. The first part of this dissertation focuses on the study of symmetric fusion categories in positive characteristic. We give lower bounds for the rank of a symmetric fusion category in characteristic $p\geq 5$ in terms of $p$. We also prove that the second Adams operation $\psi_2$ is not the identity for any non-trivial symmetric fusion category, and that symmetric fusion categories satisfying $\psi_2^a=\psi_2^{a-1}$ for some positive integer $a$ are super-Tannakian. As an application, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects. The second part of this dissertation treats symmetric categories in the context of topological quantum field theories. We construct a family of unoriented 2-dimensional cobordism theories parametrized by certain triples of sequences, and prove that some specializations of these sequences yield equivalences with an exterior product of Deligne categories. It is known that, modding out the category of 2-dimensional oriented cobordisms by the relation that a handle is the identity, and evaluating 2-spheres to $t$, produces a category equivalent to the Deligne category $\Rep(S_t)$, which generalizes the representation category of the symmetric group $S_n$ from $n\in \mathbb N$ to $t\in \textbf k$. We show an analogous story for unoriented 2-dimensional cobordisms, with a construction that recovers the category $\Rep(S_t\wr \mathbb Z_2)$. This dissertation contains previously published material.Item Open Access The Squish Map and the SL_2 Double Dimer Model(University of Oregon, 2024-08-07) Foster, Leigh; Young, BenjaminA plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a “coarser” plane partition, made of cubes of side length 2. Indeed, there are two such approximations obtained by “rounding up” or “rounding down” to the nearest cube. We relate this coarsening (or downsampling) operation to the squish map introduced by the second author in earlier work. We exhibit a related measure-preserving map between the dimer model on the honeycomb graph, and the SL_2 double dimer model on a coarser honeycomb graph; we compute the most interesting special case of this map, related to plane partition q-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity, obtain novel generating functions (some known, some new, and some conjectural) that (−1)-enumerate certain classes of pairs of plane partitions according to how their dimer configurations interact.Item Open Access Visual Aspects of Gaussian Periods and Analogues(University of Oregon, 2024-08-07) Platt, Samantha; Eischen, EllenIn this dissertation, we study Gaussian periods and their analogues from a visual perspective. Building on the work of Duke, Garcia, Hyde, Lutz, and others [BBF+14, BBGG+13, DGL15, GHL15], we introduce a more dynamical study of Gaussian periods, and we prove an explicit bound on the value of Gaussian periods using this framework. Additionally, we generalize the construction of Gaussian periods using the perspective of supercharacter theory. Using this new construction, we prove a result which greatly generalizes the main theorem of [DGL15]. We also initiate the visual study of Gaussian periods from the perspective of number theory and class field theory, and we define a generalized construction of Gaussian periods using this perspective. We discuss this class field theory analogue in depth when the base field is quadratic imaginary. The work presented here includes and expands upon a paper by this author [Pla24], which is set to appear in the "International Journal of Number Theory."Item Open Access Hermitian Jacobi Forms of Higher Degree(University of Oregon, 2024-08-07) Haight, Sean; Eischen, EllenWe develop the theory of Hermitian Jacobi forms in degree $n > 1$. This builds on the work of Klaus Haverkamp in \cite{HThesis} who developed this theory in degree $n = 1$. Haverkamp in turn generalized a monograph of Eichler and Zagier, \cite{E&Z}. Hermitian Jacobi forms are holomorphic functions which appear in certain infinite series expansions (Fourier Jacobi expansions) of Hermitian modular forms. In this work we give a definition of Hermitian Jacobi forms in degree $n > 1$, give their relationship to more classical Hermitian modular forms and construct a useful tool for studying Hermitian Jacobi forms, the theta expansion. This theta expansion allows us to relate our forms to classical modular forms via the Eichler-Zagier map and thereby bound the dimension of our space of forms. We then go on to apply the developed theory to prove some non-vanishing results on the Fourier coefficients of Hermitian modular forms.Item Open Access Reducible Dehn Surgeries, Ribbon Concordance, and Satellite Knots(University of Oregon, 2024-08-07) Bodish, Holt; Lipshitz, RobertIn this thesis we investigate knots and surfaces in $3$- and $4$-manifolds from the perspective of Heegaard Floer homology, knot Floer homology and Khovanov homology. We first investigate the \emph{Cabling Conjecture}, which states that the only knots that admit reducible Dehn surgeries are cabled knots. We study this question and related conjectures in Chapter \ref{reduciblesurgeries} and develop a lower bound on the slice genus of knots that admit reducible surgeries in terms of the surgery parameters and study when a slope on an almost L-space knot is a reducing slope. In particular, we show that when $g(K)$ is odd and $>3$, the only possible reducing slope on an almost L-space knot is $g(K)$ and in that case the complement of an almost L-space knot does not contain any punctured projective planes. In Chapter \ref{chaptersatellite} we investigate the effect of satellite operations on knot Floer homology using techniques from bordered Floer homology \cite{LOT} and the immersed curve reformulation \cites{HRW,Chen, chenhanselman}. In particular we study the functions $n \mapsto g(P_n(K)), \epsilon(P_n(K))$ and $\tau(P_n(K))$ for some families of $(1,1)$ patterns $P$ from the immersed curve perspective. We also consider the function $n \mapsto \dim(\HFKhat(S^3,P_n(K),g(P_n(K)))$, and use this together with the fibered detection property of knot Floer homology \cite{Nifibered} to determine, for a given pattern $P$, for which $n \in \Z$ the twisted pattern $P_n$ is fibered in the solid torus. In Chapter \ref{Chapterribbon} we answer positively a question posed by Lipshitz and Sarkar about the existence of Steenrod operations on the Khovanov homology of prime knots \cite[Question 3]{MR3966803}. The proof relies on a construction of a particular type of surface, called a ribbon concordance in $S^3 \times I$, interpolating between any given knot and a prime knot together with the fact that the maps induced on Khovanov homology by ribbon concordances are split injections \cite{MR3122052,MR4041014}.Item Open Access A SPECIAL ENDOMORPHISM OF THE STANDARD GAITSGORY CENTRAL OBJECT OF THE AFFINE HECKE CATEGORY(University of Oregon, 2024-03-25) Hathaway, Jay; Elias, BenUsing the combinatorial description of the standard Gaitsgory centralobject of the (extended, graded) affine type A Hecke category due to Elias, we show the existence of and explicitly describe the unique endomorphism that lifts right multiplication by the i-th fundamental weight on the i-th component of the associated graded of its Wakimoto filtration. We give work in progress on describing a conjectural program to categorify the Vershik-Okounkov approach to the representation theory of the affine Hecke algebra. Here this endomorphism will play a role. This is the affine version of the program described by Gorsky, Negut, and Rasmussen in finite type A.Item Open Access Scalar Curvature and Transfer Maps in Spin and Spin^c Bordism(University of Oregon, 2024-01-10) Granath, Elliot; Botvinnik, BorisIn 1992, Stolz proved that, among simply connected Spin-manifolds of dimension5 or greater, the vanishing of a particular invariant α is necessary and sufficient for the existence of a metric of positive scalar curvature. More precisely, there is a map α: ΩSpin → ko (which may be realized as the index of a Dirac operator) ∗ which Hitchin established vanishes on bordism classes containing a manifold with a metric of positive scalar curvature. Stolz showed kerα is the image of a transfer map ΩSpinBPSp(3) → ΩSpin. In this paper we prove an analogous result for Spinc- ∗−8 ∗ manifolds and a related invariant αc : ΩSpinc → ku. We show that ker αc is the ∗ sum of the image of Stolz’s transfer ΩSpinBPSp(3) → ΩSpinc and an analogous map ∗−8 ∗ ΩSpinc BSU(3) → ΩSpinc . Finally, we expand on some details in Stolz’s original paper ∗−4 ∗ and provide alternate proofs for some parts.Item Open Access C*-Correspondences, Hilbert Bimodules, and their L^p Versions(University of Oregon, 2024-01-10) Delfin Ares de Parga, Alonso; Phillips, N. ChristopherThis dissertation initiates the study of $L^p$-modules, which are modules over $L^p$-operator algebras inspired by Hilbert modules over C*-algebras. The primary motivation for studying $L^p$-modules is to explore the possibility of defining $L^p$ analogues of Cuntz-Pimsner algebras. The first part of this thesis consists of investigating representations of C*-correspondences on pairs of Hilbert spaces. This generalizes the concept of representations of Hilbert bimodules introduced by R. Exel in \cite{Exel1993}. We present applications of representing a correspondence on a pair of Hilbert spaces $(\Hi_0, \Hi_1)$, such as obtaining induced representations of both $\Li_A(\X)$ and $\mathcal{K}_A(\X)$ on $\Hi_1$, and giving necessary and sufficient conditions on an $(A,B)$ C*-correspondences to admit a Hilbert $A$-$B$-bimodule structure. The second part is concerned with the theory of $L^p$-modules. Here we present a thorough treatment of $L^p$-modules, including morphisms between them and techniques for constructing new $L^p$-modules. We then useour results on representations for C*-correspondences to motivate and develop the theory of $L^p$-correspondences, their representations, the $L^p$-operator algebras they generate, and present evidence that well-known $L^p$-operator algebras can be constructed from $L^p$-correspondences via $L^p$-Fock representations. Due to the technicality that comes with dealing with direct sums of $L^p$-correspondences and interior tensor products, we only focus on two particular examples for which a Fock space construction can be carried out. The first example deals with the $L^p$-module $(\ell_d^p, \ell_d^q)$, for which we exhibit a covariant $L^p$-Fock representation that yields an $L^p$-operator algebra isometrically isomorphic to $\mathcal{O}_d^p$, the $L^p$-analogue of the Cuntz-algebra $\mathcal{O}_d$ introduced by N.C. Phillips in \cite{ncp2012AC}. The second example involves a nondegenerate $L^p$-operator algebra $A$ with a bicontractive approximate identity together with an isometric automorphism $\varphi_A \in \op{Aut}(A)$. In this case, we also present an algebra associated to a covariant $L^p$-Fock representation, but due to the current lack of knowledge of universality of the $L^p$-Fock representation, we only show that there is a contractive map from the crossed product $F^p(\Z, A, \varphi_A)$ to this algebra. This dissertation includes unpublished material.Item Open Access Structures and Computations in Annular Khovanov Homology(University of Oregon, 2024-01-09) Davis, Champ; Lipshitz, RobertLet $L$ be a link in a thickened annulus. In [GLW17], Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $\exsltwo$, the exterior current algebra of the Lie algebra $\sltwo$. In this dissertation, we upgrade this result to the setting of $L_\infty$-algebras and modules. That is, we show that $\exsltwo$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $\exsltwo$. Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves. In proving the above result, we include explicit formulas to compute the higher $L_\infty$-operations. Additionally, given a morphism $I: L' \to L$ of $L_\infty$-algebras, we define a restriction of scalars operation in the setting of $L_\infty$-modules and prove that it defines a functor $I^*: L-mod \to L'-mod$. A more abstract approach to this problem was recently given by Kraft-Schnitzer. Finally, computer code was written to aid in the study of the above $L_\infty$-module structure. We discuss various patterns that emerged from these computations, most notably one relating the torsion in the annular Khovanov homology groups and the location of the inner boundary of the annulus.Item Open Access Ribbons, Satellites, and Exotic Phenomena in Heegaard Floer Homology(University of Oregon, 2024-01-09) Guth, Gary; Lipshitz, RobertWe study properties of surfaces embedded in 4-manifolds by way of HeegaardFloer homology. We begin by showing link Floer homology obstructs concordance through ribbon homology cobordisms; this extends the work of Zemke and Daemi-Lidman-Vela–Vick-Wong. In another direction, we consider the effect of satellite operations on concordances. We show that the map induced by a satellite concordance is determined by the pattern and the map induced by the original concordance map. As an application, we produce the first examples of stably exotic behavior in the four-ball, i.e. we produce exotic disks whose exotic behavior persists under many 1-handle stabilizations. As a second application, in joint work with Hayden-Kang-Park, we show that the positive Whitehead doubling pattern is injective on the class of HFK-distinguishable disks in B4: we show that for any disks D,D′ in B4 which are distinguished by their induced maps on HFK, their positive Whitehead doubles are also distinguished. In particular, Wh+(D) and Wh+(D′) are exotic.Item Open Access New A-infinity Diagonals from Contractions of the Weighted Associahedra(University of Oregon, 2024-01-09) Phillips, Bo; Dugger, DanielIn this paper, we build on the work of Lipshitz, Ozsv\'{a}th, and Thurston by constructing an algorithm that generates a weighted $A_\infty$-diagonal given a family of contractions of the weighted associahedron complexes. Using this, we exhibit a new weighted $A_\infty$-diagonal and relate it to the unweighted $A_\infty$-diagonal exhibited by Masuda-Thomas-Tonks-Vallete given by so-called ``right-moving trees.''Item Open Access Polynomial Root Distribution and Its Impact on Solutions to Thue Equations(University of Oregon, 2024-01-09) Knapp, Greg; Akhtari, ShabnamIn this study, we focus on two topics in classical number theory. First, we examine Thue equations—equations of the form F(x, y) = h where F(x, y) is an irreducible, integral binary form and h is an integer—and we give improvements to both asymptotic and explicit bounds on the number of integer pair solutions to Thue equations. These improved bounds largely stem from improvements to a counting technique associated with “The Gap Principle,” which describes the gap between denominators of good rational approximations to an algebraic number. Next, we will take inspiration from the impact of polynomial root distribution on solutions to Thue equations and we examine polynomial root distribution as its own topic. Here, we will look at the relation between the separation of a polynomial—the minimal distance between distinct roots—and the Mahler measure of a polynomial—a height function which connects the roots of a polynomial with its coefficients. We make a conjecture about how separation can be bounded above by the Mahler measure and we give data supporting that conjecture along with proofs of the conjecture in some low-degree cases.Item Open Access Lines on Cubic Threefolds and Fourfolds Containing a Plane(University of Oregon, 2024-01-09) Brooke, Corey; Addington, NicolasThis thesis describes the Fano scheme $F(Y)$ of lines on a general cubic threefold $Y$ containing a plane over a field $k$ of characteristic different from $2$. One irreducible component of $F(Y)$ is birational (over $k$) to a torsor $T$ of an abelian surface, and we apply the geometry and arithmetic of this torsor to answer two questions. First, when is a cubic threefold containing a plane rational over $k$, and second, how can one describe the rational Lagrangian fibration from the Fano variety of lines on a cubic fourfold containing a plane? To answer the first question, we apply recently developed intermediate Jacobian torsor obstructions and show that the existence over $k$ of certain classical rationality constructions completely determines whether the threefold is rational over $k$. The second question, motivated by hyperkähler geometry, we answer by giving an elementary construction that works over a broad class of base fields where hyperkähler tools are not available; moreover, we relate our construction to other descriptions of the rational Lagrangian fibration in the case $k=\bC$.Item Open Access Categorical Invariants of Graphs and Matroids(University of Oregon, 2024-01-09) Miyata, Dane; Proudfoot, NicholasGraphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to certain morphisms by realizing these invariants as functors from a category of graphs (resp. matroids). For graphs, we study invariants that respect deletions and contractions ofedges. For an integer $g > 0$, we define a category of $\mathcal{G}^{op}_g$ of graphs of genus at most g where morphisms correspond to deletions and contractions. We prove that this category is locally Noetherian and show that many graph invariants form finitely generated modules over the category $\mathcal{G}^{op}_g$. This fact allows us to exihibit many stabilization properties of these invariants. In particular we show that the torsion that can occur in the homologies of the unordered configuration space of n points in a graph and the matching complex of a graph are uniform over the entire family of graphs with genus $g$. For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base polytope of a matroid can be decomposed into a cell complex made up of base polytopes of other matroids. A valuative invariant of matroids is an invariant that respects these polytope decompositions. We define a category $\mathcal{M}^{\wedge}_{id}$ of matroids whose morphisms correspond to containment of base polytopes. We then define the notion of a categorical matroid invariant which categorifies the notion of a valuative invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon algebra is a categorical valuative invariant. This allows us to derive relations among the Orlik-Solomon algebras of a matroid and matroids that decompose its base polytope viewed as representations of any group $\Gamma$ whose action is compatible with the polytope decomposition. This dissertation includes previously unpublished co-authored material.Item Open Access A Partial Order Structure on the Shellings of Lexicographically Shellable Posets.(University of Oregon, 2024-01-09) Lacina, Stephen; Hersh, PatriciaThis dissertation has two main topics. The first is the introduction and in-depth study of a new poset theoretic structure designed to help us better understand the notion of lexicographic shellability of partially ordered sets (posets). Lexicographic shelling of posets was introduced by Bj{\"o}rner via a type of poset labeling known as an EL-labeling and was generalized by Bj{\"o}rner and Wachs to the notion of CL-labeling. We introduce and study a partial order structure on the maximal chains of any finite bounded poset $P$ which has a CL-labeling $\lambda$. We call this partial order the maximal chain descent order induced by $\lambda$, denoted $P_{\lambda}(2)$. We show that this new partial order can be thought of as the structure of the set of shellings of $P$ ``derived from $\lambda$". A motivating example is the weak order of type A. Another especially interesting class of examples produces natural partial orders on standard Young tableaux. We prove several results about the cover relations of maximal chain descent orders in general. We characterize the EL-labelings whose maximal chain descent orders have the expected cover relations, and we prove that this is the case for many important families of EL-labelings. The second main topic of this dissertation is that of determining the poset topology of two families of lattices known as $s$-weak order and the $s$-Tamari lattice.Item Open Access Metastable Complex Vector Bundles over Complex Projective Spaces(University of Oregon, 2024-01-09) Hu, Yang; Sinha, DevIn the unstable range, topological vector bundles over finite CW complexes are difficult to classify in general. Over complex projective spaces \mathbb{C}P^n, such bundles are far from being fully classified, or even enumerated, except for a few small dimensional cases studied in the 1970's using classical tools from homotopy theory, and more recently using the modern tool of chromatic homotopy theory. We apply another modern tool, Weiss calculus, to enumerate topological complex vector bundles over \mathbb{C}P^n with trivial Chern class data, in the first two cases of the metastable range.Item Open Access Accessing the Topological Properties of Neural Network Functions.(University of Oregon, 2024-01-09) Masden, Marissa; Sinha, DevWe provide a framework for analyzing the geometry and topology of the canonical polyhedral complex of ReLU neural networks, which naturally divides the input space into linear regions. Beginning with a category appropriate for analyzing neural network layer maps, we give a categorical definition. We then use our foundational results to produce a duality isomorphism between cellular poset of the canonical polyhedral complex and a cubical set. This duality uses sign sequences, an algebraic tool from hyperplane arrangements and oriented matroid theory. Our theoretical results lead to algorithms for computing not only the canonical polyhedral complex itself but topological invariants of its substructures such as the decision boundary, as well as for evaluating the presence of PL critical points. Using these algorithms, we produce some of the first empirical measurements of the topology of the decision boundary of neural networks, both at initialization and during training. We observing that increasing the width of neural networks decreases the variability observed in their topological expression, but increasing depth increases variability. A code repository containing Python and Sage code implementing some of the algorithms described herein is available in the included supplementary material.Item Open Access Composition and Cobordism Maps(University of Oregon, 2024-01-09) Cohen, Jesse; Lipshitz, RobertWe study the relationship between the algebra of module homomorphisms under composition and 4-dimensional cobordisms in the context of bordered Heegaard Floer homology. In particular, we prove that composition of module homomorphisms of type-$D$ structures induces the pair of pants cobordism map on Heegaard Floer homology in the morphism spaces formulation of the latter, due to Lipshitz--Ozsv\'{a}th--Thurston. Along the way, we prove a gluing result for cornered 4-manifolds constructed from bordered Heegaard triples. As applications, we present a new algorithm for computing arbitrary cobordism maps on Heegaard Floer homology and construct new nontrivial $A_\infty$-deformations of Khovanov's arc algebras. Motivated by this last result and a K\"{u}nneth theorem for Heegaard Floer complexes of connected sums, we also prove the existence of a tensor product decomposition for arc algebras in characteristic 2 and show that there cannot be such a splitting over $\Z$.