Mathematics Theses and Dissertations

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https://hdl.handle.net/1794/2044

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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Now showing 1 - 20 of 131

Open Access
A Categorical sl_2 Action on Some Moduli Spaces of Sheaves
(University of Oregon, 2020-09-24) Takahashi, Ryan; Addington, Nicolas
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, a global version of an action studied by Cautis, Kamnitzer, and Licata. As a corollary, we find that the moduli spaces in this sequence which are birational are also derived equivalent.
Open Access
A Categorification of the Positive Half of Quantum sl3 at a Prime Root of Unity
(University of Oregon, 2019-04-30) Stephens, Andrew; Elias, Ben
We place a differential on $\dot\UC_{\mathfrak{sl}_3}^+$ and show that $\dot\UC_{\mathfrak{sl}_3}^+$ is Fc-filtered. This gives a categorification of the positive half of quantum $\sl_3$ at a prime root of unity.
Open Access
A Characterization of Anisotropic H^1(R^N) by Smooth Homogeneous Multipliers
(University of Oregon, 2019-09-18) Hiserote, Martin; Bownik, Marcin
We extend a well known result of Uchiyama, which gives a sufficient condition for a family of smooth homogeneous multipliers to characterize the Hardy space H^1(R^N), to the anisotropic setting.
Open Access
A Partial Order Structure on the Shellings of Lexicographically Shellable Posets.
(University of Oregon, 2024-01-09) Lacina, Stephen; Hersh, Patricia
This 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.
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, Ben
Using 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.
Open Access
A Special Family of Binary Forms, Their Invariant Theory, and Related Computations
(University of Oregon, 2020-12-08) Dethier, Christophe; Akhtari, Shabnam
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 equations, and reduction theory. The Thue--Siegel method is applied to derive an upper bound on the number of solutions to Thue's equation $F(x,y) = 1$, where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in the argument to handle forms whose discriminant is small in absolute value. These results are applied to bound the number of integral points on a certain family of elliptic curves. A proof is given for an alternative classification of diagonalizable forms using the Hessian determinant. Algebraic restrictions are given on the coefficients of a diagonalizable form and divisibility conditions are given on its discriminant. A reduction theory for the family of diagonalizable forms is given. This theory is used to computationally verify that $F(x,y) = 1$, where $F$ is a quintic diagonalizable form with small discriminant, has few solutions.
Open Access
A Structure Theorem for RO(C2)-graded Cohomology
(University of Oregon, 2018-09-06) May, Clover; Dugger, Daniel
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 of the point, the RO(C2)-graded cohomology of a finite C2-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. The shifts are by elements of RO(C2) corresponding to actual (i.e. non-virtual) C2-representations.
Open Access
A(infinity)-structures, generalized Koszul properties, and combinatorial topology
(University of Oregon, 2011-06) Conner, Andrew Brondos, 1981-
Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material.
Open Access
The A-infinity Algebra of a Curve and the J-invariant
(University of Oregon, 2012) Fisette, Robert; Fisette, Robert; Polishchuk, Alexander
We choose a generator G of the derived category of coherent sheaves on a smooth curve X of genus g which corresponds to a choice of g distinguished points P1, . . . , Pg on X. We compute the Hochschild cohomology of the algebra B = Ext (G,G) in certain internal degrees relevant to extending the associative algebra structure on B to an A1-structure, which demonstrates that A1-structures on B are finitely determined for curves of arbitrary genus. When the curve is taken over C and g = 1, we amend an explicit A1-structure on B computed by Polishchuk so that the higher products m6 and m8 become Hochschild cocycles. We use the cohomology classes of m6 and m8 to recover the j-invariant of the curve. When g 2, we use Massey products in Db(X) to show that in the A1-structure on B, m3 is homotopic to 0 if and only if X is hyperelliptic and P1, . . . , Pg are chosen to be Weierstrass points. iv
Open Access
Abelian Arrangements
(University of Oregon, 2015-08-18) Bibby, Christin; Proudfoot, Nicholas
An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. We are interested in the topology of the complement of an arrangement. If the arrangement is unimodular, we provide a combinatorial presentation for a differential graded algebra (DGA) that is a model for the complement, in the sense of rational homotopy theory. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers. If the arrangement is chordal, then this model is a Koszul algebra. In this case, studying its quadratic dual gives a combinatorial description of the Q-nilpotent completion of the fundamental group and the minimal model of the complement of the arrangement. This dissertation includes previously unpublished co-authored material.
Open Access
Accessing the Topological Properties of Neural Network Functions.
(University of Oregon, 2024-01-09) Masden, Marissa; Sinha, Dev
We 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.
Open Access
Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A
(University of Oregon, 2015-08-18) Loubert, Joseph; Kleshchev, Alexander
This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.
Open Access
Algebraic Weak Factorization Systems in Double Categories
(University of Oregon, 2014-09-29) Schultz, Patrick; Dugger, Daniel
We present a generalized framework for the theory of algebraic weak factorization systems, building on work by Richard Garner and Emily Riehl. We define cyclic 2-fold double categories, and bimonads (or bialgebras) and lax/colax bimonad morphisms inside cyclic 2-fold double categories. After constructing a cyclic 2-fold double category FF(D) of functorial factorization systems in any sufficiently nice 2-category D, we show that bimonads and lax/colax bimonad morphsims in FF(Cat) agree with previous definitions of algebraic weak factorization systems and lax/colax morphisms. We provide a proof of one of the core technical theorems from previous work on algebraic weak factorization systems in our generalized framework. Finally, we show that this framework can be further generalized to cyclic 2-fold double multicategories, incorporating Quillen functors of several variables.
Open Access
An Odd Analog of Plamenevskaya's Invariant of Transverse Knots
(University of Oregon, 2020-12-08) Montes de Oca, Gabriel; Lipshitz, Robert
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, and Szabó. We show that the analog is also an invariant of transverse links and has similar properties to Plamenevskaya’s invariant. We also show that the analog invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. We demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and self-linking number.
Open Access
An Approach to the Irreducible Representations of Finite Groups of Lie Type Through Block Theory and Special Conjugacy Classes
(University of Oregon, 1980-08) Boyce, Richard Alan
Open Access
Approximate Diagonalization of Homomorphisms
(University of Oregon, 2015-08-18) Ro, Min; Lin, Huaxin
In this dissertation, we explore the approximate diagonalization of unital homomorphisms between C*-algebras. In particular, we prove that unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with tracial rank at most one are approximately diagonalizable. This is equivalent to the approximate diagonalization of commuting sets of normal matrices. We also prove limited generalizations of this theorem. Namely, certain injective unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with rational tracial rank at most one are shown to be approximately diagonalizable. Also unital injective homomorphisms from AH-algebras with unique tracial state into separable simple unital C*-algebras of tracial rank at most one are proved to be approximately diagonalizable. Counterexamples are provided showing that these results cannot be extended in general. Finally, we prove that for unital homomorphisms between AF-algebras, approximate diagonalization is equivalent to a combinatorial problem involving sections of lattice points in cones.
Open Access
AT-algebras from zero-dimensional dynamical systems
(University of Oregon, 2020-09-24) Herstedt, Paul; Phillips, N. Christopher
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 C*-algebra is an AT-algebra. Since AT-algebras of real rank zero are classifiable by K-theory, this is a large step towards a classification theorem for fiberwise essentially minimal zero-dimensional systems.
Open Access
Blocks in Deligne's category Rep(St)
(University of Oregon, 2010-06) Comes, Jonathan, 1981-
We give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an integer. Thereafter, we give a complete description of the blocks in Rep(St) for arbitrary t. Finally, we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St).
Open Access
C*-Correspondences, Hilbert Bimodules, and their L^p Versions
(University of Oregon, 2024-01-10) Delfin Ares de Parga, Alonso; Phillips, N. Christopher
This 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.
Open Access
Categorical Actions on Supercategory O
(University of Oregon, 2016-11-21) Davidson, Nicholas; Brundan, Jonathan
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 category to its so-called types A, B, and C blocks. The type A blocks were completely described in joint work with Brundan in terms of the general linear Lie superalgebra. This dissertation proves that the type C blocks admit the structure of a tensor product categorification of the n-fold tensor power of the natural sp_\infty-module. Using this result, we relate the combinatorics for these blocks to Webster’s orthodox bases for the quantum group of type C_\infty, verifying the truth of a recent conjecture of Cheng-Kwon-Wang. This dissertation contains coauthored material.