Ph.D. - Mathematics

Permanent URI for this collectionhttps://hdl.handle.net/10125/2094

Browse

Now showing 1 - 20 of 64

Pure unrectifiability of fractal sets in the plane
(University of Hawai'i at Manoa, 2025) Joshi, Shubham; Younsi, Malik; Mathematics
A set in the plane is purely unrectifiable if it intersects every curve of finite length, in a set of zero 1-dimensional Hausdorff measure. This dissertation is devoted to investigating the pure unrectifiability of certain classes of fractal sets in the plane, specifically those with infinite 1-dimensional Hausdorff measure. For Jordan curves in the plane, we study the relationship between the set of (inner) tangent points and pure unrectifiability. For quasicircles, we show that there is no relationship between its Hausdorff dimension and the quasicircle being purely unrectifiable. For a Jordan curve, we study the relationship between the 1-dimensional Hausdorff measure and harmonic measure with respect to one of the complementary components. This allows us to obtain a sufficient condition for the pure unrectifiability of a Jordan curve for which one of the complementary components is a one sided quasidisk, namely it is quasiconvex. Using these results, we give examples of purely unrectifiable Jordan curve Julia sets. In particular, we show that the Julia set of $z\mapsto z^2+1/4$ is purely unrectifiable and conclude by giving two applications of this work. We ask the question to classify all non purely unrectifiable quadratic Julia sets. In doing so, we survey various results on Julia sets containing rectifiable arcs and argue for the need to investigate the finer structure of quadratic Julia sets via pure unrectifiability. Lastly, we discuss some open questions we intend to pursue in the near future.
Algebraic statistics: Problems in phylogenetics and wasserstein distance optimization
(University of Hawai'i at Manoa, 2025) Nometa, Ikenna; Gross, Elizabeth; Mathematics
Algebraic statistics is an emerging field of applied mathematics at the intersectionof algebraic geometry, commutative algebra, and statistics, driven by the observation that polynomial equations govern many statistical models and inference problems. This dissertation contributes to the theoretical foundations of algebraic statistics by addressing three problems of contemporary interest: the structure of phylogenetic invariants for network-based models, the algebraic complexity of maximum likelihood estimation in Brownian motion tree models, and the computation of Wasserstein distance to statistical models. The first contribution characterizes the phylogenetic invariants of group-based Cavender-Farris-Neyman (CFN) models on level-1 phylogenetic networks. Building on recent work that reduces the study of invariants of arbitrary level-1 networks to that of sunlet networks, we show that the CFN model on a sunlet network admits a parameterization that factors through the space of skew-symmetric matrices. In this formulation, the associated phylogenetic variety is cut out by specific sub-Pfaffians, leading to an explicit description of the generators of their defining ideal. Second, we focus on the maximum likelihood degree (ML-degree) of Brownian motion tree models, which are Gaussian graphical models encoding trait evolution. We show that the ML-degree is invariant under root choice and derive a closed-form expression for the ML-degree in the case of star trees. Finally, we examine the algebraic complexity of computing the Wasserstein distance to statistical models, utilizing recent geometric formulations that lead to the computation of polar degrees of projective varieties. Restricting to small toric models, we compute the polar degrees of rational normal scrolls and some graphical models. Together, these results deepen our understanding of the polynomial structure and algebraic complexity of statistical models central to phylogenetics and optimization, offering new tools and directions for research in algebraic statistics.
The geometry of deviations from the asymptotic cycle
(University of Hawai'i at Manoa, 2025) Polakowski, Drew; Hadari, Asaf; Mathematics
Let w be a word in a free group, and let f be an automorphism. When theabelianization of f is the identity, it is known that lifts of the words f^k(w) to the cover corresponding to the commutator subgroup converge, in a scaled sense, to a convex polytope with rational vertices. I show that when the abelianization of f is not the identity, the lifts of the words f^k(w) to the same cover still have a notion of convergence to a geometric structure. This can be used to describe the asymptotic behavior of foliations of pseudo-Anosov homeomorphisms in the homology of punctured flat surfaces. The behavior I exhibit differs substan- tially from the known behavior of almost every other foliation. I demonstrate this difference visually with experiments that further suggest the presence of attracting geometric objects for all foliations.
Local fields, iterated extensions, and Julia Sets
(University of Hawai'i at Manoa, 2025) Lee, Pui Hang; Manes, Michelle; Mathematics
Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$, and let $f(z) = z^\ell - c \in K[z]$ be a separable polynomial. We explore the connection between the valuation $v(c)$ and the Berkovich Julia set of $f$. Additionally, we examine the field extensions generated by the solutions to $f^n(z) = \alpha$ for a root point $\alpha \in K$, highlighting the interplay between the dynamics of $f$ and the ramification in the corresponding field extensions.
Separating subgroups of Aut(F_n) using homological representations
(University of Hawai'i at Manoa, 2025) Vicente, Rico Emmanuel; Hadari, Asaf; Mathematics
Consider the free group on n generators, denoted Fn. Then Aut(Fn) is not a lineargroup, but one can construct many finite dimensional representations using finite index characteristic subgroups of Fn. We use these types of representations to distinguish consecutive terms in the Johnson filtration of Aut(Fn). This paper relates the action described from the Johnson filtration to the homology of some subgroup K ≤ Fn. These techniques can be used to separate specific types of subgroups B ≤ A ≤ Aut(Fn).
Finite-dimensional approximations and C*-algebraic K-theory
(University of Hawai'i at Manoa, 2025) Jaime, Arturo; Willett, Rufus; Mathematics
There are numerous ways in whiC*-algebras, this inC*-algebras, whiC*-algebra in half until ending up with something finite-dimensional. We relate this to other notions suC*-algebras. We show that torsion in the K1 group of a C*-algebra is an obstruction complexity rank one. Using the Kirchberg–Phillips classification theorem to compute the complexity rank of UCT Kirchberg algebras, we show that complexity rank equals one when the K1-group is torsion-free, and equals two otherwise. Finally, using the Milnor exact sequence given by the controlled picture of KK-theory by Willett and Yu, we describe some topological properties of the topological group KK(A,B) with respect to the satisfaction of Mittag-Leffler and stability conditions of certain inverse systems.
Algebraic and Combinatorial Applications in Systems and Evolutionary Biology
(University of Hawai'i at Manoa, 2024) Curiel, Maize; Gross, Elizabeth; Mathematics
Many real-world problems can be expressed as the solution set of polynomial equations. Under these constraints, these problems are addressed using tools from (computational) algebraic geometry, commutative algebra, and combinatorics. Inspired by questions in systems and evolutionary biology, three algebraic problems are addressed in this thesis: steady-state analysis of mass-action ordinary differential equations given by directed graphs, optimal phylogenetic consensus trees, and singularities arising from linear structural equation models given by acyclic mixed graphs. Specifically, for the first algebraic problem, we introduce a polyhedral geometry tool called a mixed volume to study the steady-state degree of partitionable binomial chemical reaction networks. For the second algebraic problem, a phylogenetic consensus tree is the optimal point of a minimization problem, or a point belonging to a tropical variety, and tools from tropical combinatorics are used to locate these optimal points. For the final algebraic problem, the main object of study is the covariance parameterization map arising from linear structural equation models given by an acyclic mixed graph. The concern is with understanding the parameters for which the covariance map fails to be locally injective.
CONCEPT ANALYSIS IN CATEGORIES
(University of Hawai'i at Manoa, 2023) Ferrer, Lance; Pavlovic, Dusko; Mathematics
Network computation requires concept mining and analysis to extract semantics shared between nodes. The impor- tance of this has grown immensely with the advent of the web, and the distribution of content through channels relies on concept analysis. While the methods of concept extraction vary widely, many of them appear to lend themselves to a categorical description in which the basis is an enriched matrix. The development of a categorical toolkit for concept mining and analysis is currently in its infancy. We further the development of this toolkit by focusing on set-valued matrices, constructing a category of concepts, and defining a formal concept in this new setting. Understanding this concept category is an exercise in understanding the structures of categories themselves. However, categories were introduced as a tool to understand structure without being concerned with the inner details of objects, focusing on morphisms and their composition. With the current presentation, we must carry objects along the path to the concept category, without ever appealing to their inner structure. To provide a less cumbersome description of the situation, we forget the objects of a category, and then state precisely when the objects can be recovered. This establishes an equiv- alence between certain categories and partial semigroups, similar to the theory of locales in pointless topology. This equivalence is intended to provide a supplemental approach to categories that is suitable for situations when objects prove to be unnecessary or clumsy, in particular, the concept category.
Geometric Control of a Quadcopter
(University of Hawaii at Manoa, 2023) Gray, Christopher; Chyba, Monique; Mathematics
In recent years, with the rise of affordable commercial grade quadcopters, there has been a lot of research done on modeling the motion of quadcopters, however much of it has been focused on creating robust control schemes for quadcopters. This dissertation works to bridge the gap this has left between motion planning and optimal control, with a focus mainly on two areas. First we work towards creating a model of the equations of motion as realistic as possible and connect the work done in geometric control theory in the past on underwater vehicles, to quadcopters. We also study possible simplifications, a specific control scheme and properties of basic motions. Secondly using an affine control model version of the equations of motion we study the time minimization problem with an emphasis on singular extremals. We observe numerically the possibility for optimal control to have an infinite number of switches in a finite time interval.
Applications at the Intersection of Hybrid Dynamical Systems and Interaction Networks
(University of Hawaii at Manoa, 2023) Carney, Richard Stephen; Chyba, Monique; Mathematics
We consider dynamical systems that exist within a larger network i.e. dynamical systems for which we have a set of local dynamics coupled together via pairwise connections. The particular coupling we have in mind relates to an object called the graph Laplacian. We further employ graph theoretic notions as a way of describing potential changes that the dynamical rules themselves may undergo. Specifically, we describe the notion of a hybrid dynamical system. We focus mainly on two different applications. First, we study protocols based on self-organizing principles, which are used to coordinate the movements of a set of multi-agents. These are meant to model a group of autonomous robotic vehicles operating independently of human control. Implementing a coordinated group of unmanned aerial vehicles, for example, is more efficient for many mission types including tracking, surveillance, and mapping. In this regard, there are numerous reasons why collective control of a group of UAVs is important. Second, inspired by the COVID-19 pandemic, we build a large scale model which accounts for coordination between regions each using travel restrictions in order to mitigate the rapid spread of disease. The model is designed via a family of compartmental SEIR models, and using the formalism of hybrid automata. There is a need for simulations of countries cooperating together, as travel restriction policies were taken by countries without global coordination. It is possible, for instance, that a strategy which appears unfavorable to a region at some point of the pandemic might be best for containing the global spread, or that only by coordinating policies among several regions can a restriction strategy be truly effective.
Mathematical Epidemiology: Modeling Of Infectious Disease
(University of Hawaii at Manoa, 2023) Kunwar, Prateek; Chyba, Monique; Mathematics
The ongoing coronavirus pandemic, caused by severe acute respiratory syndrome coronavirus2 was first identified in December 2019. This dissertation falls into the field of mathematical modelling of infectious diseases. It is composed of two parts: modeling the spread of Covid19, with an application to the Hawaiian archipelago; development of classification tools for the comparison of the evolution of the Covid-19 pandemic at different geographic locations. First we present a generalized discrete deterministic compartmental SEIR model for the spread of Covid-19 which incorporates competing variants of the virus, vaccination, fading of vaccine protection, the possibility of a previously infected individual becoming susceptible and travel restrictions. Using this model on the counties of the State of Hawai‘i , we study the impacts of mitigation measures and the impacts of tourism on the spread of the disease among the local population. Second, we focus on some classification tools. The notion of wave is used to describe the evolution of a pandemic but although the terminology is often used, the current literature does not have a precise definition for it. In this dissertation, we provide a mathematical definition for the notion of a wave and present an algorithm to detect waves from a given set of data. In addition, comparison of the evolution in time of different spread of a disease is also not well addressed in the literature. Here, we introduce topological structures associated to data representing the spread of daily cases or hospital data and define an orientation preserving pseudo-metric on them that can be used to compare the evolution of pandemic between different regions.
On the Distribution of Complexities of Finite Words, with Connections to Constructive Immunity
(University of Hawaii at Manoa, 2023) Birns, Samuel; Kjos-Hanssen, Bjørn; Mathematics
This thesis presents results on complexities of finite and infinite words, where ``complexity" is defined using finite state and Turing machines, respectively. Specifically, the complexity of a word $w$ is defined to be the minimum number of states in a finite state machine needed to uniquely output $w$ among all words of length $\abs{w}$. A theorem counting the number of words of length $n$ and deterministic hidden Markov model complexity $q$ is proven, giving insight into the distribution of deterministic hidden Markov model complexity. In particular, this theorem allows for the computation of deterministic hidden Markov model complexity in polynomial time, an improvement over the exponential-time naive computation. Non-deterministic hidden Markov models of various states are shown to be realized by deterministic hidden Markov models, and the strengths of languages defined by hidden Markov model complexity are investigated. Finally, analogs of these results are expanded to infinite words and to finite-state gambler complexity. When discussing complexities of infinite words in the next topic, $\Sigma^0_1$ and constructively $\Sigma^0_1$ dense sets are introduced. It is shown that these classes are distinct and that they occur in non-$\Delta^0_2$ degrees, high degrees, and c.e. degrees. Then, connections from $\Sigma^0_1$ sets to a problem on the computable dense subsets of $\Q$ is established. Finally, results on effective notions of the complexities of finite words are presented, providing a link between the two topics.
Mathematical modeling of sensing and feeding by copepods
(University of Hawaii at Manoa, 2022) Hachmeister, Julian; Takagi, Daisuke; Mathematics
The ability to feed effectively is a fundamental skill required for survival. Microscopic crustaceans called copepods are a great example of an animal that are very adept at eating. They are tiny creatures, roughly on the order of 1 mm, and so their fluid environment is not like the water that we are accustomed to, but rather, is more like that of honey. To make things even more difficult, they have poor eyesight and are only able to detect the presence of light. Instead of using vision to detect the presence of food, they rely on other sensory mechanics like chemical signals and hydro-mechanical disturbances to remotely detect their prey. Another challenge they are presented with is moving a particle to a desired location for inspection and consumption. Given the high viscosity of their environment, particle transport is difficult since small objects near the boundary of a body will tend to stick to and move with the motion of the body. In this dissertation, we first create a model for three modes of feeding: sinking, swimming and hovering. For each of the three modes, we first create the flow fields by including the antennae, a feature often neglected in previous studies. Then, we measure the magnitude of the disturbance vector induced by a spherical particle located in a plane with sensor locations along the antennae. From this, a detectable volume is generated showing what the model could theoretically detect over a given period of time. What we discover is that sinking may be a preferable mode of feeding if the copepod were surrounded by food. If the copepod is unable to detect anything, then swimming might be best as it would increase the copepod's chances of encountering food. If the copepod came across a dense cluster of food, then positioning underneath the cluster could be the best mode as it would allow the copepod to funnel the food from above to its body. In the second and third chapter, we explore the copepod's ability to transport small particles using the Weis-Fogh fling and clap mechanism. Copepods have been observed performing this motion as early as the 1980's but there have been no mathematical models for the fling and clap in the Stokes regime. We investigate the efficacy of the motion by representing a pair of appendages as either a pair of rods or a pair of plates. What we discover is that both representations lead to a positive net displacement of particles in the desired direction and that increasing the maximum slope $a$ of the appendages lead to an increase in the displacement, specifically on the order of $a^2$ for small values of $a$.
Topological Dynamics in the Study of C*-Algebras: Finite Approximations and Dynamic Dimension
(University of Hawaii at Manoa, 2022) Pilgrim, Samantha Jane; Willett, Rufus; Mathematics
We investigate properties of dynamical systems motivated by the crossed product construction, specifically finite approximation properties and the dynamic asymptotic dimension. Chapter 1 provides some motivation and historical context with a more in depth overview of each chapter. In chapter 2, we show that equicontinuous actions on Cantor sets are profinite and that equicontinuous actions by finitely generated groups are residually finite. The latter requires some background on representation theory and Lie theory, which we also provide in this chapter. In chapter 3, we show that equicontinuous actions are quasidiagonal and use this to exhibit new examples of group actions whose crossed products have the MF property. Chapter 4 gives some background on the dynamic asymptotic dimen- sion and related concepts in geometric group theory. We show the dynamic asymptotic dimension of actions on profinite completions is closely related to the asymptotic dimension of the acting group’s box spaces. Chapter 5 contains proofs for Hurewicz-type theorems for the dynamic asymptotic dimension, which we use to describe the asymptotic dimension of box spaces of elementary amenable groups. In chapter 6, we give sharp bounds for the dimension of most isometric actions, and use these to completely describe the dimension of translation actions on compact lie groups in terms of the amenability and asymptotic dimension of the acting group.
Topologies on Positive Type Functions on Groupoids, Weak Containment of Continuous Unitary Representations, and Property (T)
(University of Hawaii at Manoa, 2022) Corea, Kenneth; Willett, Rufus; Mathematics
We introduce two new topologies on the space of normalized positive type functions on a groupoid with Haar system, the weak* topology and fiberwise compact convergence topology. We demonstrate these topologies are equivalent when the groupoid is second countable and locally compact, thereby extending Raikov's theorem. Using the fiberwise compact convergence topology we introduce a notion of weak contaiment of continuous unitary representations of a groupoid. We characterize weak containment of the trivial representation with almost invariant sections which extends the idea of almost invariant vectors in group representations. This weak containment naturally suggests a definition of property (T) for groupoids which extends Kazhdan's group property (T). We briefly explore this property and attempt to generalize some of the classic results from the theory of groups.
Persistent Cohomology Of Cover Refinements
(University of Hawaii at Manoa, 2022) Markovichenko, Oleksandr; Mileyko, Yuriy; Mathematics
Topological data analysis (TDA) is a new approach to analyzing complex data which often helps reveal otherwise hidden patterns by highlighting various geometrical and topological features of the data. Persistent homology is a key in the TDA toolbox. It measures topological features of data that persist across multiple scales and thus are robust with respect to noise. Persistent homology has had many successful applications, but there is room for improvement. For large datasets, computation of persistent homology often takes a significant amount of time. Several approaches have been proposed to try to remedy this issue, such as witness complexes, but those approaches present their own difficulties. In this work, we show that one can leverage a well-known data structure in computer science called a cover tree. It allows us to create a new construction that avoids difficulties of witness complex and can potentially provide a significant computational speed up. Moreover, we prove that the persistence diagrams obtained using our novel construction are actually close in a certain rigorously defined way to persistence diagrams which we obtain using the standard approach based on Cech complexes. This quantifiable coarse computation of persistence diagrams has the potential to be used in many applications where features with a low persistence are known to be less important.
On New Notions of Algorithmic Dimension, Immunity, and Medvedev Degree
(University of Hawaii at Manoa, 2022) Webb, David J.; Kjos-Hanssen, Bjørn; Mathematics
We prove various results connected together by the common thread of computability theory. First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension. We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate how this notion of Pi^0_1-immunity is connected to other immunity notions, and construct Pi^0_1-immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a Pi^0_1-immune set. Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov-Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, i.e. there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random.
Curve Pushing Maps and Homological Representations of Mapping Class Groups
(University of Hawaii at Manoa, 2022) Flores, Daniel James; Hadari, Asaf; Mathematics
In this paper we discuss one type of finite dimensional representation of the mapping class group using push maps. Mapping class groups, denoted Mod(X), are well studied and appear in many areas of mathematics, including the study of braid groups and 3−manifolds. One area that still has much work to be done is the finite dimensional representation theory. One type of finite dimensional representation of Mod(X) is a homological representation. Homological representations are a class of finite dimensional representations of Mod(X). For every finite cover Y → X there is an associated homological representation of Mod(X). Push maps are homeomorphisms of X that push a point or a curve along the surface returning them to their original position. Since they are homeomorphisms, they belong to a class in Mod(X). This work describes the image of push maps under homological representations.
Shapes of pure prime degree number fields
(University of Hawaii at Manoa, 2021) Holmes, Erik; Harron, Robert; Varma, Ila; Mathematics
This thesis extends a result of Rob Harron, in \cite{purecubics}. Specifically, Harron studies the shapes of pure cubic number fields $K=\QQ(\sqrt[3]{m})$ and shows that the shape is a complete invariant of the family of pure cubic number fields, and that the shapes are equidistributed on one-dimensional subspaces of the space of shapes. For pure prime degree number fields, $K=\QQ(\sqrt[p]{m})$, we show that the shape is a complete invariant. For $\ell=\frac{p-1}{2}$, our main result shows that the shapes of these fields lie on one of two $\ell$-dimensional subspaces of the space of shapes and we prove equidistribution results for $p<1000$. This work uses analytic methods which differ from those used in \cite{purecubics} and we therefore obtain an alternative proof of his result as well. We also prove that the family of pure prime degree number fields is equivalently the family of degree $p$ number field with Galois group $F_p$ and fixed resolvent field $\QQ(\zeta_p)$. This allows us to rephrase our results in a manner more closely related to those in the study of number field asymptotics and specifically Malle's conjecture. This alternative also allows us to ask a very natural follow up question which we intend to investigate in future work.
Kernel-Based Galerkin Methods on Compact Manifolds Without Boundary, with an Emphasis on SO(3)
(University of Hawaii at Manoa, 2021) Collins, Patrick; Hangelbroek, Thomas; Mathematics
We develop a kernel-based Galerkin method for numerically solving elliptic partial differential equations on compact Riemannian manifolds without boundary that are \textit{mesh-free}, \textit{coordinate-free}, \textit{regularity-preserving}, and \textit{high-performing}. A fundamental challenge is to combine a theoretical solution with mesh-free \textit{quadrature} in such a way that approximation power is not lost.\\\indent We show that the approximation power of the computed (discretized) solution can be made to be on par with the approximation power of the theoretical solution, provided the \textit{oversampling exponent} is sufficiently large. We then show how to truncate a kernel on $ SO(3) $ given by a Hilbert-Schmidt series in such a way that the approximation power of both the truncated solution and the discretized truncated solution (again using quadrature) is on par with the approximation power of the theoretical solution, provided the \textit{truncation parameter} is sufficiently large.

Browse

Recent Submissions