Geometry and Topology Commons^™

Shrinking Attachment Spaces, Anastasia M. Clements

West Chester University Graduate Theses, Dissertations, and Final Projects

Gluing constructions such as pushouts and other colimits are often used to attach spaces to- gether in algebraic topology. The weak topology is a natural choice of topology for attachment spaces in the context of CW-complexes and simplicial complexes because of its universal property but is insufficient for gluing together infinitely many spaces and preserving topological properties like compactness or metrizability. In this thesis, we introduce a modification of the weak topology called the shrinking attachment topology, which is defined on a space Y constructed by attaching an infinite sequence of spaces B₁, B₁, B_{3 …}

Classifying Surfaces With Handle Decomposition, Elizabeth Sipes

Murray State Theses and Dissertations

Among topological spaces, manifolds draw a lot of interest. An
n-manifold is a space that is locally like R^n. Manifolds of dimension 2
are called surfaces. Using handle decomposition, we decompose surfaces
into k-handles, where 0< =k< =2. Techniques such as handle sliding and
handle cancellation allow us to get a more favorable representation of
our surface. We use these tools and calculation of the fundamental group
to classify all compact surfaces.

An Introduction To Modern Conversations On Knot Invariants, Stella Shah

Scripps Senior Theses

Knot Theory is a vast and diverse subfield of modern mathematics involving the classification and abstraction of knots and links. In this thesis, we wish to provide the necessary background for and an explanation of two papers in different subfields of knot theory, The Forbidden Quiver of a Link, and Biquandle Fares and Link Invariants.

In Chapter I, we begin with an introduction to knot theory and knot invariants. We continue to present the example of Fox Colorings, and conclude the chapter with an example of the Fox Coloring Number Invariant.

In Chapter II, we explore the derivation and utilization …

On Stripping And Antipodes In Motivic Steenrod Algebras, Joshua A. Peterson

Theses and Dissertations--Mathematics

We generalize the stripping process to the (mod 2) $\mathbb{C}$- and $\mathbb{R}$-motivic settings. Throughout, we include discussion on how the process changes and the difficulties moving to more general settings. We also introduce antipodes and consider what a potential $\mathbb{R}$-motivic analogue may look like. Finally, we elaborate on how the results may be used in future work to generalize a nilpotence result of Walker and Wood.

A Character Theory For Loop Representations Of Symmetric Monoidal Bicategories, Jordan Sawdy

Theses and Dissertations--Mathematics

Character theory arises in many distinct fields of mathematics, but its many instantiations often share a few key features: they arise in contexts where one object is acted on or parametrized by another, and they are often computed via "trace-like" formulas. Focusing on these properties, we present a categorical formalism for constructing such characters. We first define a notion of "loop representation" for symmetric monoidal bicategories, then build a character for such representations via the canonical symmetric monoidal trace. We then show that this character defines a symmetric monoidal functor which satisfies commutativity properties with respect to both restriction- and …

Geovig And Purevig: Geometry-Aware Architectures For Efficient Computer Vision, Omar Ismail

Theses and Dissertations (Comprehensive)

Deploying deep learning models for medical image analysis on mobile devices requires a balance between inference latency, memory footprint, and delineating anatomical boundaries with high accuracy. While Convolutional Neural Networks (CNNs) and mobile Vision Transformers (ViTs) offer efficiency, they often struggle to model the irregular, non-local geometric structures inherent in biological tissues without incurring prohibitive computational costs. In this thesis, we introduce GeoViG (Geometric Vision Graph), an architecture that bridges the gap between efficient grid-based processing and explicit Geometric Deep Learning. GeoViG introduces a novel transition from high-resolution pixel grids to low-resolution dynamic graphs via a SpreadEdgePool operator, a geometry-aware …

Near Real-Time Adaptive Isotropic And Anisotropic Image-To-Mesh Conversion For Cerebral Aneurysm Simulations, Kevin Garner, Chander Sadasivan, Nikos Chrisochoides

Computer Science Faculty Publications

This paper presents two performance optimization techniques for a mesh adaptation method that is designed to help streamline the discretization of complex vascular geometries within the numerical modeling process. This method is integrated into a pipeline with an image-to-mesh conversion tool to generate adaptive anisotropic meshes from segmented medical images. The pipeline is shown to satisfy quality, fidelity, smoothness, and robustness requirements while providing near real-time performance for medical image-to-mesh conversion. Tested with two brain aneurysm cases and utilizing up to 96 CPU cores within a single, multicore node on Purdue University’s Anvil supercomputer, the parallel adaptive anisotropic meshing method …

Arrangements Of N Planes Resulting In One Bounded Tetrahedral Chamber, Ava Knight

Williams Honors College, Honors Research Projects

This paper investigates the combinatorial geometry of plane arrangements in three-dimensional space, focusing on configurations that produce exactly one bounded tetrahedral chamber. We define T(n) as the number of face-combinatorial equivalence classes of arrangements of n planes in ℝ³ containing exactly one bounded tetrahedral chamber. Known values — T(3) = 0, T(4) = 1, and T(5) = 2 — are established through direct construction, while T(6) remains an open problem. This paper contributes experimental evidence toward resolving T(6) by systematically extending the two valid 5-plane arrangements and verifying, through a plane removal argument, that each yields a valid plane configuration …

Plumbed 3-Manifolds And Neumann Moves, Noah J. Pope

Theses and Dissertations

We give a constructive proof that every weakly negative definite plumbing tree can be transformed into a negative definite one by a finite sequence of Neumann moves. The argument combines Neumann’s plumbing calculus with the diagonalization algorithm of Duchon, Eisenbud, and Neumann, which extracts the eigenvalues of the framing matrix directly from the combinatorics of the tree. We show that any positive eigenvalues are supported on linear branches and can be eliminated systematically via controlled applications of Neumann moves. This provides an explicit algorithm reducing weakly negative definite plumbing trees to negative definite ones.

The Global Orbit ∞-Category And Applications To Assembly Maps, Zoë Pope

Electronic Theses & Dissertations (2024 - present)

We reformulate the foundations of assembly maps in the context of the global orbit ∞-category of all discrete groups. We first show that the ∞-categorical slice of the global orbit ∞-category over any fixed group G is equivalent to the orbit 1-category of G, and we also frame the subgroup 1-category of G in this global context. We then use the afore-mentioned equivalence to redefine assembly maps as counits of the adjunction between left Kan extension and restriction, and give purely ∞-categorical and conceptual proofs of known results, such as the Transitivity Principle. Additionally, we give equivalent formulations for what …

Categories, Homology And Sheaves For Hypergraphs, Robert Green

Electronic Theses & Dissertations (2024 - present)

Hypergraphs are a prominent tool for representing networks with connections among three or more entities. There is an inherent flexibility that allows hypergraphs to more naturally represent certain types of networks than graphs or simplicial complexes can on their own. This flexibility, however, comes at a cost, as there is a zoo of various categories and homology theories that are applicable to hypergraphs. The first chapter of this dissertation explores various categorical perspectives on hypergraphs, focusing on what the natural notion of morphism between hypergraphs should be. It also contains an exploration of the functoriality of vertex-edge duality in these …

Entropic Dynamics Approach To The Classical Limit Of Quantum Mechanics: Decoupling Of The Center Of Mass Motion For A Mesoscopic Particle, Fatimah Judayba

Electronic Theses & Dissertations (2024 - present)

In the Entropic Dynamics (ED) approach, quantum mechanics is derived from the principles of entropic inference and information geometry. The ED approach differs from other interpretations by making a clear commitment to distinguishing which variables are ontic (real) and which are epistemic. The classical limit for the center of mass (CM) coordinate is achieved for a large number of particles, M →∞, while Planck’s constant ℏ remains finite. Typically, the emergence of the classical limit requires decoherence through interactions with the external environment. In this work, we investigate whether the classical behavior of the CM coordinate in a mesoscopic system …

Face Value: A Computational Approach To Subjective Impressions Of Faces, Kevin Kpankou

Undergraduate Research Symposium

Various computational models of first impressions have been developed to uncover the mechanisms driving these judgments. However, the implicit notion of a singular ``human'' often overlooks meaningful individual differences in beliefs, attitudes, and associations, as well as culturally grounded group-level constructs. In this paper, we extend Cultural Consensus Theory (CCT) to estimate culturally shared beliefs about faces by incorporating latent constructs structured around interpretable facial features extracted via computer vision algorithms. We apply our model to a large-scale dataset of people’s first impressions of faces. Our approach reveals a robust mapping between facial features and culturally constructed impressions, allowing us …

There Is No Obstruction To A Euclidean Proof For The Fourth Postulate, José Gil-Férez, Piotr Błaszczyk, M. Andrew Moshier, Alberto Naibo, Jean-Michel Salanskis

Mathematics, Physics, and Computer Science Faculty Articles and Research

In Gil-Férez et al. (2025), we presented a proof of Postulate 4 using purely Euclidean techniques, against which Blåsjö (2025) raised several objections. In this note, we offer linguistic, textual, historical, and mathematical evidence that demonstrate that all these objections are baseless.

(R2137) Smarandache Curved Ruled Surfaces And Their Characterizations According To Modified Orthogonal Frame In E^3, Gülnur Şaffak Atalay

Applications and Applied Mathematics: An International Journal (AAM)

In this study, the ruled surfaces generated by Smarandache curves are expressed according to the modified orthogonal frame defined according to both curvature and torsion of the given unit speed curve in 3-dimensional Euclidean space; some special characterizations of these surfaces such as developability, striction curves, distribution parameters are given. In addition, some examples of these surfaces are given in graphical form. We use Maple to draw the graphs of the examples.

A Persistent Homology Framework For Scrna-Seq: Assessing Clustering Robustness And Quantifying Preprocessing And Integration Effects On Topological Features., Jonah Daneshmand

Electronic Theses and Dissertations

As single-cell RNA sequencing (scRNA-seq) data expands, robust methods for integrating diverse datasets are critical. This dissertation applies Persistent Homology (PH), a technique from Topological Data Analysis (TDA), to a collection of scRNA-seq datasets spanning eight tissue types to quantify how data integration affects topological features and biological interpretability. We assessed global topological structure using Betti curves, Euler characteristics, and persistence landscapes across raw, normalized, and integrated data representations. Our analysis revealed a performance inversion: while conventional methods excelled on unintegrated data, high-granularity topological methods, particularly those sensitive to global data structure, became superior after integration. This suggests a synergy …

Exploratory Study Of Semiconductor Nanomembranes In Em Applications, Grant D. Heileman

Electrical and Computer Engineering ETDs

Antenna systems are a cornerstone of modern technologies, playing an increasingly vital role in their advancement. As demand for compact, high-performance, and adaptable communication platforms grows reconfigurable antenna technologies are becoming essential. This research explores a novel front-end reconfigurable antenna system (FERAS) architecture that leverages the mechanical flexibility and photoconductive behavior of semiconductor nanomembrane (SNM) devices. By exploiting the emergent properties of ultra-thin silicon (Si) or gallium arsenide (GaAs) nanomaterials and optically exciting these samples using vertical-cavity surface-emitting laser (VCSEL) arrays, this study develops lightweight, low-cost, deployable antenna structures for satellite communications, remote sensing, GPS, and radar. Despite their significant …

Growth Mindset In The Mathematics Classroom: Studying The Effects Of A Growth Mindset Intervention On High School Students In A Geometry Classroom, Katelyn Strauss

Mathematics Graduate Theses

Having a growth mindset is crucial to being successful in a mathematics classroom. However, many students believe themselves to be either “mathematics people” or “not mathematics people”. When students believe they are not a mathematics person, they are far less likely to engage in material where they face struggles, and thus they miss out on valuable learning. A review of the literature has show that students have been able to shift their mindset from fixed to growth through a series of interventions. I used a unit in Geometry to see how a growth mindset intervention would affect the mindset of …

Matrices Induced By Scaled Hypercomplex Numbers Over The Real Field R, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we construct, and study a certain type of definite, or indefinite inner product spaces over the real field R, induced by the scaled hypercomplex numbers H_t for a fixed scale t ∈ R, and some bounded operators acting on such vector spaces. In particular, we are interested in the vector spaces H^N_t consisting of all N-tuples of scaled hypercomplex numbers of H_t, and the (N x N)-matrices acting on H^N_t whose entries are from H_t, i.e., H_t-matrices, for all N ∈ N. For an arbitrarily fixed …

Euler’S Original Derivation Of Elastica Equation, Shigeki Matsutani

Euleriana

Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of elastica and show that Euler used Noether's theorem concerning the translational symmetry of elastica, although Noether published her theorem in 1918. It is also shown that his equation is essentially the static modified KdV equation which is obtained by the isometric and isoenergy conditions, known as the Goldstein-Petrich scheme.