Geometry and Topology Commons^™

Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, Michael R. Marinelli

Dissertations, Theses, and Capstone Projects

The first focus of this dissertation is to compute Ptolemy varieties for triangulations of two infinite families of manifolds. Given an ideal triangulation of a cusped manifold, one can compute the Ptolemy variety and using it, obtain parabolic representations of the fundamental group. We compute certain obstruction classes for these manifolds, which are necessary to obtain the discrete faithful representation. This leads to our second focus of the dissertation, the twisted Alexander polynomial. The twisted Alexander polynomial (TAP) is a variation of the classical Alexander polynomial twisted by a representation of the fundamental group into a linear group. It was …

Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

Dissertations, Theses, and Capstone Projects

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …

New Class Function In Dual Soft Topological Space, Maryam Adnan Al-Ethary, Maryam Sabbeh Al-Rubaiea, Mohammed H. O. Ajam

Al-Bahir

In this paper we introduce a new class of maps in the dual Soft topological space and study some of its basic properties and relations among them, then we study and mapping.

Exploring Intraplate Seismicity In The Midwest, Alexa Fernández

Department of Earth and Atmospheric Sciences: Dissertations, Theses, and Student Research

Intraplate seismicity represents a notable occurrence within the stable North American Craton. This research explores the potential sources of stresses that could reactivate older faults and influence seismic activity within this region. Among these sources, the enduring impact of the last glacial period is considered, which includes continued glacial isostatic adjustments (GIA). During GIA the lithosphere rebounds due to the retreating ice, and the forebulge caused by far-field flexure in response to the glacial load, collapses. This results in significant faulting, fracturing, and seismic activity associated with the deglaciation phase. The adjustment of the lithosphere manifests as both near surface …

Differentiated Instruction In Geometry Using Low Floor, High Ceiling Mathematical Tasks, Franklin R. Falculan, Maria Alva Q. Aberin

Mathematics Faculty Publications

This study investigated the effects of using Low Floor High Ceiling (LFHC) mathematical tasks on students’ conceptual understanding and procedural fluency in seventh-grade Geometry by closely examining pre-test and posttest results. Two intact classes composed of thirty-two grade 7 students in each class participated in the study. The control group was taught and had practice using conventional, algorithmic tasks while the experimental group was taught and had practice using LFHC mathematical tasks. Data analysis revealed that, as compared to students exposed to algorithmic problems, students exposed to LFHC activities were much more mathematically proficient in Geometry, at the very least …

Circling The Square: Computing Radical Two, Isaiah Mellace, Joshua Kroeker

NEXUS: The Liberty Journal of Interdisciplinary Studies

Discoveries of equations for irrational numbers are not new. From Newton’s Method to Taylor Series,there are many ways to calculate the square root of two to arbitrary precision. The following method is similar in this way, but it is also a fascinating derivation from geometry that has applications to other irrationals. Additionally, the equation derived has some properties that may lead to fast computation. The first part of this paper is dedicated to deriving the equation, and the second is focused on computer science implementations and optimizations.

Are All Weakly Convex And Decomposable Polyhedral Surfaces Infinitesimally Rigid?, Jilly Kevo

Rose-Hulman Undergraduate Mathematics Journal

It is conjectured that all decomposable (that is, interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under an additional assumption of codecomposability (that is, the interior of the difference between the convex hull and the polyhedron itself can be triangulated without adding new vertices). One major set of tools for studying infinitesimal rigidity happens to be the (negative) Hessian M_T of the discrete Hilbert-Einstein functional. Besides its theoretical importance, it provides the necessary machinery to tackle the problem …

Khovanov Homology And Legendrian Simple Knots, Ryan J. Maguire

Dartmouth College Ph.D Dissertations

The Jones polynomial and Khovanov homology are powerful invariants in knot theory. Their computations are known to be NP-Hard and it can be quite a challenge to directly compute either of them for a general knot. We develop explicit algorithms for the Jones polynomial and discuss the implementation of an algorithm for Khovanov homology. Using this we tabulate the invariants for millions of knots, generate statistics on them, and formulate conjectures for Legendrian and transversely simple knots.

Bi-Harmonic Maps' Existence And P-Balanced Energy And Connections To Harmonic Maps, Lina Wu

Publications and Research

We discover a bi-harmonic map’ existence, its energy growth, and its connections to a harmonic map. First, we prove the existence of a nontrivial bi-harmonic map in a unit sphere. Second, we investigate the p-balanced energy growth of biharmonic maps. As the most important energy technical breakthroughs, we propose an innovative energy algorithm called p-balanced energy technique to break the constraints of the existing L^q-energy technique in detecting L^q-energy growth towards boundlessness. The disadvantage of the finite L^q -energy technique in the L ^q spaces is not effective in dealing with infinite L^q- energy in Non-L^q spaces. Third, we study …

Higher Diffeology Theory, Emilio Minichiello

Dissertations, Theses, and Capstone Projects

Finite dimensional smooth manifolds have been studied for hundreds of years, and a massive theory has been built around them. However, modern mathematicians and physicists are commonly dealing with objects outside the purview of classical differential geometry, such as orbifolds and loop spaces. Diffeology is a new framework for dealing with such generalized smooth spaces. This theory (whose development started in earnest in the 1980s) has started to catch on amongst the wider mathematical community, thanks to its simplicity and power, but it is not the only approach to dealing with generalized smooth spaces. Higher topos theory is another such …

Hyperbolic Groups And The Word Problem, David Wu

Master's Theses

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.

Parabolic And Non-Parabolic Surfaces With Small Or Large End Spaces Via Fenchel-Nielsen Parameters, Michael Antony Pandazis

Dissertations, Theses, and Capstone Projects

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that determine whether or not a surface X is parabolic. Fix a geodesic pants decomposition of a surface and call the boundary geodesics in the decomposition cuffs. For a zero or half-twist flute surface, we prove that parabolicity is equivalent to the surface having a covering group of the first kind. Using that result, we give necessary and sufficient conditions on the Fenchel-Nielsen parameters of a half-twist flute surface X with increasing cuff lengths such that X is parabolic. As an application, we determine whether or not each …

Dehn's Problems And Geometric Group Theory, Noelle Labrie

Master's Theses

In 1911, mathematician Max Dehn posed three decision problems for finitely

presented groups that have remained central to the study of combinatorial

group theory. His work provided the foundation for geometric group theory,

which aims to analyze groups using the topological and geometric properties

of the spaces they act on. In this thesis, we study group actions on Cayley

graphs and the Farey tree. We prove that a group has a solvable word problem

if and only if its associated Cayley graph is constructible. Moreover, we prove

that a group is finitely generated if and only if it acts geometrically …

A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The Artist, Jaimie An

Masters Theses

This thesis is an exercise in, perhaps a futile, attempt to trace just some of the ideas, stories, and musings I might meander through in my process. It’s not quite a map, nor is it a neat catalogue; it is a haphazard collection of tickets and receipts from a travel abroad, carelessly tossed in a carry-on, only to be stashed upon returning home. These ideas are derived from much greater thinkers and authors than myself; I am a mere collector or a translator, if that, and not a very good one, for much is lost. I do not claim comprehensive …

Nidus Idearum. Scilogs, Xiii: Structure / Neutrostructure / Antistructure, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this thirteenth book of scilogs – one may find topics on Neutrosophy, Plithogeny, Physics, Mathematics, Philosophy – email messages to research colleagues, or replies, notes, comments, remarks about authors, articles, or books, spontaneous ideas, and so on. It presents new types of soft sets and new types of topologies.

Exchanging ideas with Mohammad Abobala, Ishfaq Ahmad, Ibrahim M. Almanjahie, Fatimah Alshahrani, Nizar Altounji, Muhammad Aslam, Said Broumi, Victor Christianto, R. Diksh, Feng Liu, Frank Julian Gelli, Erick Gonzalez Caballero, Riad Hamido, Yaser Al-Hasan, Ahmed Hatip, Yasin Karmouta, Nivetha Martin, Preda Mihăilescu, V. Lakshmana Gomathi Nayagam, Ze Carlos Tiago de …

Canonical Extensions Of Quantale Enriched Categories, Alexander Kurz

MPP Research Seminar

No abstract provided.

On Distortion Of Surface Groups In Right-Angled Artin Groups, Lucas Bridges

Mathematical Sciences Undergraduate Honors Theses

Surfaces have long been a topic of interest for scholars inside and outside of mathe- matics. In a topological sense, surfaces are spaces which appear flat on a local scale. Surfaces in this sense have a restricted set of properties, including the behavior of loops around a surface, codified in the fundamental group.

All but 3 surface groups have been shown to embed into a class of groups called right-angled Artin groups. The method through which these embeddings are created places large restrictions on all homomorphisms from surface groups to right-angled Artin groups.

One such restriction on these homomorphisms is …

Geometric Principles In Architecture Aesthetics, Vincent Gemmiti

Architecture Undergraduate Honors Theses

This study focuses on geometric formalism in three major monuments across different architectural eras in time. The use of three distinct geometric principles to outline the use of pure or manufactured shapes and figures helps to discover and isolate an aesthetic component of architecture within which it is contained. A collection of studies are implemented and are focused on orthographic drawings from each monument in horizontality and verticality, consisting of a dual set of overlay and interpretative drawings for each type of orthographic representation for each monument. A discussion follows and highlights the changes or similarities over time the role …

Plumbing The Depths Of The Shallow End: Exploring Persistent Homology Using Small Data, R. Anne Flynn

All NMU Master's Theses

Persistent homology is a prominent tool in topological data analysis. This thesis is designed to be an introduction and guide to a beginner in persistent homology. This comprehensive overview discusses the math used behind it, the code needed to apply it, and its current place in the field. We explain and demonstrate the algebraic topology which fuels persistent homology. Homotopies inspire homology groups, which are able to determine how many holes a shape has. By visualizing data as a shape, persistent homology determines what type of holes are present.

We demonstrate this by using the package TDA in the manipulation …

On Cheeger Constants Of Knots, Robert Lattimer

Electronic Theses, Projects, and Dissertations

In this thesis, we will look at finding bounds for the Cheeger constant of links. We will do this by analyzing an infinite family of links call two-bridge fully augmented links. In order to find a bound on the Cheeger constant, we will look for the Cheeger constant of the link’s crushtacean. We will use that Cheeger constant to give us insight on a good cut for the link itself, and use that cut to obtain a bound. This method gives us a constructive way to find an upper bound on the Cheeger constant of a two-bridge fully augmented link. …