Geometry and Topology Commons^™

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.

(A,B,C) Tilings With Prescribed Symmetry Groups From Regular Triangle Or Hexagon Tiling, Mark D. Tomenes, Ma. Louise Antonette N. De Las Penas

Mathematics Faculty Publications

A tiling T of the Euclidean plane (E2) is a countable collection of closed topological disks called tiles T = {Ti : i ∈ N} that is a covering (Ui Ti = E2) as well as a packing (Int(Ti) ∩ Int(Tj) = ∅ if i ̸= j, Int(T) denotes the interior of tile T). One of the problems of interest in discrete geometry is the classification of tilings based on transitivity properties of their vertices, edges and tiles. This talk presents a family of tilings whose vertices, edges and tiles have exactly a, b and c orbits, respectively, under the …

Modern Procedural Terrain Generation Techniques And Their Background, Hunter A. Barton

2025 Symposium

Procedural terrain generation has become a staple in many digital environments, enabling the automated creation of large-scale and realistic landscapes for applications such as video games and movies. This paper provides an in-depth look at smooth noise functions and their use for terrain generation, as well as an overview of some more modern methods of generation. A method utilizing machine learning stlye transfer was reproduced for this paper with some alterations to improve visualization and realism.

Simplicial Decomposition And Realization, Matthew Ellison

Dartmouth College Ph.D Dissertations

In simplicial decomposition, we define two invariants --- V_Z and V_Q --- which represent notions of integral and rational volume of a certain class of simplicial complexes. We prove V_Z and V_Q are additive under disjoint union and connected sum, and investigate `integrality gaps' between the two quantities. We apply the theory to establish a conjecture of Sleator, Thurston, and Tarjan on tetrahedral fillings, and, as a corollary, obtain a new proof of Pournin's 2012 result on the diameter of the associahedron. In simplicial realization, we provide practical sufficient conditions and computer code to prove the existence of Euclidean embeddings …

On Diffeomorphism Groups Of Surfaces, Madeleine Goertz

Master's Theses

Let $M$ be a closed, connected, smooth manifold. What are the symmetries of $M$? From a geometric viewpoint, the symmetries of $M$ are precisely its isometries, the self maps which preserve lengths and angles. In the smooth category, the symmetries of $M$ are its diffeomorphisms, the self maps which are smooth and have a smooth inverse. Does expanding our notion of symmetries to include diffeomorphisms result in ``more'' symmetries in a meaningful sense? As a formal conjecture, the claim is that the isometry group of $M$ is a deformation retract of the diffeomorphism group of $M$. If $M$ is the …

Algorithms To Estimate Contours: Two Applications Of Analytical Tools In Differential Geometry And Topology, Mohammad Abirul Islam

Computer Science ETDs

We develop distributed robotics algorithms with analytical tools needed to define and analyze angle turned and distance traversed by robots executing geometric algorithms. We then use these analytical tools to obtain information, via sensor measurements, about an a priori unknown surface. Our contributions are threefold. First, we develop the Sketch Algorithm, which estimates the boundary of any unknown contour and is asymptotically optimal in terms of distance traversed and angle turned. Second, we present experimental field work that validates the Sketch Algorithm. Finally, we propose an approach to find multiple sources of a surface with potential applications to approximate that …

Exploration Into Homotopy Equivalence And Vietoris-Rips Complexes, Alexander J. Mobley

Honors Theses

An important question in discrete topology and geometry is how to recover the structure and characteristics of a manifold when only given a finite set of points sampled from that manifold. Thus, if mathematicians have a point cloud of data which induces a discrete metric space, they look for some structure to these points. Understanding the structure of these points often gives needed insight to solve this problem. One such way to determine structure to these points is to use these points to construct a Vietoris-Rips complex. In abstract, Latschev shows that this allows us to recover the homotopy type …

A Preliminary Study Of Hilbert–Kunz Functions: Coefficient Behavior In A Normal Affine Semigroup Ring, Jesus A. Mendiola Herrera

Theses and Dissertations

In 1890, David Hilbert published a set of notes on what now constitutes one of the bases of Commutative Algebra; his work would eventually influence the efforts of mathematicians like Ernst Kunz. In 1969, Ernst Kunz introduced a particular mapping regarding modules of regular local rings. His goal was to characterize Noetherian local rings of prime characteristic by computing the length of the composition series under Frobenius power transformations. In this thesis, the focus will be on stating the initial steps on finding the coefficients of the Hilbert-Kunz function of the normal affine semigroup ring of the form R = …

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Constructing Binary Encoding Matrices From Joined Graphs, Joshua Avalos

Electronic Theses, Projects, and Dissertations

Codes and technology are part of our daily lives and allow the modern world to function, and for us to have conveniences in our lives such as smartphones that can be used to privately call people on the other side of the planet, and for secure access to the internet. In this thesis we will explore the construction of binary codes created by vertex-edge incidence matrices of planar graphs. The Hamming (7,4) code was an incredible code that allowed the detection and correction of errors after receiving them through a transmission. We will explore the possibility of the creation of …

Dedekind-Macneille And Related Completions: Subfitness, Regularity, And Booleanness, G. Bezhanishvili, F. Dashiell Jr., M. A. Moshier, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

Completions play an important rôle for studying structure by supplying elements that in some sense “ought to be.” Among these, the Dedekind-MacNeille completion is of particular importance. In 1968 Janowitz provided necessary and sufficient conditions for it to be subfit or Boolean. Another natural separation axiom connected to these is regularity. We explore similar characterizations of when closely related completions are subfit, regular, or Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical completions, which (unlike the Dedekind-MacNeille completion) satisfy stronger forms of distributivity. The first two are widely used in pointfree topology, while the latter is of …

Circles Are So Euclidean: Exploring Convex Bodies, Norms, Symmetries, And Isometries Via The Minkowski Functional, Fanni Kertesz

Undergraduate Theses

In a real vector space, a symmetric convex body containing the origin and compact under the Euclidean norm uniquely defines a norm via the Minkowski functional, where the closed unit ball of this norm is the convex body. This establishes a one-to-one correspondence between convex bodies and norms. The norm derived from the convex body allows for the definition of a metric, enabling the study of isometries within the space. However, the symmetries of the convex body itself do not completely determine the isometries of the associated Minkowski space. Instead, the symmetries of the most symmetric shape a convex body …

New View Of Some Propositions On Hesitant Fuzzy Set And Its Application In Selecting The Best Person In Any Job, Manar Mohamed Omran Dr., Arafa A. Nasef A.Dr, Reham Abd El-Aziz Abo Khadra Dr., Mahmoud Arafa Nasef Dr.

Journal of Engineering Research

An essential part of uncertainty is played by the hesitant fuzzy set (HFS). So, it can be utilized when making decisions. The suggested use of HFS to choose the best candidate for any post is thoroughly discussed and introduces new HFS concepts

Book Vi Of The Mathematical Collection Of Pappus Of Alexandria, Translated By John B. Little, John B. Little, Pappus Of Alexandria

Holy Cross Bookshelf

Book VI of the Mathematical Collection is a collection of comments or notes about various points treated in parts of a collection of other texts sometimes known as the “Little Astronomy.” Pappus uses these works as sources and frequently quotes from them in this book. In the teaching of mathematical astronomy in late antiquity, and this would include the time of Pappus, the “Little Astronomy” is often understood to have been a follow-up to Euclid’s Elements and a preliminary to the study of the Almagest of Claudius Ptolemy (ca. 100–165 CE). The “Little Astronomy” included works by a group of …

Sobolev Regularity For The Bergman Projection On Relatively Compact Domains In Hermitian Manifolds, Philip Harrington

Mathematical Sciences Faculty Publications and Presentations

Generalizing a result of Berndtsson and Charpentier, we provide sufficient conditions for L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L< ^>2$$\end{document} Sobolev regularity of the Bergman projection acting on L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L< ^>2$$\end{document} sections of a holomorphic line bundle restricted to a relatively compact domain with Lipschitz boundary in a Hermitian manifold. We provide examples to show that our methods work for domains in Hopf manifolds endowed with a suitable Hermitian metric.

Non-Orientable Surfaces Bounded By Knots, Megan K. Fairchild

LSU Doctoral Dissertations

The non-orientable 4-genus of a knot $K$ in $S^{3}$ is defined to be the minimum first Betti number of a non-orientable surface $F$ in $B^{4}$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots, torus knots, and Whitehead doubles. We also will view obstructions to a knot bounding a M\"{o}bius band given by the double branched cover of $S^{3}$ branched over $K$. Additionally, we discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used …

White Horses In The Sky: Or The Magical Place Where Topology Meets Poetry, Sanziana Caraman, Lorelei Caraman

Journal of Humanistic Mathematics

What do Shakespearean words, doughnuts, cups of tea, and horses in the sky all have in common with a famous theorem? To discover the answer, we take you on a journey in a strange, enchanted land: the land where mathematics meets poetry. Here we explore how a fundamental literary trope, metaphor, and a fundamental result in topology, Brouver's Fixed Point Theorem, touch and illuminate one another. Delving into the symmetry of this transdisciplinary embrace, we find that it reveals, not only the beauty of poetry and mathematics, but that of life itself: the beauty hidden in ordinary things like a …

A Euclidean Proof For The Fourth Postulate, José Gil-Férez, Piotr Błaszczyk, M. Andrew Moshier, Alberto Naibo, Marco Panza, Jean-Michel Salanskis

MPP Published Research

We discuss some classical conundrums about Euclid's Fourth Postulate. Our inquire sheds lights on the role the postulate is playing within the deductive structure of Book I of the Elements and provides a proof of it fully admissible within Euclid's original setting.

Module 5 - Project, Karon Klipple, Cheryl Hedden

Module 5 Assessments

For this project, you will create a video in which you TEACH someone how to find the surface area and volume of a box.

Module 5 - Student Workbook, Karon Klipple, Cheryl Hedden

Module 5 Workbook

No abstract provided.