Geometry and Topology Commons^™

Universal Constraints Of Kleinian Groups And Hyperbolic Geometry, Hala Alaqad

Dissertations

Recent advances in geometry have shown the wide application of hyperbolic geometry not only in Mathematics but also in real-world applications. As in two dimensions, it is now clear that most three-dimensional objects (configuration spaces and manifolds) are modelled on hyperbolic geometry. This point of view explains a great many things from large-scale cosmological phenomena, such as the shape of the universe, right down to the symmetries of groups and geometric objects, and various physical theories. Kleinian groups are basically discrete groups of isometries associated with tessellations of hyperbolic space. They form the fundamental groups of hyperbolic manifolds. Over the …

Acoustic Versus Electromagnetic Field Theory: Scalar, Vector, Spinor Representations And The Emergence Of Acoustic Spin, Lucas Burns, Konstantin Y. Bliokh, Franco Nori, Justin Dressel

Mathematics, Physics, and Computer Science Faculty Articles and Research

We construct a novel Lagrangian representation of acoustic field theory that describes the local vector properties of longitudinal (curl-free) acoustic fields. In particular, this approach accounts for the recently-discovered nonzero spin angular momentum density in inhomogeneous sound fields in fluids or gases. The traditional acoustic Lagrangian representation with a scalar potential is unable to describe such vector properties of acoustic fields adequately, which are however observable via local radiation forces and torques on small probe particles. By introducing a displacement vector potential analogous to the electromagnetic vector potential, we derive the appropriate canonical momentum and spin densities as conserved Noether …

An Analysis And Comparison Of Knot Polynomials, Hannah Steinhauer

Senior Honors Projects, 2020-current

Knot polynomials are polynomial equations that are assigned to knot projections based on the mathematical properties of the knots. They are also invariants, or properties of knots that do not change under ambient isotopy. In other words, given an invariant α for a knot K, α is the same for any projection of K. We will define these knot polynomials and explain the processes by which one finds them for a given knot projection. We will also compare the relative usefulness of these polynomials.

Winding Numbers And Full Extendability In Holomorphic Motions, Yunping Jiang

Publications and Research

We construct an example of a holomorphic motion of a five-point subset of the Riemann sphere over an annulus such that it satisfies the zero winding number condition but is not fully extendable.

A Structure Theorem For Bad 3-Orbifolds, Rachel Julie Lehman

Graduate Theses and Dissertations

We explicitly construct 10 families of bad 3-orbifolds, X , having the following property: given any bad 3-orbifold, O, it admits an embedded suborbifold X ∈ X such that after removing this member from O, and capping the resulting boundary, and then iterating this process finitely many times, you obtain a good 3-orbifold. Reversing this process gives us a procedure to obtain any possible bad 3-orbifold starting with a good 3-orbifold. Each member of X has 1 or 2 spherical boundary components and has underlying topological space S2 × I or (S2 × S1)\B3.

A Novel Three-Level Isolated Ac-Dc Pfc Power Converter Topology With Reduced Number Of Switches, Obaid Aldosari

Graduate Theses and Dissertations

The three-level isolated AC-DC power factor corrected (PFC) converter provides safe and more efficient power conversion. In comparison with two-level, three-level PFC converter has the advantages of low total harmonic distortion, low device voltage rating, low di/dt, better output performance, high power factor, and low switching losses at higher switching frequencies. The high frequency transformer (HFT) grants galvanic isolation, steps up or down secondary voltage, and limits damage in case of a fault current.

The existing three-level converter based on solid-state transformer (SST) topologies convert ac power from the electrical grid to a dc load while maintaining at least the …

Geometric Realizations Of Abstract Regular Polyhedra With Automorphism Group H3, Mark L. Loyola, Jonn Angel L. Aranas

Mathematics Faculty Publications

A geometric realization of an abstract polyhedron P is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular polyhedron from its automorphism group Gamma. The method entails finding a real orthogonal representation of Gamma of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space. To demonstrate the use of the method, it is applied to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group H3.

𝑘-Plane Constant Curvature Conditions, Maxine E. Calle

Rose-Hulman Undergraduate Mathematics Journal

This research generalizes the two invariants known as constant sectional curvature (csc) and constant vector curvature (cvc). We use k-plane scalar curvature to investigate the higher-dimensional analogues of these curvature conditions in Riemannian spaces of arbitrary finite dimension. Many of our results coincide with the known features of the classical k=2 case. We show that a space with constant k-plane scalar curvature has a uniquely determined tensor and that a tensor can be recovered from its k-plane scalar curvature measurements. Through two example spaces with canonical tensors, we demonstrate a method for determining constant k-plane …

Isoperimetric Problems On The Line With Density |𝑥|ᵖ, Juiyu Huang, Xinkai Qian, Yiheng Pan, Mulei Xu, Lu Yang, Junfei Zhou

Rose-Hulman Undergraduate Mathematics Journal

On the line with density |x|^p, we prove that the best single bubble is an interval with endpoint at the origin and that the best double bubble is two adjacent intervals that meet at the origin.

The Isoperimetric Inequality: Proofs By Convex And Differential Geometry, Penelope Gehring

Rose-Hulman Undergraduate Mathematics Journal

The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of elementary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn--Minkowski-Inequality. Using this inequality, the Isoperimetric Inquality in n dimensions will be shown.

A Discrete Analogue For The Poincaré-Hopf Theorem, Savana Ammons

HMC Senior Theses

In this thesis, we develop a discrete analogue to the Poincaré–Hopf Theorem. We define the notion of a vector field on a graph, and establish an index theory for such a field. Specifically, we create well-defined indices for the nodes and “cells" formed by a planar graph. Then, we show that the sum of these indices remains constant for certain types of planar graphs, regardless of the discrete vector fields they have.

Geometry Across The Curriculum, Corey Dunn

Q2S Enhancing Pedagogy

This project is designed for a multicalculus class already familiar with computing the arc length of a parameterized curve in space. The activity asks the student to first recall basic facts about arc length, and then introduces the notion of measuring lengths of vectors differently, depending on where their initial point is. This is a foundational concept in metric differential geometry, and, this activity attempts to motivate this generalization of computing lengths of vectors through this arc length activity. The activity concludes with a short discussion of basic concepts of Lorentzian geometry, including the idea that lightlike vectors have length …

Geogebra Activities: Tracing Points, Jeremy Aikin, Corey Dunn, Jeffrey Meyer, Rolland Trapp

Q2S Enhancing Pedagogy

In this activity, we will learn how to use GeoGebra (www.geogebra.org) to trace the movement of points, which depend on the movement of other objects. The paths of these points determine curves and we will provide algebraic descriptions of these curves.

Where The Wild Knots Are, Forest Kobayashi

HMC Senior Theses

The new work in this document can be broken down into two main parts. In the first, we introduce a formalism for viewing the signed Gauss code for virtual knots in terms of an action of the symmetric group on a countable set. This is achieved by creating a "standard unknot" whose diagram contains countably-many crossings, and then representing tame knots in terms of the action of permutations with finite support. We present some preliminary computational results regarding the group operation given by this encoding, but do not explore it in detail. To make the encoding above formal, we require …

Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly

Electronic Theses and Dissertations

Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based …

Pascal's Mystic Hexagon In Tropical Geometry, Hanna Hoffman

HMC Senior Theses

Pascal's mystic hexagon is a theorem from projective geometry. Given six points in the projective plane, we can construct three points by extending opposite sides of the hexagon. These three points are collinear if and only if the six original points lie on a nondegenerate conic. We attempt to prove this theorem in the tropical plane.

Complete Bipartite Graph Embeddings On Orientable Surfaces Using Cayley Maps, Madeline Spies

Honors Program Theses

We explored how effective Cayley Maps are at embedding complete bipartite graphs onto orientable surfaces, such as spheres and tori. We embedded the graphs onto surfaces using Cayley Maps with the intent of finding rotations that result in the graphs’ optimal genera. Because there are only three groups that can be used to describe Kp,p, where p is prime, we chose to focus on this specific graph type. This paper explains how we determined the genera when using a Cayley Map, provides general theorems for surface face sizes and the Dihedral Group, and discusses our results for Kp,p up to …

Explicit Pseudo-Kähler Metrics On Flag Manifolds, Thomas A. Mason Iii

College of Graduate Studies: Theses & Dissertations

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) Kähler structure, famously used to realize the group's irreducible representations in holomorphic sections of certain line bundles (Borel-Weil theorem). Less well-known are the (indefinite) invariant pseudo-Kähler structures they also admit, which can be used to realize the same representations in higher cohomology of the sections (Bott), and whose analogues in a non-compact setting lead to new representations (Kostant-Langlands). The purpose of this thesis is to give an explicit description of these metrics in the case of the unitary group G=U_n.

Discrete Morse Theory By Vector Fields: A Survey And New Directions, Matthew Nemitz

All Graduate Theses, Dissertations, and Other Capstone Projects

We synthesize some of the main tools in discrete Morse theory from various sources. We do this in regards to abstract simplicial complexes with an emphasis on vector fields and use this as a building block to achieve our main result which is to investigate the relationship between simplicial maps and homotopy. We use the discrete vector field as a catalyst to build a chain homotopy between chain maps induced by simplicial maps.

Periodic Points On Tori: Vanishing And Realizability, Shane Clark

Theses and Dissertations--Mathematics

Let $X$ be a finite simplicial complex and $f\colon X \to X$ be a continuous map. A point $x\in X$ is a fixed point if $f(x)=x$. Classically fixed point theory develops invariants and obstructions to the removal of fixed points through continuous deformation. The Lefschetz Fixed number is an algebraic invariant that obstructs the removal of fixed points through continuous deformation. \[L(f)=\sum_{i=0}^\infty (-1)^i \tr\left(f_i:H_i(X;\bQ)\to H_i(X;\bQ)\right). \] The Lefschetz Fixed Point theorem states if $L(f)\neq 0$, then $f$ (and therefore all $g\simeq f$) has a fixed point. In general, the converse is not true. However, Lefschetz Number is a complete invariant …

Graph-Theoretic Simplicial Complexes, Hajos-Type Constructions, And K-Matchings, Julianne Vega

Theses and Dissertations--Mathematics

A graph property is monotone if it is closed under the removal of edges and vertices. Given a graph G and a monotone graph property P, one can associate to the pair (G,P) a simplicial complex, which serves as a way to encode graph properties within faces of a topological space. We study these graph-theoretic simplicial complexes using combinatorial and topological approaches as a way to inform our understanding of the graphs and their properties.

In this dissertation, we study two families of simplicial complexes: (1) neighborhood complexes and (2) k-matching complexes. A neighborhood complex is a simplicial …

"Lagrangian Disks" In M-Theory, Sebastián Franco, Sergei Gukov, Sangmin Lee, Rak-Kyeong Seong, James Sparks

Publications and Research

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G₂ holonomy spaces and to Spin(7) metrics on 8-manifolds with T² fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T[M₄] on …

The Geometry Of The Orthological Triangles, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

Plants and trees grow perpendicular to the plane tangent to the soil surface, at the point of penetration into the soil; in vacuum, the bodies fall perpendicular to the surface of the Earth - in both cases, if the surface is horizontal. Starting from the property of two triangles to be orthological, the two authors have designed this work that seeks to provide an integrative image of elementary geometry by the prism of this "filter". Basically, the property of orthology is the skeleton of the present work, which establishes many connections of some theorems and geometric properties with it. The …

Neutrosophic Local Function And Generated Neutrosophic Topology, A. A. Salama, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we introduce the notion of ideals on neutrosophic set which is considered as a generalization of fuzzy and fuzzy intuitionistic ideals studies. The concept of neutrosophic local function is also introduced for a neutrosophic topological space. These concepts are discussed with a view to find new neutrosophic topology. The basic structure, especially a basis for such generated neutrosophic topologies and several relations between different topological neutrosophic ideals and neutrosophic topologies are also studied here. Possible application to GIS topology rules are touched upon.

What's Your Sphericity Index? Rationalizing Surface Area And Volume, John A. Adam

Mathematics & Statistics Faculty Publications

Virginia Standards of Learning include mathematical content related to the surface area and the volume of various geometric objects. In the seventh grade, “Students... solve problems involving volume and surface area” In the eighth grade, “Proportional reasoning is expounded upon as students solve a variety of problems. Students find the volume and surface area of more complex three dimensional figures”. In high school geometry, “The student... use[s] surface area and volume of three-dimensional objects to solve practical problems” (Virginia Department of Education, 2016). The challenge is to find scenarios that are engaging to students and keep them interested in the …

Computing The Newton Potential In The Boundary Integral Equation For The Dirichlet Problem Of The Poisson Equation, Wenchao Guan, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier-Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has mth-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order 𝒪(n^-m log³ n), with requiring 𝒪(n log² n) number of arithmetics for …

Theory And Techniques Of Convergence Of Topological Transformation Group Actions, Murtadha Jaber Sarray

Graduate Theses, Dissertations, and Problem Reports (ETD)

n this dissertation, we present new set functions called strongly limit and strongly prolongation limit sets. We show the new sets, especially strongly prolongation limit sets, characterize proper action under an arbitrary setting. That is, we characterize proper action for wider class of proper ܩ-spaces. Also, we show the new version of the sets could be derived from strongly exceptional sets which have been used as a good technique for the characterization of a proper maps. Moreover, we review properties of well-known limit sets and prolongations and properties for the new version of limit sets under an arbitrary setting on …

Heat Kernel Voting With Geometric Invariants, Alexander Harr

All Graduate Theses, Dissertations, and Other Capstone Projects

Here we provide a method for comparing geometric objects. Two objects of interest are embedded into an infinite dimensional Hilbert space using their Laplacian eigenvalues and eigenfunctions, truncated to a finite dimensional Euclidean space, where correspondences between the objects are searched for and voted on. To simplify correspondence finding, we propose using several geometric invariants to reduce the necessary computations. This method improves on voting methods by identifying isometric regions including shapes of genus greater than 0 and dimension greater than 3, as well as almost retaining isometry.

Effects Of Aperiodicity And Frustration On The Magnetic Properties Of Artificial Quasicrystals, Barry Farmer

Theses and Dissertations--Physics and Astronomy

Quasicrystals have been shown to exhibit physical properties that are dramatically different from their periodic counterparts. A limited number of magnetic quasicrystals have been fabricated and measured, and they do not exhibit long-range magnetic order, which is in direct conflict with simulations that indicate such a state should be accessible. This dissertation adopts a metamaterials approach in which artificial quasicrystals are fabricated and studied with the specific goal of identifying how aperiodicity affects magnetic long-range order. Electron beam lithography techniques were used to pattern magnetic thin films into two types of aperiodic tilings, the Penrose P2, and Ammann-Beenker tilings. SQUID …

Invariance And Invertibility In Deep Neural Networks, Han Zhang

Theses and Dissertations

Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input …