Geometry and Topology Commons^™

Hereditary Properties Of Co-Kähler Manifolds, Giovanni Bazzoni, Gregory Lupton, John F. Oprea

Mathematics and Statistics Faculty Publications

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: dim(H^∗(M; Q)) ≥2^r, for any r-torus T^r which acts almost freely on M.

Discovering Liouville-Type Problems For P-Energy Minimizing Maps In Closed Half-Ellipsoids By Calculus Variation Method, Lina Wu

Publications and Research

The goal of this project is to investigate constant properties (called the Liouville-type Problem) for a p-stable map as a local or global minimum of a p-energy functional where the domain is a Euclidean space and the target space is a closed half-ellipsoid. The First and Second Variation Formulas for a p-energy functional has been applied in the Calculus Variation Method as computation techniques. Stokes’ Theorem, Cauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and the Bochner Formula as estimation techniques have been used to estimate the lower bound and the upper bound of the derived p-Harmonic Stability Inequality. One challenging point in …

On The Combinatorics Of Demoulin Transforms And (Discrete) Projective Minimal Surfaces, A Mccarthy, W Schief

Sciences Papers and Journal Articles

The classical Demoulin transformation is examined in the context of discrete differential geometry. We show that iterative application of the Demoulin transformation to a seed projective minimal surface generates a Z ² lattice of projective minimal surfaces. Known and novel geometric properties of these Demoulin lattices are discussed and used to motivate the notion of lattice Lie quadrics and associated discrete envelopes and the definition of the class of discrete projective minimal and Q-surfaces (PMQ-surfaces). We demonstrate that the even and odd Demoulin sublattices encode a two-parameter family of pairs of discrete PMQ-surfaces with the property that one discrete PMQ-surface …

Alice In Wonderland For G4g13, Jeremiah Farrell, Emmanuelle Malte Salvatore, Todd Wilk Estroff

Scholarship and Professional Work - LAS

Each of the ten different letters in the title is used exactly three times to form the words in the circles. Martin Gardner's famous work The Annotated Alice was first published in 1960 and we honor him in this essay.

Perihelion Precession In The General Theory Of Relativity, Charles G. Torre

Tutorials on... in 1 hour or less

This is a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. Any decent textbook on the general theory of relativity will derive this result. My analysis aligns with that found in the good old text "Introduction to General Relativity", by Adler, Bazin and Schiffer.

Sudoku Variants On The Torus, Kira A. Wyld

HMC Senior Theses

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Pattern Recognition In Stock Data, Kathryn Dover

HMC Senior Theses

Finding patterns in high dimensional data can be difficult because it cannot be easily visualized. There are many different machine learning methods to fit data in order to predict and classify future data but there is typically a large expense on having the machine learn the fit for a certain part of a dataset. We propose a geometric way of defining different patterns in data that is invariant under size and rotation. Using a Gaussian Process, we find that pattern within stock datasets and make predictions from it.

Introduction To Axiomatic Geometry, Mark Barsamian

OHIO Open Faculty Textbooks

This book presents Euclidean Geometry and was designed for a one-semester course preparing junior and senior level college students to teach high school Geometry. The book could also serve as a text for a junior level Introduction to Proofs course.

Martin Gardner Puzzle-Games, Stephen Bloom, Lacey Echols, Jeremiah Farrell, Shannon Lieb

Scholarship and Professional Work - LAS

No abstract provided.

Euler Entertainments, Jeremiah Farrell, Karen Farrell

Scholarship and Professional Work - LAS

No abstract provided.

A Gathering For Gardner Puzzle-Game, Jeremiah Farrell, Chris Morgan

Scholarship and Professional Work - LAS

Each different letter of "GATHERING FOR GARDNER" is used exactly three times in the following words: DIE, FAD, FIT, FOG, GIN, HAG, HER, HOD, NOR, RAT, TEN.

The Tea Party, Stephen Bloom, Jeremiah Farrell

Scholarship and Professional Work - LAS

No abstract provided.

Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap

Faculty Publications

Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds N_n and N^µ_n whose volume is approximately n and whose length spectra agree up to length n.

Both N_n and N^µ_n are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that …

Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap

Faculty Publications

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.

The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least …

On Independence, Matching, And Homomorphism Complexes, Wesley K. Hough

Theses and Dissertations--Mathematics

First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory.

In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the …

Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews

Honors Theses

This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.

Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar

Honors Theses

Chirality (or handedness) is the property that a structure is “different” from its mirror image. Topology can be used to provide a rigorous framework for the notion of chirality. This project examines various types of chirality and discusses tools to detect chirality in graphs and knots. Notable theorems that are discussed in this work include ones that identify chirality using properties of link polynomials (HOMFLY polynomials), rigid vertex graphs, and knot linking numbers. Various other issues of chirality are explored, and some specially unique structures are discussed. This paper is borne out of reading Dr. Erica Flapan’s book, When Topology …

Some Examples Of The Interplay Between Algebra And Topology, Joseph D. Malionek

Honors Theses

This thesis presents several undergraduate and graduate level concepts in the fields of algebraic topology and topological group theory in a manner which requires very little mathematical background of the reader. It uses non-rigorous interpretations of concepts while introducing the reader to the rigorous ideas with which they are associated. In order to give the reader an idea of how the fields of algebra and topology are closely affiliated, the paper goes over five main concepts, the fundamental group, homology, cohomology, Eilenberg-Maclane spaces, and group dimension.

Properties Of Left-Separated Spaces And Their Unions, Eric Scheidecker

Dissertations and Theses @ UNI

Left-separated spaces are topological spaces which can be well ordered such that every initial segment is closed. In this paper, we examine what topological properties imply left-separation, and under what circumstances left-separation is preserved by unions. We also introduce several known theorems regarding elementary submodels as they are one of the primary tools that we use. We prove that for a topological space X;

1. If X has a point-countable base, then X is left-separated if and only if X has closed intersection with any elementary submodel M such that X ∈ M.

2. If every elementary submodel …

The Programmatic Manipulation Of Planar Diagram Codes To Find An Upper Bound On The Bridge Index Of Prime Knots, Genevieve R. Johnson

Dissertations and Theses @ UNI

The “bridge index” of a knot is the least number of maximal overpasses taken over all diagrams of the knot. A naïve method to determine the bridge index of a knot is to perform Reidemeister moves on diagrams of the knot, and this method quickly becomes tedious to implement by hand. In this paper, we introduce a sequence of Reidemeister moves which we call a “drag the underpass” move and prove how planar diagram codes change as Reidemeister moves are performed. We then use these results to programatically perform Reidemeister moves using Python 2.7 to calculate an upper bound on …