From Sets To Metric Spaces To Topological Spaces, 2018 Ursinus College

#### From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville

*Topology*

No abstract provided.

Nearness Without Distance, 2018 Ursinus College

Toroidal Embeddings And Desingularization, 2018 California State University, San Bernardino

#### Toroidal Embeddings And Desingularization, Leon Nguyen

*Electronic Theses, Projects, and Dissertations*

Algebraic geometry is the study of solutions in polynomial equations using objects and shapes. Differential geometry is based on surfaces, curves, and dimensions of shapes and applying calculus and algebra. Desingularizing the singularities of a variety plays an important role in research in algebraic and differential geometry. Toroidal Embedding is one of the tools used in desingularization. Therefore, Toroidal Embedding and desingularization will be the main focus of my project. In this paper, we first provide a brief introduction on Toroidal Embedding, then show an explicit construction on how to smooth a variety with singularity through Toroidal Embeddings.

Interdisciplinary Fun With Knapp Chairs, Mit's Erik And Martin Demaine, 2018 University of San Diego

#### Interdisciplinary Fun With Knapp Chairs, Mit's Erik And Martin Demaine, Ryan T. Blystone

*Research Week*

No abstract provided.

A Study Of Topological Invariants In The Braid Group B2, 2018 East Tennessee State University

#### A Study Of Topological Invariants In The Braid Group B2, Andrew Sweeney

*Electronic Theses and Dissertations*

The Jones polynomial is a special topological invariant in the field of Knot Theory. Created by Vaughn Jones, in the year 1984, it is used to study when links in space are topologically different and when they are topologically equivalent. This thesis discusses the Jones polynomial in depth as well as determines a general form for the closure of any braid in the braid group B2 where the closure is a knot. This derivation is facilitated by the help of the Temperley-Lieb algebra as well as with tools from the field of Abstract Algebra. In general, the Artin braid group ...

Divergence Of Cat(0) Cube Complexes And Coxeter Groups, 2018 The Graduate Center, City University of New York

#### Divergence Of Cat(0) Cube Complexes And Coxeter Groups, Ivan Levcovitz

*All Dissertations, Theses, and Capstone Projects*

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we characterize right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. This characterization also has a direct application to the theory of random right-angled Coxeter groups. As another application of the divergence bounds obtained for cube complexes, we provide an inductive graph theoretic criterion on a right-angled Coxeter ...

On Some Geometry Of Graphs, 2018 The Graduate Center, City University of New York

#### On Some Geometry Of Graphs, Zachary S. Mcguirk

*All Dissertations, Theses, and Capstone Projects*

In this thesis we study the intrinsic geometry of graphs via the constants that appear in discretized partial differential equations associated to those graphs. By studying the behavior of a discretized version of Bochner's inequality for smooth manifolds at the cone point for a cone over the set of vertices of a graph, a lower bound for the internal energy of the underlying graph is obtained. This gives a new lower bound for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the curvature constant that appears at the cone point and the size ...

Geometry And Analysis Of Some Euler-Arnold Equations, 2018 The Graduate Center, City University of New York

#### Geometry And Analysis Of Some Euler-Arnold Equations, Jae Min Lee

*All Dissertations, Theses, and Capstone Projects*

In 1966, Arnold showed that the Euler equation for an ideal fluid can arise as the geodesic flow on the group of volume preserving diffeomorphisms with respect to the right invariant kinetic energy metric. This geometric interpretation was rigorously established by Ebin and Marsden in 1970 using infinite dimensional Riemannian geometry and Sobolev space techniques. Many other nonlinear evolution PDEs in mathematical physics turned out to fit in this universal approach, and this opened a vast research on the geometry and analysis of the Euler-Arnold equations, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. In ...

Transforming Students Through Geometric Transformations, 2018 University of Wyoming

#### Transforming Students Through Geometric Transformations, Sierra Galicia

*Honors Theses AY 17/18*

It is a common saying among teachers that the hardest grades to teach are middle school. Beginning January 3rd, 2018, I was placed in an eighth-grade mathematics classroom for student teaching with the mindset that I was going to tackle this challenge head-on. I started the experience enthusiastic and ready to teach my students with exciting and engaging lessons. Unfortunately, I soon found that many students have low self-esteem, low self-confidence and they lack the necessary perseverance needed to complete challenging assignments and to be successful in math. It quickly became apparent why people say middle school is the hardest ...

Classifications Of Resistance Distances In Simple Graphs, 2018 Carroll College

#### Classifications Of Resistance Distances In Simple Graphs, Marcellus Randall

*Carroll College Student Undergraduate Research Festival*

Within graph theory, there are multiple distance metrics which can describe the concept of “distance” between nodes on a simple graph, which are of particular interest to researchers studying link prediction and network evolution. This talk will focus on the relationship between measures of distance in simple graphs and various features of these graphs. I will discuss classifying graphs in which any edge resistance is greater than any non-edge resistance, using Katz centrality scores and classical graph theoretical features.

Analytic Geometry And Calculus I, Ii, & Iii (Dalton), 2018 Dalton State College

#### Analytic Geometry And Calculus I, Ii, & Iii (Dalton), Thomas Gonzalez, Michael Hilgemann, Jason Schmurr

*Mathematics Grants Collections*

This Grants Collection for Analytic Geometry and Calculus I, II, & III was created under a Round Six ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

- Linked Syllabus
- Initial Proposal
- Final Report

Dalton State College Apex Calculus, 2018 Dalton State College

#### Dalton State College Apex Calculus, Thomas Gonzalez, Michael Hilgemann, Jason Schmurr

*Mathematics Open Textbooks*

This text for Analytic Geometry and Calculus I, II, and III is a Dalton State College remix of APEX Calculus 3.0. The text was created through a Round Six ALG Textbook Transformation Grant.

Topics covered in this text include:

- Limits
- Derivatives
- Integration
- Antidifferentiation
- Sequences
- Vectors

Files can also be downloaded on the Dalton State College GitHub:

https://github.com/DaltonStateCollege/calculus-text/blob/master/Calculus.pdf

Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, 2018 Rose-Hulman Institute of Technology

#### Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Let *S* be a Riemann surface and *G* a large subgroup of* Aut(S)* (*Aut(S)* may be unknown). We are particularly interested in regular *n*-gonal surfaces, i.e., the quotient surface *S/G* (and hence *S/Aut(S)*) has genus zero. For various *H *the ramification information of the branched coverings *S/K -> S/H* may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of *Aut(S)*. The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy ...

Lie Sphere Geometry And Dupin Hypersurfaces, 2018 College of the Holy Cross

#### Lie Sphere Geometry And Dupin Hypersurfaces, Thomas E. Cecil

*Mathematics Department Faculty Scholarship*

These notes were originally written for a short course held at the Institute of Mathematics and Statistics, University of São Paulo, S.P. Brazil, January 9–20, 2012. The notes are based on the author’s book [17], *Lie Sphere Geometry With Applications to Submanifolds*, Second Edition, published in 2008, and many passages are taken directly from that book. The notes have been updated from their original version to include some recent developments in the field.

A hypersurface *M ^{n}*

^{−1}in Euclidean space

**R**

^{n}is proper Dupin if the number of distinct principal curvatures is constant on

*M ...*

On The Notion Of Scalar Product For Finite-Dimensional Diffeological Vector Spaces, 2018 University of Pisa

#### On The Notion Of Scalar Product For Finite-Dimensional Diffeological Vector Spaces, Ekaterina Pervova

*Electronic Journal of Linear Algebra*

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note, a way to dispense withthis issue is considered, by introducing a notion of pseudo-metric, which, said informally, is the least-degeneratesymmetric bilinear form on a given space. This notion is applied to make some observations on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in ...

The Convex Body Isoperimetric Conjecture In The Plane, 2018 Williams College

#### The Convex Body Isoperimetric Conjecture In The Plane, John Berry, Eliot Bongiovanni, Wyatt Boyer, Bryan Brown, Paul Gallagher, David Hu, Alyssa Loving, Zane Martin, Maggie Miller, Byron Perpetua, Sarah Tammen

*Rose-Hulman Undergraduate Mathematics Journal*

The Convex Body Isoperimetric Conjecture states that the least perimeter needed to enclose a volume within a ball is greater than the least perimeter needed to enclose the same volume within any other convex body of the same volume in *R ^{n}*. We focus on the conjecture in the plane and prove a new sharp lower bound for the isoperimetric profile of the disk in this case. We prove the conjecture in the case of regular polygons, and show that in a general planar convex body the conjecture holds for small areas.

Pythagorean Combinations For Lego Robot Building., 2018 Selected Works

#### Pythagorean Combinations For Lego Robot Building., Ronald I. Greenberg

*Ronald Greenberg*

This paper provides tips for LEGO robot construction involving bracing or gear meshing along a diagonal using standard Botball kits.

Pythagorean Approximations For Lego: Merging Educational Robot Construction With Programming And Data Analysis, 2018 Selected Works

#### Pythagorean Approximations For Lego: Merging Educational Robot Construction With Programming And Data Analysis, Ronald I. Greenberg

*Ronald Greenberg*

Abstract. This paper can be used in two ways. It can provide reference information for incorporating diagonal elements (for bracing or gear meshing) in educational robots built from standard LEGO kits. Alternatively, it can be used as the basis for an assignment for high school or college students to recreate this information; in the process, students will exercise skills in both computer programming and data analysis. Using the paper in the second way can be an excellent integrative experience to add to an existing course; for example, the Exploring Computer Science high school curriculum concludes with the units “Introduction to ...

Affine And Projective Planes, 2018 Missouri State University

#### Affine And Projective Planes, Abraham Pascoe

*MSU Graduate Theses*

In this thesis, we investigate affine and projective geometries. An affine geometry is an incidence geometry where for every line and every point not incident to it, there is a unique line parallel to the given line. Affine geometry is a generalization of the Euclidean geometry studied in high school. A projective geometry is an incidence geometry where every pair of lines meet. We study basic properties of affine and projective planes and a number of methods of constructing them. We end by prov- ing the Bruck-Ryser Theorem on the non-existence of projective planes of certain orders.

The Fundamental Group And Knots, 2018 University of Redlands

#### The Fundamental Group And Knots, Hannah Michelle Solomon

*Undergraduate Honors Theses*

This project will focus on studying the fundamental groups of topological spaces. The goal is to specifically use ideas from group theory to differentiate between different types of knots by using the fundamental group as a topological invariant. First, we aim to provide a background in topology, including introducing deformation retractions and homotopy types. We will then explore new algebraic concepts, such as free groups and group presentations. This will allow us to develop a general understanding of how to find the fundamental group of a topological space and how to use it to gain more insight into which spaces ...