# Geometry and Topology Commons™

## All Articles in Geometry and Topology

464 full-text articles. Page 1 of 14.

Rainich Conditions In (2+1)-Dimensional Gravity, 2016 Selected Works

#### Rainich Conditions In (2+1)-Dimensional Gravity, Dionisios Krongos, Charles G. Torre

##### Charles G. Torre

In (3 + 1) spacetime dimensions the Rainich conditions are a set of equations expressed solely in terms of the metric tensor which are equivalent to the Einstein-Maxwell equations for non-null electromagnetic fields. Here we provide the analogous conditions for (2 + 1)-dimensional gravity coupled to electromagnetism. Both the non-null and null cases are treated. The construction of these conditions is based upon reducing the problem to that of gravity coupled to a scalar field, which we have treated elsewhere. These conditions can be easily extended to other theories of (2 + 1)-dimensional gravity. For example, we apply the geometrization conditions ...

Some 2-Categorical Aspects In Physics, 2016 The Graduate Center, City University of New York

#### Some 2-Categorical Aspects In Physics, Arthur Parzygnat

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description ...

A Geometric Model Of Twisted Differential K-Theory, 2016 The Graduate Center, City University of New York

#### A Geometric Model Of Twisted Differential K-Theory, Byung Do Park

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

We construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. We use smooth U(1)-gerbes with connection as differential twists and twisted vector bundles with connection as cycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. We also construct an odd twisted Chern character of a twisted vector bundle with an automorphism. In addition to our geometric model of twisted differential K-theory, we introduce a smooth variant of the Hopkins-Singer model of differential K-theory. We prove that our model ...

2016 The Graduate Center, City University of New York

#### The Fourth Movement Of György Ligeti's Piano Concerto: Investigating The Musical-Mathematical Connection, Cynthia L. Wong

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

This interdisciplinary study explores musical-mathematical analogies in the fourth movement of Ligeti’s Piano Concerto. Its aim is to connect musical analysis with the piece’s mathematical inspiration. For this purpose, the dissertation is divided into two sections. Part I (Chapters 1-2) provides musical and mathematical context, including an explanation of ideas related to Ligeti’s mathematical inspiration. Part II (Chapters 3-5) delves into an analysis of the rhythm, form, melody / motive, and harmony. Appendix A is a reduced score of the entire movement, labeled according to my analysis.

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, 2016 The Graduate Center, City University of New York

#### On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form ...

2016 CUNY Borough of Manhattan Community College

#### Hyperplanes That Intersect Each Ray Of A Cone Once And A Banach Space Counterexample, Chris Mccarthy

##### Publications and Research

Suppose 𝐶 is a cone contained in real vector space 𝑉. When does 𝑉 contain a hyperplane 𝐻 that intersects each of the 0-rays in 𝐶\{0} exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question.We also present an example of a salient, closed Banach space cone 𝐶 for which there does not exist a hyperplane that intersects each 0-ray in 𝐶 \ {0} exactly once.

2016 The University of Western Ontario

#### On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi

##### Electronic Thesis and Dissertation Repository

In the first part of this thesis, a noncommutative analogue of Gross' logarithmic Sobolev inequality for the noncommutative 2-torus is investigated. More precisely, Weissler's result on the logarithmic Sobolev inequality for the unit circle is used to propose that the logarithmic Sobolev inequality for a positive element $a= \sum a_{m,n} U^{m} V^{n}$ of the noncommutative 2-torus should be of the form \tau(a^{2} \log a)\leqslant \underset{(m,n)\in \mathbb{Z}^{2}}{\sum} (\vert m\vert + \vert n\vert) \vert a_{m,n} \vert ^{2} + \tau (a^{2})\log ( \tau (a^2))^{1 ...

2016 Dordt College

#### Projective Geometry Hidden Inside: Can You Spot It?, Tom Clark

##### Faculty Work: Comprehensive List

In this talk Dr. Clark shared about a Math Teachers’ Circle session he recently ran centered around the children’s game Spot it! This game has some very interesting mathematics behind it and naturally begs to be explored with inquiry. He described the way he led teachers to ask questions about the game, the way the teachers then explored the topic, and the mathematics behind it all.

2016 The University of Western Ontario

#### Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega

##### Electronic Thesis and Dissertation Repository

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this thesis we use elliptic theory for edge- degenerate differential operators on singular manifolds to study general deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

Bridge Spectra Of Cables Of 2-Bridge Knots, 2016 University of Nebraska-Lincoln

#### Bridge Spectra Of Cables Of 2-Bridge Knots, Nicholas John Owad

##### Dissertations, Theses, and Student Research Papers in Mathematics

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Advisors: Mark Brittenham and Susan Hermiller

Introduction To Classical Field Theory, 2016 Department of Physics, Utah State University

#### Introduction To Classical Field Theory, Charles G. Torre

##### All Complete Monographs

This is an introduction to classical field theory. Topics treated include: Klein-Gordon field, electromagnetic field, scalar electrodynamics, Dirac field, Yang-Mills field, gravitational field, Noether theorems relating symmetries and conservation laws, spontaneous symmetry breaking, Lagrangian and Hamiltonian formalisms.

Clustering-Based Robot Navigation And Control, 2016 University of Pennsylvania

#### Clustering-Based Robot Navigation And Control, Omur Arslan

##### Departmental Papers (ESE)

In robotics, it is essential to model and understand the topologies of configuration spaces in order to design provably correct motion planners. The common practice in motion planning for modelling configuration spaces requires either a global, explicit representation of a configuration space in terms of standard geometric and topological models, or an asymptotically dense collection of sample configurations connected by simple paths, capturing the connectivity of the underlying space. This dissertation introduces the use of clustering for closing the gap between these two complementary approaches. Traditionally an unsupervised learning method, clustering offers automated tools to discover hidden intrinsic structures in ...

Patterns Formed By Coins, 2016 Formlabs

#### Patterns Formed By Coins, Andrey M. Mishchenko

##### Journal of Humanistic Mathematics

This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non- overlapping circles. The first half of the article is an exposition of the two most important facts about circle packings, (1) that essentially whatever pattern we ask for, we may always arrange circles in that pattern, and (2) that under simple conditions on the pattern, there is an essentially unique arrangement of circles in that pattern. In the second half of the article, we consider related questions, but where we allow ...

Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, 2016 Smith College

#### Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, Joseph O'Rourke

##### Faculty Publications

A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does ...

2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

##### Mathematical Sciences Technical Reports (MSTR)

Two Riemann surfaces S1 and S2 with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S1 -> S2 which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials Hq(S1) and Hq(S2). In this note we study the differences between topological equivalence and Hq equivalence of prime cyclic actions, where S1/G and S2/G have genus zero.

2016 New Mexico State University

#### The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder

##### Geometry

No abstract provided.

Octahedral Dice, 2016 Butler University

#### Octahedral Dice, Todd Estroff, Jeremiah Farrell

##### Jeremiah Farrell

All five Platonic solids have been used as random number generators in games involving chance with the cube being the most popular. Martin Gardenr, in his article on dice (MG 1977) remarks: "Why cubical?... It is the easiest to make, its six sides accomodate a set of numbers neither too large nor too small, and it rolls easily enough but not too easily."

Gardner adds that the octahedron has been the next most popular as a randomizer. We offer here several problems and games using octahedral dice. The first two are extensions from Gardner's article. All answers will be ...

The Magic Octagon, 2016 Butler University

#### The Magic Octagon, Jeremiah Farrell, Tom Rodgers

##### Jeremiah Farrell

The black nodes mark the corners of an octagon and each of these nodes in connected to four others by lines. The (rather hard) puzzle is to assign the sixteen numbers 0 through 15 to each of the sixteen lines so that each black node has a sum of 30 when the line numbers leading into it are added.

The word version of the puzzle was described in the article "Most-Perfect Word Magic", Oscar Thumpbindle, Word Ways Vol. 40(4). Nov. 2007.

Some Curious Cut-Ups, 2016 Butler University

#### Some Curious Cut-Ups, Jeremiah Farrell, Ivan Moscovich

##### Jeremiah Farrell

We have noticed a certain kind of n-gon dissection into triangles that has a wonderful property of interest to most puzzlists. Namely that any two triangles have at least one edge in common yet no two triangles need be congruent. In an informal poll of specialists at a recent convention, none of them saw immediately how this could be accomplished. But in fact it is very straightforward.

Cohomology Of Certain Polyhedral Product Spaces, 2016 Graduate Center, City University of New York

#### Cohomology Of Certain Polyhedral Product Spaces, Elizabeth A. Vidaurre

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

The study of torus actions led to the discovery of moment-angle complexes and their generalization, polyhedral product spaces. Polyhedral products are constructed from a simplicial complex. This thesis focuses on computing the cohomology of polyhedral products given by two different classes of simplicial complexes: polyhedral joins (composed simplicial complexes) and $n$-gons. A homological decomposition of a polyhedral product developed by Bahri, Bendersky, Cohen and Gitler is used to derive a formula for the case of polyhedral joins. Moreover, methods from and results by Cai will be used to give a full description of the non-trivial cup products in a ...