# Geometry and Topology Commons™

## All Articles in Geometry and Topology

666 full-text articles. Page 1 of 23.

Finite Simple Graphs And Their Associated Graph Lattices, 2018 Middle Tennessee State University

#### Finite Simple Graphs And Their Associated Graph Lattices, James B. Hart, Brian Frazier

##### Theory and Applications of Graphs

In his 2005 dissertation, Antoine Vella explored combinatorical aspects of finite graphs utilizing a topological space whose open sets are intimately tied to the structure of the graph. In this paper, we go a step further and examine some aspects of the open set lattices induced by these topological spaces. In particular, we will characterize all lattices that constitute the opens for finite simple graphs endowed with this topology, explore the structure of these lattices, and show that these lattices contain information necessary to reconstruct the graph and its complement in several ways.

Topological Detection Of The Dimension Of The Stimuli Space, 2018 Illinois State University

#### Topological Detection Of The Dimension Of The Stimuli Space, Aliaksandra Yarosh

##### Annual Symposium on Biomathematics and Ecology: Education and Research

No abstract provided.

Equivalent Constructions Of Cartan Pairs, 2018 University of New Mexico

#### Equivalent Constructions Of Cartan Pairs, Phung Thanh Tran

##### Math Theses

Feldman and Moore [4] introduce Cartan subalgebra of the von Neumann algebra M on a separable Hilbert space H from the natural subalgebra of M(R, sigma), the twisted algebra of matrices over the relation R on a Borel space (X, B, muy). They show that if M has a Cartan subalgebra A, then M is isomorphic to M(R, sigma) where A is the twisted algebra onto the diagonal subalgebra L^inf (X, muy). The relation R is unique to isomorphism and the orbit of the two-cohomology class on R in the torus T, which is the automorphism group ...

2018 The Graduate Center, City University of New York

#### Studying The Space Of Almost Complex Structures On A Manifold Using De Rham Homotopy Theory, Bora Ferlengez

##### All Dissertations, Theses, and Capstone Projects

In his seminal paper Infinitesimal Computations in Topology, Sullivan constructs algebraic models for spaces and then computes various invariants using them. In this thesis, we use those ideas to obtain a finiteness result for such an invariant (the de Rham homotopy type) for each connected component of the space of cross-sections of certain fibrations. We then apply this result to differential geometry and prove a finiteness theorem of the de Rham homotopy type for each connected component of the space of almost complex structures on a manifold. As a special case, we discuss the space of almost complex structures on ...

2018 The University of Western Ontario

#### Topological Recursion And Random Finite Noncommutative Geometries, Shahab Azarfar

##### Electronic Thesis and Dissertation Repository

In this thesis, we investigate a model for quantum gravity on finite noncommutative spaces using the topological recursion method originated from random matrix theory. More precisely, we consider a particular type of finite noncommutative geometries, in the sense of Connes, called spectral triples of type ${(1,0)} \,$, introduced by Barrett. A random spectral triple of type ${(1,0)}$ has a fixed fermion space, and the moduli space of its Dirac operator ${D=\{ H , \cdot \} \, ,}$ ${H \in {\mathcal{H}_N}}$, encoding all the possible geometries over the fermion space, is the space of Hermitian matrices ${\mathcal{H}_N}$. A distribution of ...

Syzygy Order Of Big Polygon Spaces, 2018 The University of Western Ontario

#### Syzygy Order Of Big Polygon Spaces, Jianing Huang

##### Electronic Thesis and Dissertation Repository

For a compact smooth manifold with a torus action, its equivariant cohomology is a finitely generated module over a polynomial ring encoding information about the space and the action. For such a module, we can associate a purely algebraic notion called syzygy order. Syzygy order of equivariant cohomology is closely related to the exactness of Atiyah-Bredon sequence in equivariant cohomology. In this thesis we study a family of compact orientable manifolds with torus actions called big polygon spaces. We compute the syzygy orders of their equivariant cohomologies. The main tool used is a quotient criterion for syzygies in equivariant cohomology ...

2018 Ursinus College

#### Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu

##### Mathematics Summer Fellows

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. We conclude with an example illustrating our construction.

Local Higher Category Theory, 2018 The University of Western Ontario

#### Local Higher Category Theory, Nicholas Meadows

##### Electronic Thesis and Dissertation Repository

The purpose of this thesis is to give presheaf-theoretic versions of three of the main extant models of higher category theory: the Joyal, Rezk and Bergner model structures. The construction of these model structures takes up Chapters 2, 3 and 4 of the thesis, respectively. In each of the model structures, the weak equivalences are local or ‘stalkwise’ weak equivalences. In addition, it is shown that certain Quillen equivalences between the aforementioned models of higher category theory extend to Quillen equivalences between the various models of local higher category theory.

Throughout, a number of features of local higher category theory ...

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

#### Nearness Without Distance, Nicholas A. Scoville

##### Topology

No abstract provided.

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.

2018 University of San Diego

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

##### Research Week

No abstract provided.

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 ...

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 ...

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 ...

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 ...

The Average Measure Of A K-Dimensional Simplex In An N-Cube, 2018 Missouri State University

#### The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter

Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach nk/2 times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for ...

Affine And Projective Planes, 2018 Missouri State University

#### Affine And Projective Planes, Abraham Pascoe

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.

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.