Translation Distance And Fibered 3-Manifolds, 2020 The Graduate Center, City University of New York

#### Translation Distance And Fibered 3-Manifolds, Alexander J. Stas

*All Dissertations, Theses, and Capstone Projects*

A 3-manifold is said to be fibered if it is homeomorphic to a surface bundle over the circle. For a cusped, hyperbolic, fibered 3-manifold *M*, we study an invariant of the mapping class of a surface homeomorphism called the translation distance in the arc complex and its relation with essential surfaces in *M*. We prove that the translation distance of the monodromy of *M* can be bounded above by the Euler characteristic of an essential surface. For one-cusped, hyperbolic, fibered 3-manifolds, the monodromy can also be bounded above by a linear function of the genus of an essential surface.

We ...

Convexity And Curvature In Hierarchically Hyperbolic Spaces, 2020 The Graduate Center, City University of New York

#### Convexity And Curvature In Hierarchically Hyperbolic Spaces, Jacob Russell-Madonia

*All Dissertations, Theses, and Capstone Projects*

Introduced by Behrstock, Hagen, and Sisto, hierarchically hyperbolic spaces axiomatized Masur and Minsky's powerful hierarchy machinery for the mapping class groups. The class of hierarchically hyperbolic spaces encompasses a number of important and seemingly distinct examples in geometric group theory including the mapping class group and Teichmueller space of a surface, virtually compact special groups, and the fundamental groups of 3-manifolds without Nil or Sol components. This generalization allows the geometry of all of these important examples to be studied simultaneously as well as providing a bridge for techniques from one area to be applied to another.

This thesis ...

Remote Learning Assignment, 2020 The College at Brockport: State University of New York

Fern Or Fractal... Or Both?, 2020 Concordia University St. Paul

#### Fern Or Fractal... Or Both?, Christina Babcock

*Research and Scholarship Symposium Posters*

Fractals are series of self similar sets and can be found in nature. After researching the Barnsley Fern and the iterated function systems using to create the fractal, I was able to apply what I learned to create a fractal shell. This was done using iterated function systems, matrices, random numbers, and Python coding.

K-Plane Constant Curvature Conditions, 2020 Reed College

#### 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, 2020 Nanjing International School

#### 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, 2020 Potsdam University

#### 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.

Discrete Morse Theory By Vector Fields: A Survey And New Directions, 2020 Minnesota State University, Mankato

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

*All 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.

Heat Kernel Voting With Geometric Invariants, 2020 Minnesota State University, Mankato

#### Heat Kernel Voting With Geometric Invariants, Alexander Harr

*All 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.

Periodic Points On Tori: Vanishing And Realizability, 2020 University of Kentucky

#### 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 ...

Graph-Theoretic Simplicial Complexes, Hajos-Type Constructions, And K-Matchings, 2020 University of Kentucky

#### 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 ...

A Qualitative Representation Of Spatial Scenes In R2 With Regions And Lines, 2019 University of Maine

#### A Qualitative Representation Of Spatial Scenes In R2 With Regions And Lines, Joshua Lewis

*Electronic Theses and Dissertations*

Regions and lines are common geographic abstractions for geographic objects. Collections of regions, lines, and other representations of spatial objects form a spatial scene, along with their relations. For instance, the states of Maine and New Hampshire can be represented by a pair of regions and related based on their topological properties. These two states are adjacent (i.e., they *meet* along their shared boundary), whereas Maine and Florida are not adjacent (i.e., they are *disjoint*).

A detailed model for qualitatively describing spatial scenes should capture the essential properties of a configuration such that a description of the represented ...

Enriched Derivators, 2019 The University of Western Ontario

#### Enriched Derivators, James Richardson

*Electronic Thesis and Dissertation Repository*

In homotopical algebra, the theory of derivators provides a convenient abstract setting for computing with homotopy limits and colimits. In enriched homotopy theory, the analogues of homotopy (co)limits are weighted homotopy (co)limits. In this thesis, we develop a theory of derivators and, more generally, prederivators enriched over a monoidal derivator E. In parallel to the unenriched case, these E-prederivators provide a framework for studying the constructions of enriched homotopy theory, in particular weighted homotopy (co)limits.

As a precursor to E-(pre)derivators, we study E-categories, which are categories enriched over a bicategory Prof(E) associated to E ...

Research Of Parabolic Surface Points In Galilean Space, 2019 Tashkent Railway Engineering Institute

#### Research Of Parabolic Surface Points In Galilean Space, Abdullaaziz Artykbaev, Bekzod Sultanov

*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*

The paper studies the surfaces of the Galilean space $R_3^1$. First, we consider the geometry of the surface in a small neighborhood of a point on the surface. Basically, we studied the points of the surface where at least one of the principal curvature appeals to zero. Two classes of points are defined where at least one of the principal curvature is zero. These points are divided into two types, parabolic and especially parabolic. It is proved that these neighborhoods using the movement of space is impossible to move each other. A sweep of surfaces with parabolic and especially ...

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, 2019 University of Massachusetts Amherst

#### Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

*Doctoral Dissertations*

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and ...

Loop Homology Of Bi-Secondary Structures, 2019 Illinois State University

#### Loop Homology Of Bi-Secondary Structures, Andrei Bura

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

No abstract provided.

The Energy-Spectrum Of Bicompatible Sequences, 2019 Illinois State University

#### The Energy-Spectrum Of Bicompatible Sequences, Wenda Huang

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

No abstract provided.

Diffusion And Consensus On Weakly Connected Directed Graphs, 2019 Portland State University

#### Diffusion And Consensus On Weakly Connected Directed Graphs, J. J. P. Veerman, Ewan Kummel

*Mathematics and Statistics Faculty Publications and Presentations*

Let G be a weakly connected directed graph with asymmetric graph Laplacian L. Consensus and diffusion are dual dynamical processes defined on G by x˙=−Lx for consensus and p˙=−pL for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors {γ¯i}ki=1 of the left null-space of L and a basis of column vectors {γi}ki=1 of the right null-space of L in terms of the partition of G into strongly connected components. This allows for complete characterization of the asymptotic behavior of both diffusion and ...

Torsors Over Simplicial Schemes, 2019 The University of Western Ontario

#### Torsors Over Simplicial Schemes, Alexander S. Rolle

*Electronic Thesis and Dissertation Repository*

Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups ...

Classification Of Minimal Separating Sets In Low Genus Surfaces, 2019 Portland State University

#### Classification Of Minimal Separating Sets In Low Genus Surfaces, J. J. P. Veerman, William Maxwell, Victor Rielly, Austin K. Williams

*J. J. P. Veerman*

Consider a surface *S* and let *M* ⊂ *S*. If *S* \ *M* is not connected, then we say *M* separates *S*, and we refer to *M* as a separating set of *S*. If *M* separates *S*, and no proper subset of *M* separates *S*, then we say *M* is a minimal separating set of *S*. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus *g* = 2 and *g* = 3. The classification for genus 0 and 1 was done in earlier work, using methods of algebraic topology.