Geometry and Topology Commons^™

Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, Vincent Paul Flynn

Dartmouth College Ph.D Dissertations

Recently, there has been an explosion of interest in re-imagining many-body quantum phenomena beyond equilibrium. One such effort has extended the symmetry-protected topological (SPT) phase classification of non-interacting fermions to driven and dissipative settings, uncovering novel topological phenomena that are not known to exist in equilibrium which may have wide-ranging applications in quantum science. Similar physics in non-interacting bosonic systems has remained elusive. Even at equilibrium, an "effective non-Hermiticity" intrinsic to bosonic Hamiltonians poses theoretical challenges. While this non-Hermiticity has been acknowledged, its implications have not been explored in-depth. Beyond this dynamical peculiarity, major roadblocks have arisen in the search …

Restrictions On Topological Symmetry Groups Of The 3-Rung Möbius Ladder On The Torus, Logan Willhoite

Electronic Theses and Dissertations

In this work, we discuss properties of the 3-rung Möbius ladder embedded on the surface of a torus. We present proofs on restrictions of topological symmetry groups of the Möbius ladder with and without the assumption of preserving orientation. Specifically, we show that Z₂ is the only possible non-trivial orientation-preserving topological symmetry groups, and also that Z₂ and D₂ are the only possible nontrivial topological symmetry groups.

Constructing Spanning Sets Of Affine Algebraic Curvature Tensors, Stephen J. Kelly

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we construct two spanning sets for the affine algebraic curvature tensors. We then prove that every 2-dimensional affine algebraic curvature tensor can be represented by a single element from either of the two spanning sets. This paper provides a means to study affine algebraic curvature tensors in a geometric and algebraic manner similar to previous studies of canonical algebraic curvature tensors.

A Note On The Involutive Concordance Invariants For Certain (1,1)-Knots, Anna Antal, Stanley Pritchard

Rose-Hulman Undergraduate Mathematics Journal

A knot K is a smooth embedding of the circle into the three-dimensional sphere; two knots are said to be concordant if they form the boundary of an annulus properly embedded into the product of the three-sphere with an interval. Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsváth and Z. Szabó in the early 2000's which associates to a knot a filtered chain complex CFK(K), which improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu which additionally considers a chain …

On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger

Department of Mathematics: Dissertations, Theses, and Student Research

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not …

Partitions Of R^N With Maximal Seclusion And Their Applications To Reproducible Computation, Jason Vander Woude

Department of Mathematics: Dissertations, Theses, and Student Research

We introduce and investigate a natural problem regarding unit cube tilings/partitions of Euclidean space and also consider broad generalizations of this problem. The problem fits well within a historical context of similar problems and also has applications to the study of reproducibility in randomized computation.

Given $k\in\mathbb{N}$ and $\epsilon\in(0,\infty)$, we define a $(k,\epsilon)$-secluded unit cube partition of $\mathbb{R}^{d}$ to be a unit cube partition of $\mathbb{R}^{d}$ such that for every point $\vec{p}\in\R^d$, the closed $\ell_{\infty}$ $\epsilon$-ball around $\vec{p}$ intersects at most $k$ cubes. The problem is to construct such partitions for each dimension $d$ with the primary goal of minimizing …

Gordian Distance And Complete Alexander Neighbors, Ana Wright

Department of Mathematics: Dissertations, Theses, and Student Research

We call a knot K a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from K. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases.

Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo’s unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one, which proves …

Framing How We Think About Curves, Megan Sattler

Mathematical Sciences Undergraduate Honors Theses

A frame defines a basis at a point in R^n, and we can frame a curve by placing one at every point along it. This thesis investigates adapted framed curves in R^2 and R^3 in which they are used to provide information about how a curve twists and turns. We derive differential information from the frames that describe how they change and consequently how the curve changes. We also deduce special properties of each framing and discuss how the differential information suffices to describe the shape of curves.

Prefix-Rewriting: The Falsification By Fellow Traveler Property And Practical Computation, Ash Declerk

Department of Mathematics: Dissertations, Theses, and Student Research

The word problem is one of the fundamental areas of research in infinite group theory, and rewriting systems (including finite convergent rewriting systems, automatic structures, and autostackable structures) are key approaches to working on the word problem. In this dissertation, we discuss two approaches to creating bounded regular convergent prefix-rewriting systems.

Groups with the falsification by fellow traveler property are known to have solvable word problem, but they are not known to be automatic or to have finite convergent rewriting systems. We show that groups with this geometric property are geodesically autostackable. As a key part of proving this, we …

Intrinsic Tame Filling Functions And Other Refinements Of Diameter Functions, Andrew Quaisley

Department of Mathematics: Dissertations, Theses, and Student Research

Tame filling functions are quasi-isometry invariants that are refinements of the diameter function of a group. Although tame filling functions were defined in part to provide a proper refinement of the diameter function, we show that every finite presentation of a group has an intrinsic tame filling function that is equivalent to its intrinsic diameter function. We then introduce some alternative filling functions—based on concepts similar to those used to define intrinsic tame filling functions—that are potential proper refinements of the intrinsic diameter function.

Adviser: Susan Hermiller and Mark Brittenham

Knot Equivalence, Jacob Trubey

Electronic Theses, Projects, and Dissertations

A knot is a closed curve in R3. Alternatively, we say that a knot is an embedding f : S1 → R3 of a circle into R3. Analogously, one can think of a knot as a segment of string in a three-dimensional space that has been knotted together in some way, with the ends of the string then joined together to form a knotted loop. A link is a collection of knots that have been linked together.

An important question in the mathematical study of knot theory is that of how we can tell when two knots are, or are …

Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun

Electronic Theses and Dissertations

The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we …

The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane, Kathleen Zopff

Undergraduate Theses

P-adic numbers are numbers valued by their divisibility by high powers of some prime, p. These numbers are an important concept in number theory that are used in major ideas such as the Reimann Hypothesis and Andrew Wiles’ proof of Fermat’s last theorem, and also have applications in cryptography. In this project, we will explore various visualizations of p-adic numbers. In particular, we will look at a mapping of p-adic numbers into the real plane which constructs a fractal similar to a Sierpinski p-gon. We discuss the properties of this map and give formulas for the sharp bounds of its …

A Discrete Morse Approach For Computing Homotopy Types: An Exploration Of The Morse, Generalized Morse, Matching, And Independence Complexes, Connor Donovan

Mathematics Honors Papers

In this thesis, we study possible homotopy types of four families of simplicial complexes–the Morse complex, the generalized Morse complex, the matching complex, and the independence complex–using discrete Morse theory. Given a simplicial complex, K, we can construct its Morse complex from all possible discrete gradient vector fields on K. A similar construction will allow us to build the generalized Morse complex while considering edges and vertices will allow us to construct the matching complex and independence complex. In Chapter 3, we use the Cluster Lemma and the notion of star clusters to apply matchings to families of Morse, generalized …

Translation Of: Sur Des Familles D’Hypersurfaces Isoparamétriques Des Espaces Sphériques À 5 Et À 9 Dimensions, Revista Univ. Tucuman, Serie A, 1 (1940), 5–22, By Elie Cartan, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

English title: On Families of Isoparametric Hypersurfaces in Spherical Spaces of 5 and 9 Dimensions

This is an English translation of the article "Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions" which was originally published in Revista Univ. Tucuman, Serie A, 1, pp. 5-22 (1940), by Élie Cartan.

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in French. All references should be made to the original paper.

Mathematics Subject Classification Numbers: 53B25, 53C40, 53C42

Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, Adam Wagenknecht

Dissertations

Convolutional Neural Networks (CNNs) have become one of the most commonly used tools for performing image classification. Unfortunately, as with most machine learning algorithms, CNNs suffer from a lack of interpretability. CNNs are trained by using a training data set and a loss function to tune a set of parameters known as the layer weights. This tuning process is based on the classical method of gradient descent, but it relies on a strong stochastic component, which makes the weight behavior during training difficult to understand. However, since CNNs are governed largely by the weights that make up each of the …

Translation Of: Sur Quelque Familles Remarquables D’Hypersurfaces, C.R. Congrès Math. Liège, 1939, Pp. 30–41, By Élie Cartan., Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

English title: On some remarkable families of hypersurfaces

This is an English translation of the article "Sur quelque familles remarquables d’hypersurfaces" which was originally published in C.R. Congrès Math. Liège, pp. 30–41 (1939), by Élie Cartan.

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in French. All references should be made to the original paper.

Mathematics Subject Classification Numbers: 53B25, 53C40, 53C42

Area Activity, Admin Stem For Success

STEM for Success Showcase

Lesson plan to teach students about area including an activity plan, activity description, activity video, and additional activity materials

Optimal Monohedral Tilings Of Hyperbolic Surfaces, Leonardo Digiosia, Jahangir Habib, Jack Hirsch, Lea Kenigsberg, Kevin Li, Dylanger Pittman, Jackson Petty, Christopher Xue, Weitao Zhu

Rose-Hulman Undergraduate Mathematics Journal

The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of …

Free Quasitopological Groups, Jeremy Brazas, Sarah Emery

Mathematics Faculty Publications

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group F-q(X) on a space X. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the "cross topology." Using this explicit description of F-q(X), we show that for any T-1 space X, F-q(X) is the direct limit of closed subspaces F-q(X)(n) of words of length at most n. We also prove that the natural map i(n): (sic)(n)(i=0)(X boolean OR X-1)(circle times i) - …