Physical Sciences and Mathematics Commons^™

Sum Of Cubes Of The First N Integers, Obiamaka L. Agu

Electronic Theses, Projects, and Dissertations

In Calculus we learned that 􏰅Sum^{n}_{k=1} k = [n(n+1)]/2 , that Sum^{􏰅n}_{k=1} k^2 = [n(n+1)(2n+1)]/6 , and that Sum^{n}_{k=1} k^{3} = (n(n+1)/2)^{2}. These formulas are useful when solving for the area below quadratic or cubic function over an interval [a, b]. This tedious process, solving for areas under a quadratic or a cubic, served as motivation for the introduction of Riemman integrals. For the overzealous math student, these steps were replaced by a simpler method of evaluating antiderivatives at the endpoints a and b. From my recollection, a former instructor informed us to do the value of memorizing these formulas. …

Intrinsic Curvature For Schemes, Pat Lank

Mathematics & Statistics ETDs

This thesis develops an algebraic analog of psuedo-Riemannian geometry for relative schemes whose cotangent sheaf is finite locally free. It is a generalization of the algebraic differential calculus proposed by Dr. Ernst Kunz in an unpublished manuscript to the non-affine case. These analogs include the psuedo-Riemannian metric, Levi-Civit´a connection, curvature, and various existence theorems.

Spectral Sequences For Almost Complex Manifolds, Qian Chen

Dissertations, Theses, and Capstone Projects

In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [CW18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Frolicher spectral sequence. …

Harmony Amid Chaos, Drew Schaffner

Pence-Boyce STEM Student Scholarship

We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as …

Hyperbolic Triangle Groups, Sergey Katykhin

Electronic Theses, Projects, and Dissertations

This paper will be on hyperbolic reflections and triangle groups. We will compare hyperbolic reflection groups to Euclidean reflection groups. The goal of this project is to give a clear exposition of the geometric, algebraic, and number theoretic properties of Euclidean and hyperbolic reflection groups.

Evolution Of Computational Thinking Contextualized In A Teacher-Student Collaborative Learning Environment., John Arthur Underwood

LSU Doctoral Dissertations

The discussion of Computational Thinking as a pedagogical concept is now essential as it has found itself integrated into the core science disciplines with its inclusion in all of the Next Generation Science Standards (NGSS, 2018). The need for a practical and functional definition for teacher practitioners is a driving point for many recent research endeavors. Across the United States school systems are currently seeking new methods for expanding their students’ ability to analytically think and to employee real-world problem-solving strategies (Hopson, Simms, and Knezek, 2001). The need for STEM trained individuals crosses both the vocational certified and college degreed …

An Analysis And Comparison Of Knot Polynomials, Hannah Steinhauer

Senior Honors Projects, 2020-current

Knot polynomials are polynomial equations that are assigned to knot projections based on the mathematical properties of the knots. They are also invariants, or properties of knots that do not change under ambient isotopy. In other words, given an invariant α for a knot K, α is the same for any projection of K. We will define these knot polynomials and explain the processes by which one finds them for a given knot projection. We will also compare the relative usefulness of these polynomials.

Hyperbolic Endomorphisms Of Free Groups, Jean Pierre Mutanguha

Graduate Theses and Dissertations

We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends Brinkmann's theorem that free-by-cyclic groups are word-hyperbolic if and only if they have no Z2 subgroups. To get started on our main theorem, we first prove a structure theorem for injective but nonsurjective endomorphisms of free groups. With the decomposition of the free group given by this structure theorem, we (more or less) construct representatives for nonsurjective endomorphisms that are expanding immersions relative to a homotopy equivalence. This structure theorem initializes the development of (relative) train track theory …

Compactifications Of Cluster Varieties Associated To Root Systems, Feifei Xie

Doctoral Dissertations

In this thesis we identify certain cluster varieties with the complement of a union of closures of hypertori in a toric variety. We prove the existence of a compactification $Z$ of the Fock--Goncharov $\mathcal{X}$-cluster variety for a root system $\Phi$ satisfying some conditions, and study the geometric properties of $Z$. We give a relation of the cluster variety to the toric variety for the fan of Weyl chambers and use a modular interpretation of $X(A_n)$ to give another compactification of the $\mathcal{X}$-cluster variety for the root system $A_n$.

Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat

Honors Theses

Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this paper, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the torsion subgroups of elliptic curves. In particular, we will examine two methods used to classify these subgroups. We finish by employing these methods to categorize the torsion subgroups for a specific family of elliptic …

A Multi Centerpoint Theorem Via Fourier Analysis On The Torus, Yan Chen

Senior Projects Spring 2020

The Centerpoint Theorem states that for any set $S$ of points in $\mathbb{R}^d$, there exists a point $c$ such that any hyperplane goes through that point divides the set. For any half-space containing the point $c$, the amount of points in that half-space is no bigger than $\frac{1}{d+1}$ of the whole set. This can be related to how close can any hyperplane containing the point $c$ comes to equipartitioning for a given shape $S$. For a function from unit circle to real number, it has a Fourier interpretation. Using Fourier analysis on the Torus, I will try to find a …

Pascal's Mystic Hexagon In Tropical Geometry, Hanna Hoffman

HMC Senior Theses

Pascal's mystic hexagon is a theorem from projective geometry. Given six points in the projective plane, we can construct three points by extending opposite sides of the hexagon. These three points are collinear if and only if the six original points lie on a nondegenerate conic. We attempt to prove this theorem in the tropical plane.

Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes

All Graduate Theses, Dissertations, and Other Capstone Projects

This paper will explore the use and construction of Gröbner bases through Buchberger's algorithm. Specifically, applications of such bases for solving systems of polynomial equations will be discussed. Furthermore, we relate many concepts in commutative algebra to ideas in computational algebraic geometry.

Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, William Trok

Theses and Dissertations--Mathematics

This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points.

We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' …

Harmonic Morphisms With One-Dimensional Fibres And Milnor Fibrations, Murphy Griffin

UNF Graduate Theses and Dissertations

We study a problem at the intersection of harmonic morphisms and real analytic Milnor fibrations. Baird and Ou establish that a harmonic morphism from G: \mathbb{R}^m \setminus V_G \rightarrow \mathbb{R}^n\setminus \{0\} defined by homogeneous polynomials of order p retracts to a harmonic morphism \psi|: S^{m-1} \setminus K_\epsilon \rightarrow S^{n-1} that induces a Milnor fibration over the sphere. In seeking to relax the homogeneity assumption on the map G, we determine that the only harmonic morphism $\varphi: \mathbb{R}^m \setminus V_G \rightarrow S^{m-1}\K_\epsilon$ that preserves \arg G is radial projection. Due to this limitation, we confirm Baird and Ou's result, yet establish …

Phylogenetic Networks And Functions That Relate Them, Drew Scalzo

Williams Honors College, Honors Research Projects

Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate …

Heat Kernel Voting With Geometric Invariants, Alexander Harr

All Graduate 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.

Scrollar Invariants Of Tropical Chains Of Loops, Kalila Joelle Sawyer

Theses and Dissertations--Mathematics

We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and compute these invariants for a much-studied family of tropical curves. Our examples highlight many parallels between the classical and tropical theories, but also point to some substantive distinctions.

Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj

Theses and Dissertations--Mathematics

In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.