Integrable Systems On Symmetric Spaces From A Quadratic Pencil Of Lax Operators, 2023 Technological University Dublin
Integrable Systems On Symmetric Spaces From A Quadratic Pencil Of Lax Operators, Rossen Ivanov
Conference papers
The article surveys the recent results on integrable systems arising from quadratic pencil of Lax operator L, with values in a Hermitian symmetric space. The counterpart operator M in the Lax pair defines positive, negative and rational flows. The results are illustrated with examples from the A.III symmetric space. The modeling aspect of the arising higher order nonlinear Schrödinger equations is briefly discussed.
Explorations In Well-Rounded Lattices, 2023 Claremont Colleges
Explorations In Well-Rounded Lattices, Tanis Nielsen
HMC Senior Theses
Lattices are discrete subgroups of Euclidean spaces. Analogously to vector spaces, they can be described as spans of collections of linearly independent vectors, but with integer (instead of real) coefficients. Lattices have many fascinating geometric properties and numerous applications, and lattice theory is a rich and active field of theoretical work. In this thesis, we present an introduction to the theory of Euclidean lattices, along with an overview of some major unsolved problems, such as sphere packing. We then describe several more specialized topics, including prior work on well-rounded ideal lattices and some preliminary results on the study of planar …
Cayley Map Embeddings Of Complete Graphs With Even Order, 2023 Rollins College
Cayley Map Embeddings Of Complete Graphs With Even Order, Michael O'Connor
Honors Program Theses
German mathematician Claus Michael Ringel used voltage graphs to embed complete graphs onto orientable surfaces such that none of the graph's edges cross each other. Cayley maps do the same whilst being simpler to work with. The goal is to determine the efficiency of Cayley maps in embedding complete graphs onto orientable surfaces. This article focus on complete graphs of even order with an emphasis on graphs whose orders are congruent to 6 modulo 12 and 0 modulo 12. We establish 12 distinct classes that each have their own unique qualities. Through the generalization of a previous technique, we prove …
Studies On Depth And Torsion In Tensor Products Of Modules, 2023 West Virginia University
Studies On Depth And Torsion In Tensor Products Of Modules, Uyen Huyen Thao Le
Graduate Theses, Dissertations, and Problem Reports
This dissertation represents an in-depth exploration of two distinct yet interconnected research topics within commutative algebra: one centered around a conjecture of Huneke and R. Wiegand and the other concerns a depth inequality of Auslander. It consists of the following three papers as well as the author's work under the direction of Professor Olgur Celikbas:
- Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology, Journal of Mathematical Society of Japan Advance Publication. (joint work with Olgur Celikbas, Hiroki Matsui, and Arash Sadeghi).
- An extension of a depth inequality of Auslander, Taiwanese Journal of Mathematics, …
An Inquiry Into Lorentzian Polynomials, 2023 Harvey Mudd College
An Inquiry Into Lorentzian Polynomials, Tomás Aguilar-Fraga
HMC Senior Theses
In combinatorics, it is often desirable to show that a sequence is unimodal. One method of establishing this is by proving the stronger yet easier-to-prove condition of being log-concave, or even ultra-log-concave. In 2019, Petter Brändén and June Huh introduced the concept of Lorentzian polynomials, an exciting new tool which can help show that ultra-log-concavity holds in specific cases. My thesis investigates these Lorentzian polynomials, asking in which situations they are broadly useful. It covers topics such as matroid theory, discrete convexity, and Mason’s conjecture, a long-standing open problem in matroid theory. In addition, we discuss interesting applications to known …
Long Increasing Subsequences, 2023 Claremont Colleges
Long Increasing Subsequences, Hannah Friedman
HMC Senior Theses
In my thesis, I investigate long increasing subsequences of permutations from two angles. Motivated by studying interpretations of the longest increasing subsequence statistic across different representations of permutations, we investigate the relationship between reduced words for permutations and their RSK tableaux in Chapter 3. In Chapter 4, we use permutations with long increasing subsequences to construct a basis for the space of 𝑘-local functions.
Permutations, Representations, And Partition Algebras: A Random Walk Through Algebraic Statistics, 2023 Claremont Colleges
Permutations, Representations, And Partition Algebras: A Random Walk Through Algebraic Statistics, Ian Shors
HMC Senior Theses
My thesis examines a class of functions on the symmetric group called permutation statistics using tools from representation theory. In 2014, Axel Hultman gave formulas for computing expected values of permutation statistics sampled via random walks. I present analogous formulas for computing variances of these statistics involving Kronecker coefficients – certain numbers that arise in the representation theory of the symmetric group. I also explore deep connections between the study of moments of permutation statistics and the representation theory of the partition algebras, a family of algebras introduced by Paul Martin in 1991. By harnessing these partition algebras, I derive …
A Visual Tour Of Dynamical Systems On Color Space, 2023 Claremont Colleges
A Visual Tour Of Dynamical Systems On Color Space, Jonathan Maltsman
HMC Senior Theses
We can think of a pixel as a particle in three dimensional space, where its x, y and z coordinates correspond to its level of red, green, and blue, respectively. Just as a particle’s motion is guided by physical rules like gravity, we can construct rules to guide a pixel’s motion through color space. We can develop striking visuals by applying these rules, called dynamical systems, onto images using animation engines. This project explores a number of these systems while exposing the underlying algebraic structure of color space. We also build and demonstrate a Visual DJ circuit board for …
The Multiset Partition Algebra: Diagram-Like Bases And Representations, 2023 Dartmouth College
The Multiset Partition Algebra: Diagram-Like Bases And Representations, Alexander N. Wilson
Dartmouth College Ph.D Dissertations
There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur--Weyl Duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. My thesis begins with a construction of an analogous basis for the multiset partition algebra. It continues with applications of this basis to constructing the irreducible …
Quasisymmetric Functions Distinguishing Trees, 2023 CNRS, Université de Bordeaux
Quasisymmetric Functions Distinguishing Trees, Jean-Christophe Aval, Karimatou Djenabou, Peter R. W. Mcnamara
Faculty Journal Articles
A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the P-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.
Mathematical Structure Of Musical Tuning Systems, 2023 Bard College
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Senior Projects Spring 2023
Over the course of history, western music has created a unique mathematical problem for itself. From acoustics, we know that two notes sound good together when they are related by simple ratios consisting of low primes. The problem arises when we try to build a finite set of pitches, like the 12 notes on a piano, that are all related by such ratios. We approach the problem by laying out definitions and axioms that seek to identify and generalize desirable properties. We can then apply these ideas to a broadened algebraic framework. Rings in which low prime integers can be …
Algebraic Tunnelling, 2023 Leipzig University
Algebraic Tunnelling, Gaurab Sedhain
2023 REYES Proceedings
We study the quantum phenomenon of tunnelling in the framework of algebraic quantum theory, motivated by the tunnelling aspects of false vacuum decay. We see that resolvent C*-algebra, proposed relatively recently by Buchholz and Grundling rather than Weyl algebra provides an appropriate framework for treating the dynamics of non-free quantum mechanical system as an algebraic automorphism. At the end, we propose to investigate false vacuum decay in algebraic quantum field theoretic setting in terms of the two-point correlation function which gives us the tunneling probability, with the corresponding C*-algebraic construction.
The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, 2023 Liberty University
The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna
Senior Honors Theses
The ability to communicate securely across the internet is owing to the security of the RSA cryptosystem, among others. This cryptosystem relies on the difficulty of integer factorization to provide secure communication. Peter Shor’s quantum integer factorization algorithm threatens to upend this. A special case of the hidden subgroup problem, the algorithm provides an exponential speedup in the integer factorization problem, destroying RSA’s security. Robert McEliece’s cryptosystem has been proposed as an alternative. Based upon binary Goppa codes instead of integer factorization, his cryptosystem uses code scrambling and error introduction to hinder decrypting a message without the private key. This …
Elliptic Curves Over Finite Fields, 2023 Colby College
Elliptic Curves Over Finite Fields, Christopher S. Calger
Honors Theses
The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of Fq-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …
The Lie Algebra Sl2(C) And Krawtchouk Polynomials, 2023 University of North Florida
The Lie Algebra Sl2(C) And Krawtchouk Polynomials, Nkosi Alexander
UNF Graduate Theses and Dissertations
The Lie algebra L = sl2(C) consists of the 2 × 2 complex matrices that have trace zero, together with the Lie bracket [y, z] = yz − zy. In this thesis we study a relationship between L and Krawtchouk polynomials. We consider a type of element in L said to be normalized semisimple. Let a, a^∗ be normalized semisimple elements that generate L. We show that a, a^∗ satisfy a pair of relations, called the Askey-Wilson relations. For a positive integer N, we consider an (N + 1)-dimensional irreducible L-module V consisting of the homogeneous polynomials in two variables …
Strong Homotopy Lie Algebras And Hypergraphs, 2023 Virginia Commonwealth University
Strong Homotopy Lie Algebras And Hypergraphs, Samuel J. Bevins, Marco Aldi
Undergraduate Research Posters
We study hypergraphs by attaching a nilpotent strong homotopy Lie algebra. We especially focus on hypergraph theoretic information that is encoded in the cohomology of the resulting strong homotopy Lie algebra.
Understanding And Advancing College Students' Mathematical Reasoning Using Collaborative Argumentation, 2023 Missouri State University
Understanding And Advancing College Students' Mathematical Reasoning Using Collaborative Argumentation, Rachel Kay Heili
MSU Graduate Theses
This study explored students’ mathematical reasoning skills and offered supports to advance them through a collaborative argumentation framework in a college intermediate algebra class. The goals of this study were to make observations about student reasoning, identify specific actions to address those observations, and document student growth in reasoning as a result of those actions. An iterative analysis, mixed method study was conducted in which the researcher engaged students in responding to questions that required conceptual understandings using a collaborative argumentation framework as a tool to identify and code components of their responses—claim, evidence, and reasoning. After coding and analyzing …
On Covering Groups With Proper Subgroups, 2023 Missouri State University
On Covering Groups With Proper Subgroups, Collin B. Moore
MSU Graduate Theses
In this paper, we explore groups that can be expressed as a union of proper subgroups. Using “covering number” to denote the minimal number of proper subgroups required to cover a group, we explore the nature of groups with covering numbers 3 and 4, while also finding covering numbers for p-groups, dihedral, and generalized dihedral groups.
Q-Polymatroids And Their Application To Rank-Metric Codes., 2023 University of Kentucky
Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany
Theses and Dissertations--Mathematics
Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …
Higher Spanier Groups, 2023 West Chester University
Higher Spanier Groups, Johnny Aceti
West Chester University Master’s Theses
When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …