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Articles 1 - 20 of 20
Full-Text Articles in Algebra
Explorations In Well-Rounded Lattices, Tanis Nielsen
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 …
An Inquiry Into Lorentzian Polynomials, Tomás Aguilar-Fraga
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, Hannah Friedman
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, Ian Shors
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, Jonathan Maltsman
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 …
On Rank-Two And Affine Cluster Algebras, Feiyang Lin
On Rank-Two And Affine Cluster Algebras, Feiyang Lin
HMC Senior Theses
Motivated by existing results about the Kronecker cluster algebra, this thesis is concerned with two families of cluster algebras, which are two different ways of generalizing the Kronecker case: rank-two cluster algebras, and cluster algebras of type An,1. Regarding rank-two cluster algebras, our main result is a conjectural bijection that would prove the equivalence of two combinatorial formulas for cluster variables of rank-two skew-symmetric cluster algebras. We identify a technical result that implies the bijection and make partial progress towards its proof. We then shift gears to study certain power series which arise as limits of ratios of …
The Complexity Of Symmetry, Matthew Lemay
The Complexity Of Symmetry, Matthew Lemay
HMC Senior Theses
One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial …
On The Inverse Hull Of A One-Sided Shift Of Finite Type, Aria Beaupre
On The Inverse Hull Of A One-Sided Shift Of Finite Type, Aria Beaupre
HMC Senior Theses
Let S be the semigroup constructed from a one-sided shift of finite type. In this thesis, we will provide the construction of H(S), the inverse hull of S, explore the properties of H(S), and begin to characterize the structure of H(S). We will also focus on a kind of one-sided shift of finite type, Markov shifts, and prove an invariant on isomorphic inverse hulls of Markov shifts.
On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers
On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers
HMC Senior Theses
Interpolation Jack polynomials are certain symmetric polynomials in N variables with coefficients that are rational functions in another parameter k, indexed by partitions of length at most N. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable k and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in k. In this …
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
HMC Senior Theses
Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …
Enhancing The Quandle Coloring Invariant For Knots And Links, Karina Elle Cho
Enhancing The Quandle Coloring Invariant For Knots And Links, Karina Elle Cho
HMC Senior Theses
Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.
Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri
Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri
HMC Senior Theses
Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through …
Toric Ideals, Polytopes, And Convex Neural Codes, Caitlin Lienkaemper
Toric Ideals, Polytopes, And Convex Neural Codes, Caitlin Lienkaemper
HMC Senior Theses
How does the brain encode the spatial structure of the external world?
A partial answer comes through place cells, hippocampal neurons which
become associated to approximately convex regions of the world known
as their place fields. When an organism is in the place field of some place
cell, that cell will fire at an increased rate. A neural code describes the set
of firing patterns observed in a set of neurons in terms of which subsets
fire together and which do not. If the neurons the code describes are place
cells, then the neural code gives some information about the …
Graph Cohomology, Matthew Lin
Graph Cohomology, Matthew Lin
HMC Senior Theses
What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the …
Convexity Of Neural Codes, Robert Amzi Jeffs
Convexity Of Neural Codes, Robert Amzi Jeffs
HMC Senior Theses
An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …
Realizing The 2-Associahedron, Patrick N. Tierney
Realizing The 2-Associahedron, Patrick N. Tierney
HMC Senior Theses
The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.
A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.
A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.
HMC Senior Theses
The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.
Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott
Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott
HMC Senior Theses
Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe …
Structured Matrices And The Algebra Of Displacement Operators, Ryan Takahashi
Structured Matrices And The Algebra Of Displacement Operators, Ryan Takahashi
HMC Senior Theses
Matrix calculations underlie countless problems in science, mathematics, and engineering. When the involved matrices are highly structured, displacement operators can be used to accelerate fundamental operations such as matrix-vector multiplication. In this thesis, we provide an introduction to the theory of displacement operators and study the interplay between displacement and natural matrix constructions involving direct sums, Kronecker products, and blocking. We also investigate the algebraic behavior of displacement operators, developing results about invertibility and kernels.
Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston
Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston
HMC Senior Theses
Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool …