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Undergraduate research and projects
Undergraduate research and projects
Over four years at Kalamazoo College, I was blessed to be a part of many research groups and independent studies. Here is a brief rundown on research and independent study projects I completed in my undergraduate career. Please see my resume/CV for some further details on project activities.
Math Research and Projects
Math Research and Projects
Konstant's Weight q-Multiplicity Formula for sp_6(C), Summer 2021
Article+Recorded Presentation+Slides:
**Article submitted to Applicable Algebra in Engineering, Communication and Computing
**Recording submitted to Young Mathematicians Conference 2021
Abstract:
"Kostant’s weight q-multiplicity formula is an alternating sum over a finite group known as the Weyl group and whose terms involve the q-analog of Kostant’s partition function, which is a polynomial-valued function that counts number of ways the weight z of a Lie algebra G can be written as a sum of exactly i positive roots of G . The evaluation of the q-multiplicity formula at q=1 recovers the multiplicity of a weight in a irreducible highest weight representation of G. In this paper, we specialize to the Lie algebra sp_6(C) and we provide a closed formula for the q-analog of Kostant's partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of sp_6(C), and use these results to provide a closed formula for the q-multiplicity for any pair of dominant integral weights of sp_6(C). Throughout this work, we provide code to facilitate these computations. "
Link to recorded presentation for YMC 2021
On the Minimal Generating Sets of Invariants for Finite Abelian Groups, Summer 2021
Recorded presentation:
**Submitted to Young Mathematicians Conference 2021
Abstract:
"Given some arbitrary polynomial ring, an invariant polynomial is a polynomial that is unchanged by the action of a group G. We investigate the ring of invariant polynomials under the action of some abelian groups with the goal of finding generators for this ring. When considering an abelian group, we can always find a basis such that the action is diagonal, so there exist monomial generators m_i for the invariant ring. By Noether's degree bound, the minimal set of generating monomials m_1,...,m_k is finite and the degree of each generating monomial m_i is less than G. Motivated by the previous work of Gandini and Derksen, we present a new approach to find invariants for Z_p x Z_p and show that this approach can fail for Z_n x Z_n when n is not a prime."
On Mutually Unbiased Bases and Hadamard Matrices,
Winter and Spring 2021
Recorded presentation:
**Presented at Michigan MAA Section Meeting 2021
**Presented at Kalamazoo College Math Physics SIP Symposium 2021
Abstract:
"Two orthonormal bases are called mutually unbiased if the norm of the inner product squared between arbitrary basis vectors from distinct bases are equal--more precisely, equal to the inverse of the dimension of the vector space in which they live. In prime-power dimensions of vector spaces over the reals and the complex we can find maximal sets of mutually unbiased bases, however, the question about the maximality of sets of mutually unbiased bases (MUBs) is still open. We focus our attention on complex spaces so we can exploit familiar tools such as roots of unity. Using the language of complex Hadamard matrices, we provide a standard construction of maximal sets of MUBs in prime dimensions over the complex and observe how it fails in the first composite dimension, 6."
Thesis paper:
Abstract:
"The goal of this paper will be to study the behavior of mutually unbiased bases in complex dimensions by looking at equivalent objects that preserve the mutual unbiasedness property. Of the many equivalent objects, this paper will focus on the unitary operators and complex Hadamard matrices. We will thoroughly examine the standard constructions of mutually unbiased bases using these two approaches. By understanding why the existing constructions work in prime power dimensions (and fail in non-prime dimensions) we aim to motivate new methods to tackle their existence non-prime power dimension--in particular, in the first composite dimension, six."
On Mutually Unbiased Bases and Hadamard Matrices thesis (2021)
**not my image of the bloch sphere <--I wish I had the citation!
at SIP Fest 2021 (April 14th)
Rational Mappings and Relative Finiteness, Winter 2021
Final paper+recording+slides:
Abstract:
"For this paper, we have two primary goals in extending our previous knowledge on mappings on varieties and quotient rings. The first aim is to extend our understanding of polynomial mappings on varieties to rational mappings onto varieties. The second aim is to extend our ideas on polynomial quotient rings as vectors spaces to k-algebras."
Link to a draft of a scholarly paper written for MATH-398:
Link to short video presentation on mappings on varieties:
Link to presentation slides:
Finite Geometries and Quantum Systems, Fall 2020.
Final paper+recording+slides:
**Presented at Kalamazoo College Math Colloquium, Winter 2021
Abstract:
"We aim to understand the relationship between mutually unbiased bases and their relation to finite geometries, namely finite affine planes. The first section of the paper will be dedicated to exploring incidence structures and projective planes which are key to understanding affine planes. In the second section, we will lay out the necessary algebraic tools such as inner product, orthogonality, and orthonormal bases which will help us understand the property of mutual unbiasedness and hence sets of mutually unbiased bases. In the third section, we will wrap up by discussing how these geometrical and algebraic objects are intertwined, and end by suggesting some applications and directions to continue."
Link to an unpublished expository paper written for MATH-395:
Finite Geometries and Quantum Systems paper (2020)
Link to a video presentation that will serve as a math colloquium credit in winter 2021 (also provided on the right): Video presentation (2020)
Cyclotomic Polynomials, Winter 2019-2020.
J-term project paper:
Abstract:
"We would like to discuss nth roots of unity, primitive nth roots of unity, and cyclotomic polynomials. Theorems, elementary properties, proofs, and examples will be provided. The ultimate goal will be to recreate the proof of the irreducibility of cyclotomic polynomials over the rationals."
Link to an unpublished expository paper written for j-term project:
Irreducibility of Cyclotomic Polynomials over the Integers (2020)
Right: Visualizations of the 6th and 8th roots of unity; primitive roots are highlighted in the 6th roots of unity graphic.
Physics research and projects
Physics research and projects
Solar Flare Research Group, Kalamazoo College, Spring 2018-on hold
Project Description:
In this project, we worked with abelian sandpile models (Bak-Tang-Wiesenfeld) and self-organized criticality to produce a 2-D cellular automata model of energy redistribution and solar flares on the surface of the sun. We later produced a statistical analysis package to determine Dragon-King significance in Python. This project is still incomplete and has been left on hold.
Right: some unpublished cute test results of the stats package
Electron Bubble Research Group, Brown University, Summer 2018
Project Description:
For this project, we modeled one-dimensional sedimentation in different flow regimes of two-dimensional fluids while considering the effects of the Basset force. We also dealt with the Navier stokes' equation and differential equations models to simulate fluid dynamics in Fortran and python using RK methods and Forward Difference Hybrid.
Right: Basset force equation
Student government projects
Student government projects
Student Affairs Committee Manuscript (KCCSR 2020-2021)
Project Description:
For this project, we wrote a 61-page long in-depth review of K's "Strategic Plan" and the Student Climate. For each problem we point out, we offer a handful of tangible solutions that the Kalamazoo College administration can reasonably implement.
A Review of the "Strategic Plan" and the Student Climate
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