Articles and Projects

Projects, both personal and academic, that I have completed during my time at Brown.

Maximal Number of k-cycles in a d-regular Graph

We construct the d-regular graph with the maximum number of k-
cycles for k = 5, 6.
Using a M ̈obius inversion relation between graph homomorphism numbers and injective homomorphism numbers, we reframe the problem as a
continuous optimization problem on the eigenvalues of G by leveraging
the fact that the number of closed walks of length k is tr(A^k ).
For k = 5 and d > 3, we show G is a collection of disjoint K_{d+1} graphs.
For d = 3, disjoint Petersen graphs emerge. For k = 6 and d large enough,
G consists of copies of K_{d,d}.
Additionally, we introduce and give formulas for non-backtracking homomorphism numbers and backtracking homomorphism numbers, respectively. Moreover, we find the d-regular graph on n vertices with the most non-backtracking closed walks of length k by considering an optimization problem on the non-backtracking spectrum of G. We also solve the same problem, but for backtracking closed walks. Lastly, a corollary gives formulas for the number of 4-cycles and 5-cycles of a graph with respect to its spectrum, regardless of regularity.

We conjecture that for odd k and sufficiently large d, the optimal G is a collection of K_{d+1}, while for even k with sufficiently large d, the optimal G consists of K_{d,d}

Non-universality in Clustered Ballistic Annihilation:
Electronic Communications in Probability 28: 1-12 (2023)

Summer 2022

I did research at the NYC Discrete Mathematics REU at Baruch College. My mentor was Prof. Matthew Junge. Cynthia Rivera Sanchez and Lily Reeves also worked with us.

Abstract: Ballistic annihilation is a stochastic interacting particle system. Infinitely many particles with randomly assigned velocities move across the real line and mutually annihilate upon contact. We introduce a variant with superimposed clusters of multiple stationary particles. Our main finding is that the critical initial cluster density to ensure species survival depends on both the mean and variance of the cluster size. In other words, particle clustering affects particle survival. This result contrasts with recent ballistic annihilation universality findings with respect to particle spacings. A corollary of our theorem resolves an open question for coalescing ballistic annihilation.

I have given talks on this topic at Duke University's New Connections in Mathematics conference, Brown University's Undergraduate Math Colloquium, and Brown's Symposium for Undergraduates in the Mathematical Sciences (SUMS) .

Partially supported by NSF DMS #2115936

Paper

Senators Project

Fall 2020

I was part of Brown's Applied Math Directed Reading Program (DRP) for the Fall 2020 semester. My project was on the Singular Value Decomposition and Its Applications.

I used the SVD to analyze Puerto Rican Senators' voting patterns. The image to the left is an example of how senators of the same political party (color) vote similarly.

I continued working on this project in 2021 as part of a project for my Computational Linear Algebra course and plan to repeat the analyses for the next term.

Mathematical analysis on Puerto Rican politics is rarely available to the general public. This makes this project meaningful to me.

In 2023, I gave a talk about this project at Brown University's Symposium for Undergraduates in the Mathematical Sciences (SUMS).

Video of the presentation from 2020 is available in the videos section of the website.

Proposal

2017-2020 controversial SVD_edited_edited.jpg

Geometry of Gerrymandering

Spring 2021

This paper discusses fair bisectors, and fair partitions. A proof of how every convex polygon of n sides can be partitioned into regions of equal area and perimeter.

Different definitions of compactness for a region are analyzed, compared, and applied to voting districts of Puerto Rico using a Python script.

I propose a method for finding unfairly gerrymandered districts by measuring compactness.

Although this data is not highly accurate, it is the only study of its kind. and cannot be
signicantly improved without accessible measurements and data of the districts.

Paper

Schwarz-Christoffel Formula

Fall 2021

Slides for my talk on the Schwarz-Christoffel Formula. I discuss conformal mapping, the Riemann Mapping Theorem, polygonal domains, introduce the formula, provide some examples and applications, and have a computer illustration.
The video can be seen in the videos section of this website.

Slides