My research interests are still developing and narrowing. I am most interested in combinatorics and discrete math.
Adviser: Dr. Steven J Miller
Group: Probability and Number Theory
Project: The Fibonacci Quilt Game
Abstract: Zeckendorf proved that every positive integer can be expressed as the sum of non-consecutive Fibonacci numbers. This theorem inspired a beautiful game, the Zeckendorf Game. Two players begin with n 1's and take turns applying rules inspired by the Fibonacci recurrence, Fn+1 = Fn + Fn-1, until a decomposition without consecutive terms is reached; whoever makes the last move wins. We look at a game resulting from a generalization of the Fibonacci numbers, the Fibonacci Quilt sequence. These arise from the two-dimensional geometric property of tiling the plane through the Fibonacci spiral. Beginning with 1 in the center, we place integers in the squares of the spiral such that each square contains the smallest positive integer that does not have a decomposition as the sum of previous terms that do not share a wall. This sequence eventually follows two recurrence relations, allowing us to construct a variation on the Zeckendorf Game, the Fibonacci Quilt Game. While some properties of the Fibonaccis are inherited by this sequence, the nature of its recurrence leads to others, such as Zeckendorf's theorem, no longer holding; it is thus of interest to investigate the generalization of the game in this setting to see which behaviors persist. We prove, similar to the original game, that this game also always terminates in a legal decomposition, give a lower bound on game lengths, show that depending on strategies the length of the game can vary and either player could win, and give a conjecture on the length of a random game.
Paper: https://arxiv.org/abs/1909.01938 - Fibonacci Quarterly Volume 58 Number 2 - pdf
Conference for New England REU - July 23rd 2019 - pdf
Young Mathematicians Conference - August 9th 2019 - pdf
Women in Mathematics in New England - September 21st 2019 - pdf
Undergraduate Mathematics Symposium - November 2nd 2019 - pdf
Joint Mathematics Meeting - January 2020 - pdf
Advisers: Dr. Robert Davis, Dr. Rajinder Mavi
Project: Combinatorial Neural Codes
Abstract: In the 1970s, O’Keefe and Dostrovsky discovered that certain neurons, called place cells, in an animal’s brain are tied to its location within its arena. A combinatorial neural code is a collection of 0/1-vectors which encode the patterns of co-firing activity among the place cells. Gross, Obatake, and Youngs have recently used techniques from toric algebra to study when a neural code is 0- 1-, or 2-inductively pierced: a property that allows one to reconstruct a Venn diagram-like planar figure that acts as a geometric schematic for the neural co-firing patterns. This article examines their work closely by focusing on a variety of classes of combinatorial neural codes. In particular, we identify universal Gröbner bases of the toric ideal for these codes.
Paper: https://arxiv.org/pdf/1904.10127 - pdf
SUMMR Conference (MSU) - July 17th 2018
MathFest - August 3rd 2018 -pdf
Joint Math Meeting - January 18th 2019- pdf