Summary

Over the summer of 2024, Dr. Arvind Kumar designed a five-week undergraduate research project titled “Binomial Edge Ideals of crown graphs.” he supervised undergraduate student Joshua Pomeroy and graduate student Le Tran. During these five weeks, they focused on studying graphs to determine when the projective dimension of the quotient of the binomial edge ideals matches the big height of that ideal. Additionally, they explored the Vasconcelos number of binomial edge ideals for cycles and crown graphs and provided a proof of a conjecture related to the Vasconcelos number of binomial edge ideals for cycles. Paper titled: "Binomial Edge Ideals of Crown Graphs "

 

"I started this research project looking to learn what math research looked like. Even though I was unfamiliar with any graph theory concepts or much of algebra, I learned a lot and could understand and prove many of the results in the
paper. I became proficient in reading other papers and integrating their results to go further, and I enjoyed the
challenge of proving results that were not known. It was great to gain experience in math research and further my
understanding of mathematics. I would highly recommend a research project to anyone who's looking to go further in
Mathematics or learn about research."

Joshua Pomeroy

 

"Engaging in this summer project has been a wonderful opportunity and a deeply rewarding experience. I have gained
a comprehensive understanding of commutative algebra, especially through the lens of combinatorics. This journey
has not only enhanced my appreciation for the elegance of mathematics but also sharpened my analytical and
problem-solving skills. Working closely with a postdoc and an undergraduate as a graduate student has provided
valuable mentorship and fostered a sense of community. Each challenge I encountered contributed to my personal
growth, teaching me resilience and perseverance. For students passionate about mathematics, this journey is
incredibly worthwhile."

Le Tran