2025 Villanova-Puerto Rico Research Retreat (VPR^3)

Description

Villanova-Puerto Rico Research Retreat (VPR^3) is an 8-week summer research program (June 2nd - July 25th) at Villanova University under the direction of Dr. Alexander Diaz-Lopez. The goal of VPR^3 is to provide a math research opportunity for students from the University of Puerto Rico and from Villanova University, connect them to the broader math community, and provide them with the necessary skills and knowledge to be successful in a graduate program if they choose to enter one.  The deadline to apply is March 15, 2025.

Expected Background:

• At least one proof-based mathematics course such as: Foundation of Mathematics (Intro to Proof), Linear Algebra, Abstract Algebra, Advanced Calculus/Real Analysis, proof-based Discrete Math, Combinatorics, or any other proof-based course.

• Desire to work on mathematics research problems.

Expectations:

• Travel to Villanova University (we provide travel funding, room, meals, and a $4,000 stipend) from June 1st to July 26th, 2025.

• Treat every participant and faculty with respect.

• Attend daily meetings, engage with fellow participants, and work on the project.

• Participate in Saturday outings meant to build community among the participants. 

• Attend a series of colloquium talks.

• Attend workshops aimed at developing skills and techniques needed for research careers in the mathematical sciences.

• Give a presentation and write a technical report at the end of the project.

After the summer, participants will have the opportunity to attend a national math meeting to present their research findings. 

Projects:

• Project 1: Arithmetical Structures on Graphs

  • An arithmetical structure on a graph is a integer labeling of its vertices such that the label at each vertex divides the sum of the labels of its neighbor vertices. Recently, a lot of work has been done counting the number of arithmetical structures on a given graph (such as path graphs, cycle graphs, complete graphs, star graphs, and cycles with a double edge), studying their critical groups, and their spectral radius. In this project, we will look at some of these problems for different families of graphs that have not yet been studied. No background is assumed other than a proof-based course.

• Project 2: Metrics on subsets of permutations

  • Metrics are functions that allow us to measure the difference/distance between objects based on certain criteria. When our objects are permutations, there are several ways to measure distance (Hamming metric, ell-infinity metric, Kendall-tau metric, among many others). In this project, we will study these and many other metrics on subsets of permutations based on certain properties they have in common.  No background is assumed other than a proof-based course.

Calendar:

• Weeks 1 and 2:

  • The program will start Monday June 2nd with introductions, NSF-mandated trainings, and a campus tour.

  • The remaining of week 1 and beginning of week 2 will be devoted to learning the background material needed to work on the projects.

  • The end of week 2 will be devoted to a literature search about what has been done in the past regarding these projects.

• Weeks 3-7:

  • Weeks 3-7 will mostly be devoted to working on the research problems in two groups of three students each.

• Week 8:

  • Week 8 will be devoted to finishing a written report with all the work done during the summer and preparing for a final presentation on Friday July 25, 2025.

Outings:

• Every Saturday, we will go to community building outings. Potential outings are: Amish Tour and Dinner, Philadelphia Tour, Valley Forge Park, Adventure Aquarium, Longwood Gardens, Phillies game, Ocean City Boardwalk, Philly Museum of Arts, among others. Here are some pictures from the 2024 VPR^3 program.

Participants from 2024: Chloe Abo (Villanova), Juliana Castrodad (UPR Mayagüez), Diego Estrada (UPR, Rio Piedras), Jean Garcia  (UPR Mayagüez), Katie Noonan (Villanova), and Katie Shattuck (Villanova).