The banter could be heard up and down the halls. It was perhaps the only sound emanating from the third floor of Villanova's St. Augustine Center on a hot, hazy July morning just before 9 a.m.

Its source was six undergraduate students - three from Villanova University and three from the University of Puerto Rico. One of those students, Juliana Castrodad-García, had just defeated Alexander Diaz-Lopez, PhD, associate professor of Mathematics and Statistics, in Rummikub. She and her peers were making sure he knew it wasn't the first time.

The friendly math-based competition was how the student participants of the Villanova-Puerto Rico Research Retreat (VPR^3) warmed up for a day of mathematics a bit more complicated in nature: research on combinatorics, which deals with studying finite structures.

The research was the focal point for the inaugural VPR^3, an eight-week summer program in which the six students work under the guidance of Dr. Diaz-Lopez, where they are provided valuable research opportunities and benefit from assorted colloquia and additional connections to the broader math community. Dr. Diaz-Lopez started the program himself with a National Science Foundation Grant, underscoring his commitment to undergraduate math research opportunities. He had already co-founded The Community of Mathematicians and Statisticians Exploring Research (Co-MASTER) program on campus, facilitating research opportunity by pairing eager students with faculty researchers engaging in various mathematics and statistics projects. This was his next project.

"Undergrad research is one of the high-impact practices that is known to help students have a deeper learning of their material and prepare them beyond their college career and into their workforce," said Dr. Diaz-Lopez. "Working on challenging problems and developing useful skills are valued quite a bit by both academia and industry. I completed two summer research programs myself as an undergraduate that were crucial for me in deciding that I wanted to continue working in a research career in mathematics."

His experiences were also crucial in shaping VPR^3.

A Program That Hits Home

Dr. Diaz-Lopez, who hails from Puerto Rico, spent his childhood on the east side of the island, in the town of San Lorenzo. He remained there for college, pursuing an undergraduate degree in mathematics from the University of Puerto Rico, Mayagüez.

While studying, he discovered that although opportunities to advance his mathematics education existed beyond the classroom, being geographically isolated on a small island made it difficult to access them.

"My first year and a half at school, I didn't know what was out there," he said. "I was doing my work and seeing my faculty, but I was completely unaware of all the different opportunities, or even that I could come to the United States for grad school."

That was, until a faculty mentor guided him toward potential summer programs in the U.S. He participated in two of them - one at Williams College, and the other at the Simons Laufer Mathematical Sciences Institute (SLMath) in Berkeley, Calif. They opened his eyes to the different pathways to which mathematics could lead, helping him ultimately decide what he wanted to pursue.

"The mentors and research supervisors I met there were really encouraging," he said. "You need to have someone that says, 'Hey, take a look at this. Go try that.'"

Years later, after obtaining his PhD from the University of Notre Dame and joining Villanova's Department of Mathematics and Statistics, his mission was to provide similar opportunities and guidance for students like him. So, after obtaining the NSF grant, he carefully designed the VPR^3 program based off his lived experiences, down to the granular details. Dr. Diaz-Lopez then visited the University of Puerto Rico twice, building relationships and trust with faculty and sharing the opportunity so they could pass along the application information to prospective students.

"It was easier for me to set up because I lived it [as a student]," Dr. Diaz-Lopez said. "I went to a summer program and I remember thinking, 'Oh shoot, I need a hoodie here in California.' Even packing for a whole summer, bringing things like sheets - I wanted to ensure that all of these details, however minor, were taken care of and for folks to realize I'm someone who cares about the students and runs a program that will really benefit them."

The program was designed to include three students from the University of Puerto Rico and three from Villanova. They would lodge together in one of Villanova's residence halls and work in two mixed teams of three to help build rapport and comfortability. The bulk of the workweek was for their research projects, starting off with a few weeks of background on conducting and communicating mathematics research, as it was a new endeavor for all of the participants.

"I didn't really know what to expect," said Katie Shattuck, '26 CLAS. "I was kind of scared that it was going to be hard, or I wasn't going to get it. But we spent a lot of time on background, so I felt like I had a good foundation when the research began."

They caught on quickly, with one group diving into a project related to graphs and matrices and the other on counting arithmetical structures. Dr. Diaz-Lopez provided the framework, but the "motivation was all within them," he said.

As the weeks passed, both Dr. Diaz-Lopez and the students saw marked change in the confidence of their work. He expects two research papers to result from their summer studies.

"They saw tremendous growth in terms of being able to write and address mathematics," he said. "The way you do that requires practice and effort, and it's so different in a big research project where you might not know the answer, as opposed to a problem solved in class."

"We're still developing the intuition, and I feel like [Dr. Diaz-Lopez's] support made it a lot easier for us to go through certain ideas and just work on them ourselves," said Diego A. Estrada-Rivera.

Once per week, the participants also attended colloquia that focused on an array of mathematical research - a component of the program each student cited as a top highlight. These talks ranged in topic from theory to research on origami and were intended for the students to be able to explore various avenues of mathematics and expand their relationships in the field.

Estrada-Rivera, for instance, studies math related to computer programs and coding in Puerto Rico. But he was particularly struck by a colloquium on topology.

"That was my favorite, because it really allowed me to see a different perspective and how similar themes are applied across various fields."

"What I've learned in this whole program, from the research to the colloquia, has helped me come to know what I like, what I don't like and what I want to do in my future," said Castrodad-García.

More Than Mathematics

There are, of course, benefits to Dr. Diaz-Lopez's program beyond mathematics and networking. To him, one of the most important is to hopefully increase the representation of Latine individuals - and more broadly, the BIPOC community - in mathematics.

"Representation is one of those things that is important when somebody joins any specific community," he said. "To be able to see someone like yourself fosters feelings of 'I can achieve this, I can do this,' particularly in mathematics, because traditionally, it's a field in which BIPOC have been underrepresented. That makes it harder as a student to see yourself in that community.

"Now that I have the opportunity to facilitate programs like this, I want to show them that beyond work, practice and mentorship, there is nothing overtly special that got me here. If I can do it, you can too. And regardless of your end goal, there are mentors and people willing to guide you. This is something that I am very passionate about."

He has been for some time, even early on in his academic career. In 2016, Dr. Diaz-Lopez co-founded Lathisms, a resource site showcasing the contributions of Hispanic and Latine mathematicians during Hispanic Heritage Month.

This, he feels, is a more "concrete way" to help increase representation within that same community and give back specifically to young Puerto Rican students interested in math, the way that his mentors did for him.

"Working with students is a different way of achieving that same goal of increasing representation, and one that I've wanted to engage in for a long time," he said.

Perhaps hand-in-hand with increasing representation is building community. An appropriate saying in this case might be "there is power in numbers."

From the onset, even in the design of the program and the split of the research groups, there was concerted effort to help foster relationships. Even the quick little games of Rummikub add up to something after eight weeks. It was all part of the calculus of the program.

On Saturdays, the group would get together for an outing. They visited Center City, went bowling, wandered Adventure Aquarium and walked Longwood Gardens. Many of the students mentioned one particular outing - a visit to Lancaster County - as their favorite, because it exposed them to a new culture.

"I think that trip had an underlying importance to this program," Diaz-Lopez said. "It's easy to have all of stereotypes about peoples and communities. And when you meet someone and you have that connection you realize that while we all have our own cultural differences, we're all people."

The sum of the components, by all accounts, led to a meaningful first iteration of the retreat. Estrada-Rivera looks forward to applying knowledge he learned over the summer to his work at school and being able to look at smaller details with a different lens. Castrodad-Garcia highlighted the freedom Dr. Diaz-Lopez gave them, and how it helped with her career path clarity. For Dr. Diaz-Lopez, who is already considering how he can improve the program, a sense of pride stems from something each of the students told him during their interviews at the conclusion of the retreat.

"I don't want this to end."

"That made me really happy," he said. "But the success of the program is because the students came in wanting to do this, with really strong skills and work ethic. The talent is always there. It's just a matter of providing the opportunity."