Dr. Juan Gil and Jessica Tomasko

Listen to the Math

Juan Gil, professor of mathematics, and senior math major Jessica Tomasko have teamed up for a number of research projects resulting in published papers and, in one case, an undergraduate award for Tomasko.
By: Therese Boyd

Two heads, it’s said, are better than one. Solving a problem is often easier if two people can talk it over. Applying that aphorism are Juan Gil, professor of mathematics at Penn State Altoona, and senior math major Jessica Tomasko. The two have teamed up for a number of research projects resulting in published papers and, in one case, an undergraduate award for Tomasko.

Gil says he saw Tomasko’s potential when she took two of his calculus courses. At the time, she was intending to be a biomedical engineering major, but “it was clear to me she had some talent for mathematics. I mentioned to her, ‘it appears to me that you are a math major waiting to happen.’” But he didn’t ask her then to be a student researcher. Gil explains his approach is “to offer to students the opportunity, and if they want to do some research with me, they should let me know.”

Tomasko was considering research but hadn’t yet committed to anything. “One of my friends was doing research with Prof. Michael Weiner,” she says, “and so I was hearing about research from his perspective. It intrigued me.” Once she was committed to the idea of research, she still had to decide what discipline. “I talked to a chemistry professor, but the chemistry research didn’t interest me.” But mathematics did.

Before Tomasko came on board, Gil had a different approach to undergraduate research that he says “wasn’t that successful. So, this time, I did it differently. We started building some basic combinatorics skills before going into research-level material.” Working with Tomasko, “we explored Fibonacci colored compositions, worked out some examples, and found some things that were new.” Their results were published online in 2021 as “Fibonacci Colored Compositions and Applications.”

An opportunity to showcase that success came with the Undergraduate Research Poster Exhibition hosted by Penn State’s Eberly College of Science in October 2021. With just a week to work, Gil says, Tomasko created a poster, presented at the exhibition, and won first prize. He is clearly not exaggerating when he says, “She is very disciplined.”

During the summer of 2021 Gil and Tomasko changed their research focus. “We were exploring pattern-avoiding permutations,” Gil says. Tomasko’s availability to work with him was in part because she was in the inaugural class of students to benefit from the Altoona Summer Undergraduate Research Fellowship program. “She received two grants to focus on research,” Gil says. “It was a very, very intensive summer. We met every other day—it was work, work, work.”

Tomasko describes the math research process: “With math you come in with knowledge, things you know and have learned, asking questions that don’t have answers or looking for a new perspective. There’s not really a set way to do math research.” She gives an example of their work: “One common thing we’ve done is, if you have two sets of objects, and they’re counted by the same numbers, you want to connect these objects through a bijective map.”

“These topics can be complicated,” Gil admits. “Usually what happens is that you start working on an interesting question, you follow some intuitive path, and hope that the math will show itself.” For their summer project, he says, “We wanted to study Boolean permutations, but within a few weeks, we found another direction. You have to listen to the math. We went in that direction and ended up exploring pattern avoidance for Grassmannian permutations. We found a lot of different things.” So much so that “I decided to split our findings into two parts.” The first paper of that two-part project, entitled “Restricted Grassmannian Permutations,” was submitted for publication in December 2021.

“The second part of the summer project had a natural division into two types of patterns,” he continues. “Jessica will write her honors thesis on one of them. In the meantime, we want to work together to formalize and write down what is left. My hope is that by the end of the spring, we can combine all the remaining pieces and make them into a third paper.”

Gil feels strongly that, even though Tomasko is a student, “it’s important to me that she owns the work. I want to see her as a collaborator, not a follower. If you work with a sophomore, it’s okay to expect less independence. Because she’s a senior, I wanted her to be able to say, ‘I want to go this way.’” He knows Tomasko is up to the challenge. “Our first research paper had essentially four sections. One of the two main sections she did completely on her own. She was the one who solved the problem.”

Tomasko is quite modest about her participation. She describes their collaboration as “we sit and talk. We read papers. A lot of it is sitting down in front of a chalkboard and talking, trying to interpret or understand something—other people’s interpretations of things, what they thought, and developing our own. Having this experience showed me how fun and interesting research can be, and it confirmed that it’s something I want to pursue in the future.” She has no regrets about her chosen research path: “I’m glad I got into it.”