Florian Frick Mentors Mathematicians Through Geometry REU

When Carnegie Mellon University Associate Professor of Mathematical Sciences Florian Frick introduces undergraduate students to research, it can be the start of a long collaboration.

"Instead of being shown mathematics, which is what a lot of classes do for undergraduates, they see an actual research problem that's open and nobody has solved. Then they learn tools for how to approach it and actually get to solve the problem in the end, hopefully," Frick said. "This is important for them to experience — what it is like to do mathematics — especially before they decide whether they want to go to graduate school."

Frick, whose research is at the confluence of combinatorics, geometry and topology has won two National Science Foundation grants, including funders for research experiences for undergraduates in his latest grant. While he previously hosted hybrid opportunities during the height of the COVID-19 pandemic and in-person ones previously at Cornell University, 2023 was his first summer to work with visiting undergraduate researchers in person at Carnegie Mellon.

For six weeks, four students with mathematical sciences backgrounds worked with him and Zoe Wellner, a doctoral student in mathematical sciences, on geometric and topological projects. While geometry focuses on quantitative measurements, properties and relationships of shapes, topology considers qualitative properties such as the shape of the universe or the shape of a data set with hidden parameters.

Meenakshi McNamara, is a rising senior double majoring in mathematics and physics at Purdue University. She was curious if topology was a field to pursue in graduate school.

"I had previously conducted mathematical research in a couple of different areas, but I had never done any research in geometry or topology," McNamara said. "I have always been a very visual thinker and the more geometrical aspects of my prior research had been some of my favorite parts, so I was very excited about the chance to work in this area."

The students worked on two projects during their time in Pittsburgh. McNamara said the rest of the students used topological methods to study zero-sum Ramsey theory. Each student also worked on a second problem of their choosing.

"In particular, we were interested in proving a theorem showing that given enough numbers it must be possible to choose some of them so that they sum to a multiple of n, and generalizations of this theorem show a required amount of structure even in random sets. To do this we built topological spaces which encoded the information about the numbers in their shapes, and then used topological tools to show the theorems," she said.

Jacob Lehmann Duke, a rising senior at Williams College in Massachusetts met Frick via Zoom before committing to the program.

"His quiet intelligence and down to earth-ness instantly made him someone I wanted to work with. I also liked the way the REU was part of the larger science program, since it meant there would be ample social opportunities in addition to intellectual ones," Lehmann Duke said.

Students participating in summer research offered by professors in the Mellon College of Science often participate in the Summer Scholars Program (SSP), which provides opportunities to learn more about educational and career pathways in science. SSP participants also lived in the same dorm and could participate in social activities and field trips.

Lehmann Duke said they made progress on the zero-sum Ramsey theory problem.

"We came up with three interesting new topological proofs of the original theorem, each of which led to new generalizations, and we are currently working on a paper with our results, some of which we will also present at the Young Mathematicians Conference at the Ohio State University in mid-August," Lehmann Duke said.

Frick said that all of the students are working on writing papers or preparing presentations. He stays in touch with his students throughout their academic careers, oftentimes writing recommendation letters for graduate school applications.

"The vast majority have gone on to graduate school in mathematics," Frick said, including two students who are currently at Carnegie Mellon.

Hannah Park-Kaufmann, a rising senior at Bard College, said this was her third summer research experience, but the first in which she lived and worked in close proximity to other students in the same program. She and McNamara shared a room, with the other two geometry students nearby.

"With previous REUs we split up the work. Here we came together at the blackboard and worked through problems together at all times of the day," Park-Kaufmann said.

Steven Raanes, a rising senior at Vassar College, was familiar with Carnegie Mellon's Department of Mathematical Sciences before applying to work with Frick this summer. His sister, Cat, is a doctoral candidate there.

"Working with Florian was an absolute treat," Raanes said. "He was very kind and down to earth and spent lots of time just hanging out with us and making jokes, but it was also great to witness how his mind worked and the connections he was able to make that we all missed."

One of Raanes' projects for the summer involved graph coloring, such as, given some vertices connected by edges, how can you color the vertices such that no two vertices of the same color are connected by an edge?

"There has been some exciting progress in this area by translating these problems into geometric problems. The translation goes both ways, too: some geometric problems can be turned into graph coloring problems," Raanes said. "While geometry and graph coloring are both very difficult, it often happens that hard problems in geometry become easy graph coloring problems, and hard graph coloring problems become easy geometric problems. We investigated some new ways of translating problems back and forth between these areas."

Lehmann Duke said that along with a deeper understanding of topological methods he came away from the experience with an appreciation for the field and an eye on his future.

"The most useful thing I'll take away from the program is a clarified sense of what graduate research in mathematics will be like," Lehmann Duke said. "I found the grad school panels the program organized to be very helpful, and also really appreciated getting grad school advice from Florian and our grad student mentor Zoe."