# Mathematics undergraduates gain intensive research experience during summer

**08-09-2016**

**By: **Tim Brouk

*Purdue Mathematics Prof. Edray Goins (left) talks research with University of Rochester visiting student Dionel Jaime (middle) and Mathematics graduate student Mark Pengitore.*

Several visiting mathematics students were PRiME-d and ready to dive into research over the summer.

Led by professors Edray Goins and Jonathon Peterson, the group of visiting math majors from universities as far away as California, New York and Florida joined a few of their Purdue counterparts for the Purdue Research in Mathematics Experience (PRiME). The research project was one of Purdue’s successful Research Experience for Undergraduates (REU) sessions. The projects were presented at a conference in Indianapolis in July as well as showing at Purdue. In August, the students sat in at the Mathematical Association of America’s national meeting (MAA MathFest) in Columbus, Ohio. The research was a ticket into the mathematics community -- where mathematicians interact and exchange ideas worldwide.

“Most of us want to go to grad school and be actual mathematicians,” said Dionel Jaime, a junior from University of Rochester, “and that entails research. This is a great way to start.”

Purdue Mathematics REU’s are intra-disciplinary; they touch on many mathematic sub disciplines. Number theory, algebraic geometry, complex analysis, topology, real analysis, probability theory and more were explored this summer.

Summer research projects are common in the College of Science and the Department of Mathematics is a popular home for such endeavors. Life sciences research grabs many headlines but mathematics research is just as crucial.

However, students must be experts in order to help make research breakthroughs in a mathematical field. The toughest classes must be completed before the undergrads are ready to research.

“You’re attempting to do math that has never been done before,” said Purdue math graduate student Mark Pengitore, who assisted with Goins’ research project.

Eight visiting students were divided into two groups. Goins’ group focused their efforts on graphs on elliptic curves, which can be drawn on the geometrical doughnut-like torus while Peterson’s students studied “excited random walks,” a model in probability theory of self-interacting random motion.

*Notes by Caitlin Lienkaemper, a visiting researcher from Harvey Mudd College, explore elliptical curves.*

**Learning curves**

Goins’ students created a database of toroidal graphs associated to elliptic curves. Their goal was to find as many examples of “dessin d’enfants” (“children’s drawings” in French) and “Belyi pairs” as possible. The inspiration for the project was of contemporary interest. The graphs are products of a 20^{th} century master.

One of the most revered mathematicians of this past century, Alexander Grothendieck was looking at these natural graphs that show up. He called these graphs “dessin d'enfants” because he thought the scribblings were deceptively simple yet contained a lot of information about the deep mathematics underlying the objects at hand, Goins said.

The graphs are three dimensional and go way beyond the high school ink and paper y=x^{2 }graphs most people that aren’t mathematicians are more accustomed to. They look at vertices and edges. Goins’ students set out to map and explain these complex graphs.

Goins and Peterson started their students off with intensive lectures six hours a day – Monday through Friday -- for the first two weeks. Welcome to the rigors of Purdue Mathematics.

Then it was time to roll up the sleeves and get into the technical aspects of mathematical research. Learning more about elliptic curves took much coding and simulation. The work was divided up and the results were presented during a statewide conference for other mathematics REU’s held at Indiana University-Purdue University Indianapolis as well as a Purdue Mathematics panel on Aug. 2.

**Random research**

The “excited random walk” model that Peterson’s students studied also involved learning some advanced mathematics beyond what the undergraduate typically encounters. Random walks are a classical model in probability theory for random motion, and excited random walks are a generalization which introduces dependencies that make the mathematical analysis much more difficult.

“Imagine you’re standing on one of the squares of an infinite sidewalk,” began Owen Levin, a junior from University of Minnesota, “you have some probability of moving forward to the next square or backwards to the previous square. Your path that you take, whether you go forward or backward, that transition probability depends on how you got to the square you’re on.”

The focus of the project was in studying the speed of the excited random walk – i.e., quantifying the rate at which the pedestrian moves along the sidewalk. Giving a precise answer to this question is made more difficult by the complex dependence of the model. Unlike a sequence of coin flips where the outcome of one flip has no effect on the outcome of the other flips, all the previous steps of the walk influence the steps of excited random walks.

Levin described the research process in probability as a lot of trial and error before the “glimmer of hope” that led to positive results. Peterson was pleasantly surprised at the estimates for the walker’s limiting speed calculated for the project.

“The estimates were much, much better than I had ever hoped,” he said. “I’m proud of how hard they worked. I’m especially proud of how my group worked together. They came from five different universities and never knew each other before. They had to be able to work together really quickly.”

The students put their results in a paper and are hoping to get it published – another tremendous opportunity resulting from the Research Experience for Undergraduates.

Goins believes mathematics research experience should revolve around a current or contemporary problem or concept. Some mathematicians may think undergraduates aren’t ready for research or they should just read a book on well-known mathematics and write a paper on it. That’s fine but that doesn’t go beyond the level of just taking notes in class. Research experience gives the students more.

“The concept of an REU is to try to make it as contemporary as possible,” Goins explained. “We do new, novel research -- something that is cutting edge or hasn’t been though about before. We want to do something that you can’t do in class.

“It’s really just learning more about the whole mathematical culture. What does it mean to do research? It’s not just sitting in a room or working on a computer. It’s actually speaking about it and presenting it to others.”

*Purdue Mathematics Prof. Jonathon Peterson (left) checks University of Minnesota junior Owen Levin's probability research during the Purdue Research in Mathematics Experience. *

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