Stacks of papers clutter Becca Feiden’s room. Using recycled Teach for America flyers for scrap paper, Feiden has spent the past eight months brainstorming and sketching proofs for her thesis in mathematics. With little use for a carrel, Feiden worked all over campus, transporting her papers in a big green folder.

At the beginning of the year, Feiden had no intention of writing a senior thesis, let alone the 41-page thesis entitled “Once-Reinforced Random Walks on Z and The Doubly Infinite Ladder” that she ended up submitting on April 14. However, in order to get honors in mathematics, the department requires that student either write a thesis and then present on it, or else take graduate classes while completing independent research.

When her former and current mathematics professor, Michael Keane, encouraged her to write a thesis, Feiden reconsidered.

“It was a great honor [that he asked], and an opportunity that I couldn’t pass up,” she said. “It kind of came out of nowhere, but I’m very happy I did it.”

Feiden’s thesis includes three proofs, all centered on the probability concept of random walks. When explaining her thesis to others, Feiden tells them to imagine they are lost in New York City and have to make a random choice at every intersection—they can either walk forwards, backwards, or turn right or left.

Now, Feiden says, imagine the city grid is infinite. The question she poses is, “What is the probability you will return to your starting place?”

Feiden’s thesis centers on reinforced random walks, which are random walks that account for one’s memory of where they have been, assuming this influences their decision of where to walk. She explored these concepts in one, two, and three-dimensional spaces.

Feiden explained that her work is very applicable to the outside world, including modeling tumor growth, polymer shapes, and learning behaviors.

“Clearly humans don’t necessarily act in the same way as random walkers,” she said. “But in some ways it does simulate the motion of particles in space.”

Unlike humanities theses, which require much library-based research, Feiden’s only formal research was for her introduction. The bulk of her work, however, was done old-fashioned trial-and-error on scrap papers, as she spent five hours per week working in the beginning and up to 45 hours during the last week.

“Most of those ideas on those papers, I abandoned as soon as I wrote them,” she said. “It was a lot of writing down things then realizing there’s a better way to approach them.”

Many of the changes Feiden made to her proofs involved explaining her work more explicitly—a skill she greatly improved through her work.

“I would look for flaws in my arguments and try to make them sound,” she said. “If it was a solid idea, I would then have to go back and explicitly describe an enormous amount of small things.”

This process of fleshing out her ideas in utmost detail was one of the most challenging and frustrating aspects of Feiden’s process.

“With these tiny little things, I would think, ’This obviously makes sense. Why do I have to write it down?’” she said. “But I had to write it down.”

As Feiden constantly re-worked her proofs, her thesis began to evolve on its own.

“The project continued to change as I worked on it,” she said. “I wrote every chapter of my thesis at least 10 times, easily.”

In fact, during the last week of her project, Feiden e-mailed her so-called “final draft” to her advisor six times.

Throughout the process, Feiden and Keane met twice a week, walking through the proofs together.

“It was an amazing experience to work with somebody who has been in math so long and is just so incredibly smart and creative,” she said of Keane. “He could be an outside objective eye on what I had done and its validity.”

Similarly, Keane praised Feiden for all her hard work.

“She is not only very talented as a young mathematician, but has a less common talent of being able to communicate her ideas to an audience, giving them pleasure in the process,” he said. “In my opinion she would do well to adopt mathematics as a career.”

Although Feiden is unsure if she will pursue math in the future, working on her thesis has certainly made her consider it.

“This experience has definitely made me think seriously about continuing to study mathematics,” she said. “It’s very fun to have a problem in front of you and pick it apart every little piece. It’s like a great logic problem.”

While Feiden admitted that there were a few times she wanted to give up, not once did she seriously consider not finishing her thesis.

“Sometimes I would think, ’I have got to stop this, this is absurd,’” she said. “You feel like your brain is walking in circles. All in all, it was a really positive experience with a couple of horrific nights.”