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Marquis Scholar Brian Kronenthal ’07 (Yardley, Pa.) doesn’t know if it’s possible to solve the research problem he’s been given, but he’s doing his best to find out.
Conducted under the guidance of Derek Smith, assistant professor of mathematics, Kronenthal’s EXCEL research involves finding an answer to a problem rooted in linear algebra and vector space.
In Lafayette’s distinctive EXCEL Scholars program, students conduct research with faculty while earning a stipend. The program has helped to make Lafayette a national leader in undergraduate research. Many of the more than 160 students who participate each year share their work through articles in academic journals and/or conference presentations.
“What I’m doing is an extension of research done by Professor Smith and another student, which was tested in coordination with a professor from New Mexico State University,” Kronenthal says. “One of the results that they came up with was a result for a value that was greater than some number, say M. If M is greater than some number, it proves a certain result. Professor Smith is interested in knowing if M is less than some number, what does that mean?”
Smith says if anyone is able to solve the problem, it will be Kronenthal.
“He is exceptionally careful and does not take anything for granted,” says Smith. “What impresses me with him as much as any student I’ve worked with is just how important the details are. In fact, just the other day I received proofs of the paper that’s about to be published and the first thing I said when I saw him that day was, ‘Brian, I just received the proof, this is the last chance to make corrections before it goes to print.’ I can think of no one on the project better suited to find any possible problems that could pop up.”
No doubt the work is complex and difficult, says Kronenthal, a mathematics major and French minor.
“The biggest challenge has been understanding this high-level math. I have to thoroughly understand the concepts before I can begin working on the problem,” he says.
Noting that it can be difficult at times to balance conducting research and completing course work, Kronenthal spent the first half of the semester solely reading the results of work already completed on the problem. But the most demanding aspect of the work is determining if he will be able to find some way to figure out a previously unsolved problem.
“In some ways, this is less stressful than solving a problem that a professor expects you to get,” he explains. “It is much harder and you can’t go to anyone for help, but I guess I’m going to just try it and hopefully it will come out for me.”
The research is not only teaching Kronenthal more about high-level math, but about problem-solving in general.
“The main skill that he’s acquiring is an ability to approach a problem with a solution that is unknown,” Smith says. “It’s a problem that isn’t guided, doesn’t have a directed set of examples, but one where the question that he’s trying answer is whether it’s possible to improve a known result into something that may be better. The answer may be no, and if so, you have a little sense of loss but you know you’ve given it your best shot. If you succeed, the result is known and you’ve made a contribution to the field of math and you’ve been able to say something that others have not been able to prove before.”
For a student of math, these sorts of problems are necessary to remain sharp.
“It’s one thing to study via course work and another to study via an open-ended challenging problem – the two examples are completely different,” Smith adds. “You need both to sustain you.”
Even if it turns out that the problem is unsolvable, Kronenthal’s time as an EXCEL Scholar will have been well spent.
“He’s already been a success with it,” Smith says. “Success on this project is not gauged, nor should any math research be gauged, on the ability to solve a problem. What’s been learned along the way and throughout this project has filled some holes in his knowledge of linear algebra and that new knowledge will pay off down the road whether in coursework or in future research.”
Kronenthal is president of Math Club, a peer tutor, a former writer for The Lafayette, and a participant in intramural tennis and golf. He graduated high school from George School.
Chosen from among Lafayette’s most promising applicants, Marquis Scholars like Kronenthal receive special financial aid and distinctive educational experiences and benefits, including a three-week, Lafayette-funded study-abroad course during January’s interim session between semesters. Marquis Scholars also participate in cultural activities in major cities and on campus, and mentoring programs with Lafayette faculty.
As a national leader in undergraduate research, Lafayette sends one of the largest contingents to the National Conference on Undergraduate Research each year. Thirty-nine students were accepted to present their research at this year’s conference.