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Research that mathematics major Jordan Tirrell ’08 (West Grove, Pa.) has performed over the last few years with Cliff Reiter, professor of mathematics, has led to presentations at numerous national conferences and publication in an academic journal.
The research has uncovered a proof for generalizing Pythagorean triples in multiple dimensions. He made this finding as Reiter’s EXCEL scholar during the spring and summer of 2006.
Tirrell and Reiter published an article about their findings titled, “Matrix Generation of the Diophantine Solutions to Sums of 3 ≤ n ≤ 9 Squares that are Square,” in the JP Journal of Algebra, Number Theory and Applications (June, 2007).
Tirrell has presented his research at the annual Joint Mathematics Meeting of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA) on Jan. 5-8, 2007, in New Orleans, La., and Jan. 6-9, 2008, in San Diego, Calif. He will also present at the AMS Sectional Meeting on May 3-4 in Claremont, Calif.
Their current research is an extension of another project they worked on in 2005 using integers and sums of squares in an attempt to create a perfect parallelepiped. The research resulted in a proof for three dimensions and their findings were published in the same journal.
The goal of their recent work was “to generalize our previous work to higher dimensions. Specifically, we were aiming at four dimensions,” explains Tirrell.
Tirrell performed computer searches to get ideas for what matrix generators might work for their theorem, and, after testing numerous possibilities, he found one in which all the solutions in dimensions up to nine can be generated by using a small number of matrices.
“Within only a couple hours, I had a short proof that worked for three to nine dimensions (our previous proof for three dimensions was much longer and more complicated),” he continues.
After such a significant finding, Reiter decided that Tirrell’s name should appear on the article first, as opposed to standard alphabetical order.
“The theorem really is Jordan’s in the sense that he found the key identity that causes the argument to work so long as the dimension is not too high,” says Reiter. “This identity gives an algebraic structure to integer solutions to equations. The hope would be that the algebraic structure could give insights into forms of possible solutions, especially in other contexts, such as the perfect parallelepiped problem. His idea is a very nice independent insight and resulted in covering more dimensions than I expected could be solved.”
Tirrell is grateful for the prospects that a small, liberal arts college like Lafayette has to offer, such as being able to work so closely with esteemed scholars like Reiter.
“Professor Reiter has done a ton of research with students and it shows,” says Tirrell. “He is an excellent research advisor and knew what direction our research should go. I am very fortunate to have had the opportunity to work with someone so experienced in undergraduate research.”
Upon graduating, Tirrell hopes to enter a one-year masters program at the University of Cambridge, or immediately begin pursuing a PhD in mathematics. He plans to become a professor.
Tirrell was a recipient of the national Goldwater Scholarship in 2006. Awarded for academic merit, the Goldwater is the premier undergraduate award of its type in the fields of mathematics, science, and engineering.
Along with his EXCEL research, Tirrell studied abroad in Budapest, Hungary, in the fall of 2007 and participated in the Research Experience for Undergraduates (REU) program at Penn State University in the summer of 2006 and again at Claremont University in the summer of 2007. He has participated in several math competitions sponsored by the Mathematical Association of America and serves as secretary for Pi Mu Epsilon and Vice President of the College’s Math Club. Tirrell is a teaching assistant for the math department, a math lab proctor, and a peer tutor as well as a resident assistant and tour guide for the Admissions Office. He is a member of the Tennis Club, Table Tennis Club, Roller Hockey Team, and Calculus Cavalry.