Teaching the Art of Mathematics: Hertz Fellow Maria Gillespie

Mathematics isn’t just a discipline for Maria, it’s an art form.

The 2010 Hertz Fellowship recipient recently finished up her PhD at UC Berkeley in mathematics, specifically in algebraic combinatorics, which brings together group theory, representation theory and combinatorial (counting) techniques involving discrete, finite mathematical structures.

Her thesis, A Combinatorial Approach to the q, t-Symmetry Relation in Macdonald Polynomials, a version of which was recently published by the Electronic Journal of Combinatorics, takes the combinatorial approach to Macdonald polynomials, a type of symmetric function (polynomials with several variables where interchanging any variables leaves the polynomial the same), described by British mathematician Ian Macdonald in the 1970s.

“You can write down Macdonald polynomials using this amazing combinatorial formula,” Gillespie said. “Each monomial comes about by looking at a way of filling boxes with numbers and measuring certain statistics that tell you how mixed up the numbers are. It boils down some very complicated constructions in algebra involving these symmetric polynomials down to much simpler objects to study.”

Gillespie, who describes herself as a pure mathematician, recently began her National Science Foundation postdoctoral work at UC Davis after spending part of the summer with the Prove it! Math Academy. Two years ago, she helped create the camp with her father, University of Scranton math professor Ken Monks, her mother Gina, a faculty member in math at Penn State, brothers Kenneth and Keenan, and husband Bryan Gillespie.

The purpose of the two-week camp, held this year at Colorado State University in Fort Collins, Colorado, is to give high school students a place to learn rigorous proof techniques while also inspiring them to get excited about higher mathematical concepts.

“There are not a lot of programs out there that teach you how to go from doing computational problem solving to proving a theorem,” Gillespie said. “That’s a transition that kids either struggle to make and then make, or have trouble with it and end up not doing math in the future. It’s a transition that we need to focus on more. There are so many techniques they need to know that aren’t taught in school.”

Teaching the wonders of math has become a passion for Gillespie, who has been an instructor at various camps and math circles such as the Berkeley Math Circle. In grade school, bored by memorization and a lack of challenge, she was convinced she hated math. But after being homeschooled in the subject by her father, Gillespie grew to love it, taking part in numerous competitions and learning from online programs such as the Art of Problem Solving. She attended MIT planning to be an astrophysicist, but the elegance of math kept luring her back.
“I do math for the beauty of it, and if it happens to be applicable, that’s great,” Gillespie said. “It’s like music, it’s an art form in some sense… You’re just driven by the human desire to always explore, to see what’s over the next mountain pass.”

At UC Davis, Gillespie will focus on continuing her research, particularly in trying to find combinatorial proofs for the Macdonald positivity and n! (factorial) conjectures, and expanding her doctoral thesis in the more algebraic and geometric directions of representation theory and Schubert calculus. In the long-term, she wants to become a professor, and help students fill gaps in their math education.

“I can’t imagine not teaching math,” Gillespie said. “You can’t just do all this research and write a paper that five people in the world can understand, and not tell anyone else about it. What’s the fun in that? (Teaching) helps distill my ideas from the higher level and not only clarify what I’m doing for myself but spread the joy of mathematics to lots of other people.”