Here's a question for our Spire readers: which primes are the sum of two rational cubes, p = x3 + y3? For example, 7 = 2^3 + (-1)^3, 13 = (7/3)^3 + (2/3)^3, ...
If you aren't sure, just ask John Voight, one of the newest assistant professors in the Department of Mathematics and Statistics.
"Methods for finding integer solutions to algebraic equations can be traced back to the Greeks and, quite spectacularly, to this day many interesting open questions remain. For the problem of primes represented as the sum of two rational cubes, Samit Dasgupta (now at the University of California, Santa Cruz) and I were quite excited to give a partial answer to the question. It may be surprising to hear that there is anything new at all to say in such an ancient subject, but in fact we can prove today only a fraction of what we believe to be true. Particularly now, with computational tools at our disposal, new questions of an algorithmic nature tend to arise."
Voight was born in Georgia, and was raised in South Carolina and Washington state not unusual for the son of an Air Force major. Music was his first passion, starting with the piano at age five, then continuing to viola, cello, and string bass. Along with his musical pursuits still an interest for Dr. Voight he spent his high school years participating in debate and forensics.
"We would be given a resolution for the year (for example, 'Resolved: That the United States government should substantially strengthen regulation of immigration to the United States') and would compete in weekend tournaments, alternately affirming or negating the resolution," says Voight. "Our team would spend many long hours after school writing blocks and preparing negative strategies. My senior year, my partner and I were state champions, arguing on the affirmative for the rights of Hong Kong asylee seekers. I also enjoyed individual events on these same weekends, I would deliver an oratory or an expository speech on a topic like Spam, a luncheon meat with quite an interesting history, as it turns out."
The fact that Dr. Voight ended up with a career in mathematics is not unusual, given another childhood interest: numbers. (Also telling his mother is still the coach of the "Math is Cool" team for Farwell Elementary in Spokane, Washington, and before he passed away, his father taught mathematics and computer science at some tribal colleges.) This interest was further piqued when he took a linear algebra class at Gonzaga University while in high school; he stayed there to pursue an undergraduate degree after being recruited by the debate team. "Gonzaga is a Jesuit school and follows the liberal arts playbook, which was very appealing to me," says Voight.
After graduating summa cum laude from Gonzaga, Voight went on to receive his Ph.D. in Mathematics from the University of California, Berkeley, in 2005. He continues to work in the area of arithmetic geometry: "Although my research begins in the conceptual realm," says Dr. Voight, "many of the problems number theorists contemplate are related to modern cryptographic systems, so studying these questions will have quite practical implications for our day-to-day lives, even if this impact is delayed."
When asked about his teaching philosophy, Voight is very clear: "One of my favorite quotes about teaching comes from Terrel Bell, former U.S. Secretary of Education. Bell said, 'There are three things to remember about education. The first is motivation. The second one is motivation. The third one is motivation.' One of my highest priorities in teaching is exactly that, to foment mathematical curiosity in my classroom to the extent that I am able. I try to help students see beyond whatever difficulty lies on the surface and to be excited by the mathematical beauty underneath."
Dr. Voight can be contacted at: John.Voight@uvm.edu