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Mathematician Discovers Lemniscate Proof

Issue | Fri, 05/18/2012 - 13:31

Photo courtesy of Trevor Hyde

I first met Trevor Hyde at practices for the William Lowell Putnam Competition, an annual intercollegiate math competition in which math students from around the country compete for scholarship prizes. I’d done math competitions before, but Hyde assured me that this one was different. Scored out of 120 points (10 for each of 12 questions), it is administered over a six-hour period, three hours in the morning and three hours in the afternoon on the day of the test. The median score out of 120 is usually between zero and two points. More importantly, he explained, the Putnam exam tests “real” math, not counting or clever tricks with combinatorics.

Hyde, as I quickly found out, excels at real math, the subjects that draw lines between high-school-level and college-level math, where numbers take a back seat to the many more abstract features of the subject. On Putnam that year, Hyde did well, but he didn’t rank among the best. Though a high score would have helped in applying to graduate school in math, he took it as encouragement to work harder in his classes and to apply himself more to the more direct determinants of graduate-school admissions.

**From Far ‘Afield’ to Amherst**

Hyde hails from the small town of Fairfield, Iowa, home to a large group of the followers of Maharishi Mahesh Yogi, best known as the spiritual leader of the Beatles. As he describes it, the town features a divide between the “townies” and the “rus” (short for “gurus”) who follow the Maharishi’s teachings. This dynamic proves to be a persistent source of differences between the two groups on matters of town governance. Hyde explained that despite being a “townie,” through his job at a local organic grocery store, he was exposed to those on the other side, summarizing, “I consider myself pretty neutral — this whole thing is sort of childish, that there are still disputes between these groups of people.” Though the movement has lost some steam since its height in the 1960s, it retains a foothold in Fairfield and, until just a few years ago, in Amherst as well, where the movement maintained an information center downtown.

An important point for Hyde in arriving both at Amherst and at the math department was a summer spent at a math camp, at the urging of his computer science teacher. He explains, “I didn’t really know what I was getting myself into… I was sort of thrown into this camp with these really brilliant people who had already studied abstract algebra and analysis while they were in high school, and I didn’t really know what a proof was yet.” While at the camp, he met a professor from Williams College who was a very inspiring lecturer, and after learning more about Williams, decided to apply to both Williams and the similar Amherst College. After being accepted to both, Amherst invited Hyde to visit and, though, set on Williams, he decided to take Amherst up on its offer. After visiting, his decision was made: he was coming to Amherst.

By the time he arrived at Amherst, Hyde was certain that he would be a math major. After briefly considering physics and more lengthily considering computer science as a second major, he determined that even though these were interests, they were not worthy of the focus of a major. His focus determined, he let little stand in the way of his pursuit of math, taking during his time at Amherst 17 math courses, an impressive number well beyond the nine required for a major. Excepting Putnam, he did few extracurricular activities, recounting, “I did the Frisbee team for the first two weeks of my freshman year, and then I remember the reason I quit was because I got some really interesting problem for my number theory class, and when it was time to go to practice, I was just like, I just don’t want to stop doing this to play Frisbee. And that was the end of my Frisbee career.” It was this single-minded focus on math that drove Hyde to take the comprehensive exam as a junior in order to qualify for writing a thesis in his senior year.

**The Genesis of Thesis Project**

Hyde’s thesis involved the lemniscate, a curve in two-dimensional space best described as “the infinity symbol,” though more formally described by a mathematical equation, (x2+y2)2 = 2a2(x2−y2). He took up an analysis of this curve after taking a reading course with Professor David Cox, using Cox’s book about Galois Theory. While the book begins with the foundations of Galois Theory, the later chapters explore a number of different topics, including one on the lemniscates, which both interested Hyde and contained several unsolved problems, a happy intersection for a junior looking to write a thesis the following year. The problem in question was to find the size of a certain Galois group, but the benefit of doing so was a more elegant proof of a theorem by Abel about the division of a lemniscate into equal lengths using a ruler and compass. The theorem states that the lemniscate can be divided into n equal lengths if n can be expressed as a power of two multiplied by distinct Fermat primes, primes of the form 22^m+1, where m is some positive integer, which is the same formula for the equivalent operation on a circle.

Initially, Hyde and Cox headed in the direction of number theory in order to apply the tool of class field theory, which easily gave the size of a similar Galois group. Despite learning a significant amount of class field theory, however, Hyde was unable to extend this result to find the size of his desired group. Cox describes: “We were at this dead end with class field theory, so Trevor said, let me try to come up with a lemniscates analogue of the cyclotomic polynomial, and if [we] can prove that’s irreducible, then we have an isomorphism, and then good things happen. Trevor called this a ‘lemnotomic polynomial.’ It was really a genuinely good idea, and it worked.” From this key insight, it was a matter of making things fit to achieve the full proof. With some write-up, it is likely to be a publishable work, as it is an elegant and original proof.

Cox also explained that the dead-end that Hyde encountered might have caused his thesis to be very different: “It could have been an expository thesis, [with] a very nice introduction to class field theory, how it applies to this case, and how it gives Abel’s theorem.” This easier approach, he said, would still be doing math at the graduate level and would have been a “very nice” thesis. However, “by coming up with [a] genuinely original approach,” Hyde was able to create an “exceptional” thesis.

**To the Future: (more) Math!**

With this exceptional thesis under his belt, Hyde plans to go to graduate school in mathematics after taking a year to work as the Quantitative Center fellow at Amherst. As the Q Center fellow, he will provide the fairly rare service of helping students with upper-level math courses, something that they have been generally less able to provide to students. In addition, he will have to time to continue independent research and apply to graduate schools, with the eventual goal of teaching at the college level. When asked about the relative importance of teaching and research, he cited Richard Feynman, the famed physicist, who, he explaind, after returning from a research sabbatical, “realized, when he went back to teaching, that his teaching inspired and informed his research, and that having students was a crucial aspect of his research career.” This insight has and will inspire Hyde’s learning experiences working in the Q Center.

Asked if he had to choose between teaching and researching, he confirmed that he’d choose teaching, balking, like Feynman, at the prospect of research without teaching. QianQian Chen ’13 agreed that teaching has become an important skill for Hyde, stating that he “got much better at teaching at the Q Center.” She recalled that while freshman year she “couldn’t understand what [he] was talking about, […] this semester [… he was] really good at teaching.”

By working at the Q Center for another year, Hyde is delaying his inevitable departure from Amherst. Afterwards, he will go on to apply his math skills in new and interesting ways elsewhere, but not before helping dozens more Amherst students achieve their own insights in the Q Center.