It was at a high-school girls' basketball game that I met Yuanqi "Kylie" Zhang, who had recently won the Gold Prize in the Shing-Tung Yau High School Mathematic Awards. The competition pitted over 400 teams from Europe, Singapore, China, Taiwan, and the United States. The second-year international (as opposed to foreign exchange) student at St. Gregory College Prep was, due to circumstance, a team of one. She cruised through the preliminaries and then presented her findings in front of an international panel of world-class mathematicians at Tsinghua University in Beijing this past December, where she won the top award.
During a 2013 summer program at Harvard, she met Professor Shing-Tung Yau, a winner of the Fields Medal (the equivalent of the Nobel Prize for math) and the person who developed the competition. Kylie explained her ideas for potential research and he introduced her to one of his former students, Xianfeng David Gu, who teaches at State University of New York at Stony Brook. Gu is doing groundbreaking work with conformal geometry (www.cs.sunysb.edu/~gu/), which he describes as "the intersection of algebraic geometry, differential geometry, complex analysis, and algebraic topology." Or, in laymen's terms, Party at the Math Place.
When Kylie first arrived in the U.S. from Hengshui, Hebei, China, teachers and students were having trouble pronouncing her name. She went online to adopt an American name, eventually settling on Kylie because it would be easier for her parents back in China to pronounce.
She is quick to deflect any suggestion that she is some sort of math prodigy, arguing instead that any success she might achieve is the product of application and dedication. "I never say that I have a special gift for math. There are many other students around the world who are much better in math than (I). I am just one of the hard-working ones. I found my passion for mathematics and physics while in middle school. Last May, when working with Prof. Gu, I discovered the beauty of pure mathematics. I want to continue exploring that world."
Put as simply as possible, her work is in the field of 3-D scanning, like that that is employed in the burgeoning new field of 3-D printing. There are many fascinating (and more than a few frightening) possibilities in 3-D printing, but in these early stages, it is limited by challenges associated with the initial scanning process. Creating a 3-D copy of something that has broad, flat surfaces is relatively simple. However, when the subject has curves and/or uneven surfaces, the difficulty in producing an accurate scan increases exponentially.
Her approach is to "flatten" the original 3-D image using conformal mapping. In her words, "Convert surfaces embedded in three-dimensional Euclidean space to two families of transversal foliations." This creates a Riemann surface, which is a geometrical representation of a function of a complex variable that is depicted on several planes that are connected at points where the function takes on multiple values.
Since her new, "flattened" image is a Riemann surface, it allows for holomorphic differentiations, which, somewhat simplistically, means that the flattened-out surface can be expressed mathematically.
The more complex the original image, the more complex the math needed to complete the mapping. For example, if the original subject is a human face, the conformal module employed in the mapping onto a planar (flat) rectangle is a straightforward ratio of height divided by width. But, if the mouth were open, that would create a cylindrical situation. The mapped image would be that of a distorted torus (donut) shape, with the conformal module being the logarithm of the ratio between the two radii. And, it gets increasingly more complex after that.
Kylie's insight involves what to do with the planar conformal images after the mapping is done. She decided to cut the image into mathematically defined horizontal and vertical strips and then painstakingly weave the strips back into a 3-D image. This process allows the original subject to be mapped onto a plane, then re-assembled into a 3-D reproduction.
Even the math involved in the creation and definition of the strips is daunting. She used a mixed boundary condition, which involves the simultaneous use of a Dirichlet boundary condition and a Neumann boundary condition. The committee of mathematicians found it to be excellent, especially since she hadn't had time to assemble a team to help power through all of the complex mathematics that are involved.
Kylie is most appreciative of the process involved in the achievement. "Many times, I would speak to my advisor and propose some mathematical approach. He would say 'No!' most emphatically and send me off in another direction. I had to learn about Dirichlet, Neumann, Riemann surfaces, and LaPlace transforms. "That is why I say I am not a prodigy. I have to attack the problem and learn those things, step by step."
Kylie recently presented her work to an assembly in front of the entire St. Gregory student body and faculty. The reaction was universal: Wide-eyed, mouths-open, blanks stares. According to Jim Carlson, who teaches Latin at the school, "Someday, that kid is going to be running the world and we're all going to be working for her."
