The grand prize winner of this year's Intel Science Talent Search was math wizard Eric K. Larson, 17, of Eugene, Ore. His mathematics project, "The Classification of Certain Fusion Categories," garnered $100,000.

Last December, Larson's project won second place—a silver medal and $50,000—at the 2008 Siemens Competition.

Fusion categories belong to a branch of algebra known as category theory, an abstract way of dealing with mathematical structures and expressing relationships between them. Larson helped advance the classification of fusion categories.

Earlier work had classified those categories of dimensions *p*, *pq*, and *pqr*, where *p*, *q*, *r* are distinct prime numbers. Larson classified those of dimension *pq*^{2}, where *p* and *q* are again distinct primes. Unlike the previous cases, this class contains examples not arising from group theory. He also classified some other families of graded fusion categories. Larson has coauthored a paper, with graduate student David Jordan of the Massachusetts Institute of Technology, on the subject.

*Eric Larson (right), 17, of Eugene, Ore., wins top honors at the 2009 Intel Science Talent Search and a $100,000 scholarship from the Intel Foundation. Larson congratulates second and third place winners William Sun (middle), and Philip Streich (left).*

At South Eugene High School, Larson is active in the math, chemistry, and programming clubs. In 2007, as a 10th-grader, he was a winner of the USA Mathematical Olympiad, then competed in the International Mathematical Olympiad in Hanoi, Vietnam, where he received a silver medal. An accomplished classical pianist, he is a four-time gold medalist at the Oregon Junior Bach Festival. Larson enjoys conducting his own investigations and thinking about questions that nobody knows how to answer. He hopes to pursue a career as a mathematician at Harvard or MIT.

Larson's project wasn't the only winning mathematics entry. Noah M. Arbesfeld, 17, of Lexington, Mass., took sixth place with an algebra research project. He titled it "On the Structure of Lower Central Series Quotients of a Free Associative Algebra."

A free algebra has no special properties; as such, it is important because all other algebras can be built from it. Using both theoretical and computational tools, Arbesfeld studied the dimension of certain subspaces inside quotients of the lower central series. In the case of two generators, he improved a known bound from quadratic to linear, and he also gave a precise description of the first few quotients. Arbesfeld has coauthored a paper with MIT's David Jordan on the subject, highlighting new results.

At Lexington High School, Arbesfeld is active in the multicultural club and the Chinese Exchange. The recipient of numerous awards in math, debate, and science, he participated in the 2006, 2007, and 2008 USAMO competitions and was at the 2006 Mathematical Olympiad Summer Program. Arbesfeld enjoys playing the saxophone in the school band. In his spare time, he also enjoys amateur astronomy, geography, and improvisational theater. Arbesfeld plans to continue researching mathematics at MIT or Harvard, and hopes that his work will help communicate complex ideas to a nonscientific audience.

Among the 40 STS finalists, chosen from more than 1,600 entries, five other mathematics students and their projects merit mention.

Maxim Rabinovich, 17, a student at Shorecrest Preparatory School in St. Petersburg, Fla., submitted a mathematics project determining the scaling limits of certain anisotropic models. Titled "The Scaling Limit of a Generalized Divisible Sandpile Model," his work addressed a random process of diffusion in which a finite set of particles start out on points on a grid, and those particles that share a site move randomly to unoccupied points until no site has more than one particle. Researchers had earlier demonstrated that the shape of the expected result would be rectangular under certain assumptions. Under a more general assumption, Rabinovich showed that the resulting shape is an ellipsoid.

Anissa Y. Mak, 18, of Stuyvesant High School, in New York, submitted a project that involved graphs and algorithms. She called it "A Certifying Algorithm for the Modular Decomposition of Undirected Graphs."

A fundamental question that arises about an algorithm concerns its implementation: How can we know that the computation is free of bugs? A certifying algorithm gives evidence that the result of the computation is correct. The modular decomposition of a graph is a useful way of describing a graph in terms of a certain tree. Mak provided a certification algorithm for modular decomposition.

Tong Zhan, 16, of Mason, Ohio, investigated rainbow Ramsey theory, which asks about combinatorial properties of the natural numbers. A rainbow coloring of a set of numbers is one in which each of the numbers is assigned a different color. Zhan showed that a coloring of the natural numbers in which each of three colors is assigned sufficiently often must contain a rainbow coloring *a*, *b*, *c* such that *a* – *b* = *c*^{2}. Earlier Rainbow Ramsey results involved solutions of linear equations; Zhan improved certain ones of these en route to his results on quadratic conditions. The project was titled "On Rainbow Solutions to Equations with Quadratic Terms." Zhan competed in the 2007 and 2008 USAMO contests.

Adam B. Sealfon, 17, of Brooklyn, N.Y., submitted a computer science project called "Complexity Gap between Adaptive and Nonadaptive Algorithms for Property Testing of Hypergraphs." He explored graphs and hypergraphs, which have applications in biology, particle physics, and Internet searches. A graph is a collection of vertices (points) in which certain pairs are connected by edges (lines). Sealfon studied algorithms for testing properties of *k*-uniform hypergraphs—generalizations of graphs in which each line connects exactly *k* points, where *k* is greater than 2. Sealfon competed in the 2008 USAMO.

Marianna Y. Mao, 17, of Fremont, Calif., submitted a physics project titled "Gravitational Radiation from Encounters with Compact Binaries in Globular Clusters." She used perturbation theory and the laws of general relativity to model gravitational radiation from binary star systems. Mao was a member of the first U.S. national team that competed in 2007 in the China Girls Math Olympiad, where she earned a bronze medal. She also took part in the 2007 and 2008 USAMO contests.

The top 10 winners in the 2009 Intel Science Talent Search were announced at a gala awards ceremony in Washington, D.C., on March 10, 2009. On the previous day, the 40 finalists had met with President Obama at the White House to highlight the importance of math and science.

Source: Society for Science & the Public, March 10, 2009; *Science News*, March 10, 2009; *Scientific American*, March 10, 2009.