Created to encourage talented high school students to pursue careers in science, mathematics, engineering, or medicine, the Intel Science Talent Search (STS) honors seniors for scientific and mathematical research. Among this year's 40 finalists, five presented mathematics projects, and three of them ended up among the winners, announced on March 13.

In second place, John Pardon, 17, of Chapel Hill, N.C., solved an open problem in differential geometry. He showed that a finite-length closed curve, no matter how convoluted, can be unfolded and made convex in a continuous manner without bringing any two points of the curve closer together during the transformation. Pardon's work extends to continuous curves an earlier proof by Erik Demaine, Robert Connelly, and Günter Rote that any polygon, no matter how crinkled, can be unfolded in two dimensions without any sides crossing each other.

Dmitry Vaintrob, 18, of Eugene, Ore., placed third for his investigations in a new area of mathematics known as string topology. He studied ways to associate algebraic structures to topological spaces, proving that loop homology and Hochschild cohomology coincide for an important class of spaces. The resulting formula for describing the way certain mathematical shapes combine may lead to applications in theoretical physics, particularly research on string theory and mirror symmetry. His mentor was mathematician Pavel Etingof of the Massachusetts Institute of Technology. Vaintrob had earlier won the grand prize in the Siemens Competition in Math, Science & Technology.

Gregory Brockman, of Thompson, N.D., earned sixth place for an analysis of certain Ducci sequences. A Ducci sequence (or the N-number game) is the result of an iterative process that involves taking differences of consecutive pairs from a finite set of integers, *x*_{1} . . . *x*_{n}, then taking differences of the resulting integers, and so on. Brockman applied the Ducci map to vectors of four real numbers (the four-number game) to determine what happens asymptotically to such sequences. In many instances, such sequences converge to (and often arrive at) all zeros.

The top award went to Mary Masterman, 17, of Oklahoma City for a physics project. She built a remarkably accurate instrument for detecting Raman spectra of vibrating molecules. Her homemade spectrometer cost about $300 to build (commercial units can run from $20,000 to $100,000).

Brian Lawrence, 17, of Kensington, Md., was an STS finalist for a project on the classification of minimal finite groups: "Finite Groups with *p*^{2} -1 Elements of Order *p*." He defined the notions of centrally and absolutely minimal groups, listed examples of such groups, proved a bound on the possible size that any example could have, then showed his list to be complete. In 2005, Lawrence earned a gold medal for achieving a perfect score at the 46th International Mathematical Olympiad. Last year, he won the Clay Olympiad Scholar Award for the most creative solution to a problem posed in the U.S.A. Mathematical Olympiad.

Sohan Mikkilineni, 18, of Bloomfield Hills, Mich., was an STS finalist for a project that analyzed determinantal sequences: "A Finiteness Property for Integral Points in a Family of Conics." A determinantal sequence is a sequence *a*_{n} of nonzero integers, one for each integer *n*, where the determinant *a*_{n}*a*_{n+3} - *a*_{n+1}*a*_{n+2} takes on the same fixed nonzero value *d* for all choices of *n*. Mikkilineni's mentor was Brian Conrad of the University of Michigan.—*I. Peterson*