9407503 CHRISTIANSEN The Mathematical Sciences Postdoctoral Research Fellowships are awards to recent recipients of doctoral degrees in the mathematical sciences. These awards are a means of contributing to the future vitality of the scientific effort of the nation. As researchers in the mathematical sciences expand their interactions with other disciplines, and as the interplay increases between the various areas of mathematics itself, opportunities for postdoctoral research and training are becoming increasingly important. The fellowships are designed to permit awardees to choose research environments that will have maximal impact on their future scientific development. The fellowship is designed to provide 24 months of support divided into 18 academic-year months and 3 periods of two summer months. The recipient may choose (1) the Research Fellowship option which allows for full-time support for any 18 academic-year months in a three-year period, in interval not shorter than 3 consecutive months or (2) the Research Instructorship option which allows the 18 months of academic year support to be taken as 9 months of full-time support and 18 months of half-time support. Not more than two months of summer support from this Fellowship may be received in any calendar year. Tanya Christiansen received her doctoral degree from the Massachusetts Institute of Technology, and will pursue research under the guidance of Charles Epstein, at the Johns Hopkins University and the University of Pennsylvania. The research will be conducted in spectral and scattering theory for the Laplacian on manifolds with cylindrical ends.