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INTRODUCTION The mathematical sciences provide the tools and the foundation on which the structure of science and technology is built. This has been especially true for the development of computational science in recent decades. The mathematical sciences contribute fundamentally to the development of hardware, software, and networks for computational science, and to the development of human resources for that field--in other words, to all the major components of high-performance computing and communications. Mathematics is also critical to the modeling that translates important problems into computable form, to the numerical analysis enabling the computation, and to the statistical analysis for assimilating physical data and numerical output. The United States is now embarking on a program directed at building the high-performance computing and communications infrastructure necessary for the solution of "Grand Challenge" problems of major scientific and societal importance.] This report documents the essential role of the mathematical sciences both in building this infrastructure, according to the plan of the federal High Performance Computing and Communications (HPCC) program, and in ultimately rising to the grand challenges of computational science. A broad collection of mathematical tools will be required, and the continued development of new mathematics must be an integral part of both the HPCC program and ongoing attacks on grand challenge problems. The primary conclusions of this report are as follows: . The goals of the HPCC program cannot be met challenge problems cannot continue, without the mathematical sciences community. Therefore. , and progress against grand active involvement of the ~ , Therefore, relevant mathematical sciences research should be a significant part of the heightened research effort associated with the HPCC program. 1The 1991 report Grand Challenges: High Performance Computing and Communications, from the White House Office of Science and Technology Policy, offers this definition: "A Grand Challenge is a fundamental problem in science and engineering, with broad economic and scientific impact, whose solution could be advanced by applying high performance computing techniques and resources." There is no exhaustive list of such problems, but examples such as weather and climate prediction, structural biology, semiconductor design and modeling of turbulence are indicative of the range and complexity of the targeted problems. 1

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. The mathematical research topics listed in Section 5 are particularly germane to the HPCC program, as evidenced by their frequent mention in Sections 2 through 4. . Research and education relative to high-performance computing and communications and grand challenge problems must link a broad range of physical and life sciences, the mathematical sciences, and computational science. Therefore multidisciplinary settings offer the best chance for successes. Many of the advances profiled in Sections 2 through 4 were achieved in such settings. Section 2 below identifies where the mathematical sciences have contributed in the past to progress in high-performance computing and communications, Section 3 surveys potential near-term advances relevant to the goals of the HPCC program and beyond, and Section 4 samples the role of mathematical research in selected grand challenge problems. 2