PI: Theodore A. Slaman
The 1995 National Research Council ranked the Mathematics Department at the University of California, Berkeley as the top research-doctorate program in mathematics in the United States. It has 53 permanent faculty members, over 25 postdoctoral faculty members, and over 200 graduate students. The Berkeley Mathematics Department is unique among the top research departments in that applied and pure fields form a seamless web, with active collaboration between many individuals in pure and applied fields. Within the Department, advances in computational power mean increases in the applicability of numerical methods and increases in the complexity of systems that can be directly analyzed.
For example, in computational algebraic geometry, Professors David Eisenbud and Bernd Sturmfels use the symbolic computation packages Macaulay and Maple, among others, to analyze Groebner bases, toric varieties, and related topics. The basic algorithms of Groebner basis theory have many applications. For example, Eisenbud and Sturmfels are collaborating with Persi Diaconis and Susan Holmes and using the geometric theory to give accurate methods of estimation for specific problems of great statistical interest. A second example is from the research of Professor John Strain, who designs efficient numerical algorithms for solving the partial differential equations of phase field models of solidification. The processes of solidification and growth of crystalline materials are important in technological areas as diverse as steel, semiconductors, photography and oil pipelines. These processes present phase transition problems whose analysis requires deep methods from partial differential equations and geometric measure theory. Numerical modeling of solidification processes, the subject of Strain's research, is at the cutting edge of research in fast algorithms and approximation theory.
The proposed equipment donation by Intel would enable significant advances in both of these efforts. Allyson Reeves, a former postdoc of Eisenbud who is now at NASA, has written a parallel version of the symbolic computation program Macaulay, but as of recently, there was only one publicly available supercomputer which supported Reeves's program. This fall, Eisenbud will come to Berkeley as director of Mathematical Sciences Research Institute (MSRI), and in the fall of 1998, Sturmfels will co-organize a special semester at MSRI on Symbolic Computation in Geometry and Analysis. This is a unique opportunity to integrate research efforts in symbolic computation at UC Berkeley with expertise available through MSRI visitors, including Reeves. The fall of 1998 will be very exciting, if the Intel network is available for that talented and experienced group. In addition to porting Macauly and Maple to the NOW, we will see substantial refinement and development of the algorithms of Groebner base theory to realize the potential of parallelization. In the general methodology which Strain is developing, he expects to work with a mesh of up to two million points to be necessary, 1024^2 in two dimensions or 128^3 in three. Visualization and postprocessing data sets of this size will require gigabytes of storage and gigaflop computing capabilities, and is only feasible with a resource of the capacity of the proposed network of Pentium workstations.
Other Matematics projects to be pursued with the proposed Intel equipment donation include the work of Professor F. Alberto Grunbaum (algorithms and software for medical imaging devices, X-ray crystallography, wave propagation in dispersive media) and the work of Assistant Professor Bjorn Poonen (estimating the number of real solutions for a degree n polynomial with random coefficients, tabulating the rational points on genus 2 curves, with potential applications to cryptography).
February 1999