This Small Grant for Exploratory Research will support mathematical research on general mathematical structures for fast Fourier transforms algorithms. The focus will be on computing the discrete Fourier transform for the general case. Applications to data possessing crystallographic symmetries will be made. Studies on related fast convolution algorithms will also be made. Data transformation and permutation algorithms will be produced. These be necessary for transforming data between stages of the Fourier transforms and for reducing and expanding data for input and output. Algorithms will be adapted to machine architectures including those of vector, parallel and hierarchical memory machines. They will take account of data flow, data permutations and data access in large scale computations. Group theoretical methods will play an important part in this effort. Included in the work is the goal of a systematic classification of the symmetries of all crystallographic groups. Algorithms and programs for others parts of the crystallographic calculation, particularly the refinement problem, will be produced. Implementation of this work in one large general crystallographic program package and report on the efficiency is planned.