The Interplay of Regularity Method and Computer Science

In this report, we review the connections between regularity theory, low rank approximation of matrices and tensors, and approximate constraint satisfaction problems. Our presentation of the algorithmic aspects of regularity theory follows parts of [1, 2, 7]; wherever possible we cast the material in the modern language advocated by Tao [15]. One main motivation is to understand the power and limitations of the classical regularity theory, and to identify new algorithmic possibilities offered by the new developments in regularity theory (i.e. the relative Szemerédi theorem which first appeared in Green-Tao [11], and hypergraph regularity of Gowers [9], and Rödl et al [14]).