【报告摘要】: This talk will primarily involve two phases. In the first half, we will focus on describing how the polynomial density Hales-Jewett conjecture, a central open problem in combinatorics, provides an independent external motivation to a succession of three results, where the proof of each result uses the next result as a black box: an internal approximation theorem on sets defined by polynomial conditions, a local generalisation of the theorem of Green and Tao on the equidistribution of polynomials, and the existence of some kinds of high-rank subtensors of high-rank tensors. In the second half, we will present additional results within the two nascent areas encompassing the previous results: the basic properties of the ranks of tensors, and the structure of objects defined in terms of linear forms and polynomials restricted to power sets such as the Boolean cube.