On inverse theorems, constraint satisfaction problems and combinatorial lines

Suppose f_1, ..., f_k are functions that have a non-negligible correlation over a product distribution \mu^n; what structure can be deduced about the functions f_1, ..., f_k? While the initial motivation for this problem stemmed from theoretical computer science, and more precisely from approximation algorithms for constraint satisfaction problems, it turns out to appear in many different settings. In this talk we will discuss some preliminary results and their applications. In particular, we will discuss the solution to the case that k=3 and its relation to the density Hales-Jewett problem of length 3. Based on joint works with Amey Bhangale, Subhash Khot and Yang P. Liu