Computational complexity and decidability of tiling problems

Can a set of tiles (think polyominoes) tile a region? This decision problem is (computationally) hard in general, while special cases, such as tiling by dominoes, are well understood. We will consider some interesting results in the theory of tilings. In particular, in the finite setting, we will survey some problems that look similar but exhibit vastly different behaviours. Several classes of infinite problems will also be presented, most of which are undecidable. Joint work with Igor Pak.