Another major result from this year was Rachel Greenfeld and Terry Tao's disproof of the periodic tiling conjecture. The conjecture essentially said that if one has a single tile which aperiodically tiles $\mathbb{R}^n$ then it also periodically tiles $\mathbb{R}^n$. This conjecture was known to be true for $n=2$, when restricted to $\mathbb{Z}^2$ but open for $\mathbb{R}^n$ for all other dimensions. The question touches upon issues not just in geometry but also in logic, since a lot of classes of basic tiling problems in their general form turn out to be undecidable (and in fact equivalent to the Halting problem). Greenfeld and Tao constructed a tile in a massive number of dimensions (they don't work it out explicitly, but estimating it seems to involve a tower of exponentials) where there is a tile which aperiodically tiles that space but does not periodically tile it. It is likely that their construction though can be brought down to a much lower dimension, especially given that they disproved the lattice version of the problem and even showed that the lattice version is false in $\mathbb{Z}^2 \times G$ for some finite abelian group $G$.
The preprint is here and there is a good Quanta article explaining some of what went into it.
