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 $R^n$ then it also periodically tiles $R^n$. This conjecture was known to be true for $n=2$, but open 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 abelian group $G$. 

The preprint is [here although it is more of an announcement without all the details][1] and there is a [good Quanta article explaining some of what went into it][2].

 


  [1]: https://arxiv.org/abs/2209.08451
  [2]: https://www.quantamagazine.org/nasty-geometry-breaks-decades-old-tiling-conjecture-20221215/