Puzzling through exact sequences: A bedtime story with pictures is a beautiful visual explanation of spectral sequences (and subobjects and quotients) by Ravi Vakil. I guess that some (many?) mathematicians have similar pseudo-visual calculi for such operations (at least, I do) but I cannot put it into actual pictures the way Ravi just did. I don't know whether seeing someone else's visual intuition is actually helpful when you have no idea what's going on, but I might try and use some of the visualisations in my own teaching later.
Nikolas Kuhn, Devlin Mallory, Vaidehee Thatte, Kirsten Wickelgren: An explicit self-duality is an expository writeup on an explicit form of Grothendieck duality in a specialised setting.
This is the first chapter (on the arXiv) of a proceedings volume that Johan de Jong, Wei Ho and myself are putting together, with contributions related to the 2 Stacks project workshops we've had. I've seen the other contributions, and I'm looking forward to sharing them (when they are published as preprints by the authors at some point, and in their final book form) with you!
Manjul Bhargava: Galois groups of random integer polynomials and van der Waerden's Conjecture proves van der Waerden's conjecture, which gives the asymptoptic upper bound for the number of monic integer polynomials in a "large box" (with coefficients bounded by some bound $H$) whose Galois group is not the full symmetric group.
I have fond memories to 2011 when I was learning sieve methods and experimented with this problem myself, and somehow it has cemented itself as one of my favourite aspects of Galois theory. It's great to see the problem finally solved!