Breakthroughs in mathematics in 2021 This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important breakthroughs in 2021 in different mathematical disciplines?
 A: Since other answers mention works published in 2021, I think one can add to the list the proof of triviality of the $\phi^4$ quantum field theory in four dimensions:
Michael Aizenman, Hugo Duminil-Copin,
"Marginal triviality of the scaling limits of critical 4D Ising and $\lambda\phi_4^4$ models", Ann. of Math. (2) 194(1): 163-235 (July 2021).
A: Advancing mathematics by guiding human intuition with AI, Nature 600, 70 (2021), stands out because it represents the first significant advance in pure mathematics generated by artificial intelligence.
More newsworthy items (each item has a link to a blog on Quanta magazine for an informal discussion of its significance):

*

*A counterexample to the
unit conjecture for group rings, Giles Gardam, Ann. of Math. 194, 967 (2021). [Quanta link]

*Tadayuki Watanabe solved the last open case of the Smale conjecture. (still unpublished in 2021)

*$MM^{++}$ implies $(\ast)$, David Asperó and Ralf Schindler, Ann. Math. 193, 793 (2021). [Quanta link]

*Proof of the p-adic formula for Brumer–Stark units, Samit Dasgupta and Mahesh Kakde. [Quanta link]

*Geometrization of the local Langlands correspondence, Laurent Fargues and Peter Scholze. [Quanta link]

*Proof of Arnold's conjecture for cyclical number systems, Mohammed Abouzaid and Andrew Blumberg. [Quanta link]

A: Strictly speaking this is not a new mathematical result (meaning no new proof), but let me mention the Liquid Tensor Experiment, the verification in Lean of a very recent theorem by Clausen and Scholze.
Here is the original post by Scholze, here the story six months later, the canonical quanta link and, last but not least, a nature article.
PS: I participated in the project, so my opinion about its importance is surely biased.
A: One of the most exciting developments in combinatorics in 2021 is the proof of the Erdos-Faber-Lovasz Conjecture on the chromatic index of hypergraphs. There is a good article in Quanta magazine about the proof.
A: In (analytic) number theory Paul Nelson's recent preprint (https://arxiv.org/abs/2109.15230) solved the subconvexity problem for a huge class of L-functions in the t-aspect.
More precisely subconvexity bounds for $L(\frac{1}{2}+it, \pi, St)$ are established for cuspidal automorphic representations of $GL_n$.
This is a huge breakthrough and also the methods are very exciting and promising.
Edit: Now there is an article on this result on Quanta magazine.
A: Having just listened to some of Jacob Tsimerman's Minerva lectures, I became aware of the recent arXiv preprint, Canonical Heights on Shimura Varieties and the André–Oort Conjecture, by Jonathan Pila, Ananth N. Shankar, Jacob Tsimerman, Hélène Esnault, and Michael Groechenig.  Assuming the paper is correct, it gives the first unconditional (i.e., not assuming the Generalized Riemann Hypothesis) proof of the full André–Oort Conjecture.  The proof builds on a lot of previous work and knits together a wide variety of techniques and ideas, but one thing that I find personally appealing is that the theory of o-minimality plays a key role behind the scenes.  A priori, one might not guess that model theory has much to say about counting rational points, but it does!
A: Dmitri Pavlov and Daniel Grady released a preprint containing the first complete proof of the Cobordism Hypothesis, and in fact they prove a significant generalization to cobordism categories with geometric structure. Their article has a good discussion of prior work on this problem.
