[Math] Breakthroughs in mathematics in 2021

Question

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?

Answer 1

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 _{(still unpublished in 2021)}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]