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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?

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    $\begingroup$ Mathematics tend to be slow. It's rare to see a breakthrough "in that year" go from preprint announcement, to review, to accepted, to actually published. It's not impossible, but those tend to be shorter, smaller, e.g. some counterexample of some finite conjecture. Otherwise, things tend to take time. That's a good thing. $\endgroup$
    – Asaf Karagila
    Dec 24, 2021 at 15:29
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    $\begingroup$ Also, 2021 is not over. I still have a week to do something big! $\endgroup$
    – Kimball
    Dec 24, 2021 at 23:03
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    $\begingroup$ @Kimball: We're working on it!!! $\endgroup$
    – Asaf Karagila
    Dec 25, 2021 at 1:03
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    $\begingroup$ A similar question was asked about ten years ago: Noteworthy achievements in and around 2010? $\endgroup$ Dec 26, 2021 at 21:00
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    $\begingroup$ From the answers, it seems that a good way to notice important results outside of our specialty is to read Quanta magazine $\endgroup$
    – user551504
    Dec 29, 2021 at 3:56

8 Answers 8

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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):

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    $\begingroup$ Perhaps it would be better to post different answers for different breakthroughs? I think it might be useful to see how much each one is upvoted. $\endgroup$
    – Will Sawin
    Dec 24, 2021 at 13:28
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    $\begingroup$ The first paragraph is very debatable. $\endgroup$ Dec 24, 2021 at 13:56
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    $\begingroup$ Watanabe's Arxiv preprint is 3 years old. Is there something new, e.g. is the paper accepted somewhere? $\endgroup$
    – abx
    Dec 24, 2021 at 14:17
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    $\begingroup$ I presume this applies to all substantial math research, that there is not a single moment in time when it happens; I would argue that the date of publication in a refereed journal is the date it enters into the mathematical body of knowledge. $\endgroup$ Dec 24, 2021 at 15:47
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    $\begingroup$ @CarloBeenakker oh, like IUT? That got published this year in a refereed journal ;-) $\endgroup$
    – David Roberts
    Dec 25, 2021 at 1:41
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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.

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    $\begingroup$ Not to be confused with the Liquid Tension Experiment, who released a new album this year. $\endgroup$ Dec 25, 2021 at 2:12
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    $\begingroup$ With a song titled "Solid Resolution Theory" :D $\endgroup$
    – Ricky
    Dec 25, 2021 at 2:17
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    $\begingroup$ @Ricky I interpreted that as a loose word-for-word antonym of the band name, but it is eerily relevant... $\endgroup$ Dec 25, 2021 at 10:53
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    $\begingroup$ @R. van Dobben de Bruyn Sure, the name of the project was related to the name of the band, but "Solid Resolution Theory" arrived later! $\endgroup$
    – Ricky
    Dec 25, 2021 at 11:46
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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.

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    $\begingroup$ ... for large enough vertex numbers. I don't get why everyone is so cavalier about this. $\endgroup$ Jan 9 at 2:20
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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.

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My favourite theorems in mathematics are the ones that at the same time great and have easy-to-understand formulation. To put aside various P=NP claims in arxiv, I will concentrate on theorems that where peer-reviewed and published in 2021. Most of them appeared in arxiv before.

So, the greatest easy-to-understand theorems published in 2021 are:

I am sorry if you think that this list is too long but in my opinion all these theorems are both great and beautiful, so I will let you to choose your own 3-5 favourite ones.

Finally, you may want to look at my book https://link.springer.com/book/10.1007/978-3-030-80627-9 with the descriptions of all such theorems published from 2001 until now.

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    $\begingroup$ I got your book from the library and have been reading it voraciously. It is excellent! $\endgroup$ Feb 5 at 2:33
  • $\begingroup$ Thank you! I am very happy that you like it! $\endgroup$ Feb 5 at 7:58
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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!

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  • $\begingroup$ Of course an important link between model theory and counting rational points is the Pila-Wilkie theorem! $\endgroup$
    – nombre
    Jan 16 at 11:06
  • $\begingroup$ And here is the link to the expected Quanta article. $\endgroup$ Feb 5 at 2:29
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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.

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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).

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