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?

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. — Asaf Karagila, Dec 24 '21 at 15:29
Also, 2021 is not over. I still have a week to do something big! — Kimball, Dec 24 '21 at 23:03
What do you mean by important breakthroughs? For understanding my question see these levels in math research: ordinary paper < good quality paper < systematic papers answering some important conjectures < works that lead to a new sub-branch of math (like Ricci flow inside of differential geometry/topology) < works that discover/introduce a new branch of math (such k-theory and category theory) and so on. There is also a breakthrough prize in math. Which one is related to your post? — C.F.G, Dec 25 '21 at 06:53
A similar question was asked about ten years ago: Noteworthy achievements in and around 2010? — Timothy Chow, Dec 26 '21 at 21:00
For a good story on The Year in Math and Computer Science, see also here — Nicky Hekster, Dec 27 '21 at 11:48
From the answers, it seems that a good way to notice important results outside of our specialty is to read Quanta magazine — user551504, Dec 29 '21 at 03:56
It's definitely possible to learn something from the answers to this kind of question, but it's worth bearing in mind that answers are pretty much limited to various papers in one or two prestige journals, and/or those papers which Quanta writers chose to pay attention to. So it's hard to get a balanced perspective. — Quarto Bendir, Jan 16 '22 at 18:10

Carlo Beenakker · Answer 1 · 2021-12-24T22:34:54.780

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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):

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]

edited Dec 24 '21 at 22:34

answered Dec 24 '21 at 13:16

Carlo Beenakker

177,695

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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. – Will Sawin Dec 24 '21 at 13:28
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The first paragraph is very debatable. – Sam Hopkins Dec 24 '21 at 13:56
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Watanabe's Arxiv preprint is 3 years old. Is there something new, e.g. is the paper accepted somewhere? – abx Dec 24 '21 at 14:17
@abx --- it's still unpublished and I have crossed it out from the list. – Carlo Beenakker Dec 24 '21 at 14:55
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The proof that $\sf MM^{++}$ implies $(*)$ is not from 2021. It was published in 2021, yes. But it is definitely older. The preprint on arXiv is from 2019, and I know that they took a few months to write this properly. David told me about this breakthrough as it was happening, but I don't remember if it was late 2018 or early 2019 right now. – Asaf Karagila Dec 24 '21 at 15:07
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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. – Carlo Beenakker Dec 24 '21 at 15:47
Do you have the Quanta Magazine links as well? – Christopher King Dec 24 '21 at 21:36
added the Quanta Magazine links – Carlo Beenakker Dec 24 '21 at 22:35
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Watanabe very recently posted this https://arxiv.org/pdf/2109.01609.pdf . In the acknowledgement he also thanked the referee for their comments. Me and one of my collaborator read some part of his paper while writing our very recent paper. I am not in a state to judge if it is correct or not, but I haven't met anyone who doubted his proof yet. It's just a very ground breaking technique. And we (me and Jianfeng Lin) able to prove that gauge theory cannot detect such things. – Anubhav Mukherjee Dec 25 '21 at 01:39
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@CarloBeenakker oh, like IUT? That got published this year in a refereed journal ;-) – David Roberts Dec 25 '21 at 01:41
@SamHopkins Speaking of which, anything in Proof Systems? – Mark C Dec 27 '21 at 10:53
The media hype around the AI work seems highly debatable, see e.g. https://arxiv.org/abs/2112.04324 (a direct response to the Nature article linked to in the answer) – Quarto Bendir Jan 16 '22 at 18:00

Ricky · Answer 2 · 2021-12-25T01:10:31.723

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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.

edited Dec 25 '21 at 01:10

answered Dec 25 '21 at 00:53

Ricky

3,674

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Not to be confused with the Liquid Tension Experiment, who released a new album this year. – electronpusher Dec 25 '21 at 02:12
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With a song titled "Solid Resolution Theory" :D – Ricky Dec 25 '21 at 02:17
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@Ricky I interpreted that as a loose word-for-word antonym of the band name, but it is eerily relevant... – R. van Dobben de Bruyn Dec 25 '21 at 10:53
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@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! – Ricky Dec 25 '21 at 11:46

score 32 · Answer 3 · edited Dec 24 '21 at 22:33

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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.

edited Dec 24 '21 at 22:33

Nike Dattani

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answered Dec 24 '21 at 17:40

James Tuite

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... for large enough vertex numbers. I don't get why everyone is so cavalier about this. – darij grinberg Jan 09 '22 at 02:20

Windom Earle · Answer 4 · 2022-01-16T09:13:50.973

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.

score 25 · Answer 5 · answered Dec 28 '21 at 18:29

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:

Theorem of Annika Heckel about non-concentration of the chromatic number of a random graph https://www.ams.org/journals/jams/2021-34-01/S0894-0347-2020-00957-2/
A dramatic progress on sunflower conjecture by Alweiss, Lovett, Wu and Zhang https://annals.math.princeton.edu/2021/194-3/p05
The resolution of the rectangular peg problem for smooth Jordan curves by Greene and Lobb https://annals.math.princeton.edu/2021/194-2/p04
Characterization of fields of values of odd-degree irreducible characters by Navarro and Tiep https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/fields-of-values-of-characters-of-degree-not-divisible-by-p/0FE327DAE6088EC9B43A0CFAFC8E8D52
Proof that the Dehn function of $\mathrm{SL}_4({\mathbb Z})$ is quadratic by Leuzinger and Young https://annals.math.princeton.edu/2021/193-3/p02
Proof that the group $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for all $n \geq 6$ by Kaluba, Kielak and Nowak https://annals.math.princeton.edu/2021/193-2/p03
Characterization of $\frac{3}{2}$-generated groups by Burness, Guralnick and Harper https://annals.math.princeton.edu/2021/193-2/p05
A counterexample to the unit conjecture for group rings by Gardam https://annals.math.princeton.edu/2021/194-3/p09
Proof of the universal optimality of the ${E}_8$ and Leech lattices by Cohn, Kumar, Miller, Radchenko and Viazovska https://annals.math.princeton.edu/articles/17703
New bounds on the density of lattice coverings by Ordentlich, Regev and Weiss https://www.ams.org/journals/jams/2022-35-01/S0894-0347-2021-00984-0/home.html
Determining, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle by Jiang, Tidor, Yao, Zhang and Zhao https://annals.math.princeton.edu/2021/194-3/p03
The isoperimetric inequality for minimal surfaces by Brendle https://www.ams.org/journals/jams/2021-34-02/S0894-0347-2021-00969-4/
A connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan and Park https://annals.math.princeton.edu/2021/194-2/p02
An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture by Chen https://link.springer.com/article/10.1007/s00039-021-00558-4
An $O(n \log n)$ algorithm for multiplication of $n$-digit integers by Harvey and van der Hoeven https://annals.math.princeton.edu/2021/193-2/p04
An explicit construction of polynomials with optimal condition number by Beltran, Etayo, Marzo and Ortega-Cerd https://www.ams.org/journals/jams/2021-34-01/S0894-0347-2020-00956-0/

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.

I got your book from the library and have been reading it voraciously. It is excellent! — Timothy Chow, Feb 05 '22 at 02:33

score 22 · Answer 6 · answered Jan 09 '22 at 01:48

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!

Of course an important link between model theory and counting rational points is the Pila-Wilkie theorem! — nombre, Jan 16 '22 at 11:06

score 20 · Answer 7 · answered Dec 27 '21 at 16:52

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.

score 11 · Answer 8 · answered Jan 09 '22 at 19:50

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).

score 1 · Answer 9 · answered Dec 25 '23 at 22:34

The negative answer by counterexample to the Modular Isomorphism Problem for group rings (that is, the question whether for $p$-groups $G$ and $H$, the group rings ${\mathbb F}_p G$ and ${\mathbb F}_p H$ are isomorphic only if $G$ and $H$ are isomorphic) by Garcia-Lucas, Margolis and del Rio.

García-Lucas, Diego; Margolis, Leo; del Río, Ángel, Non-isomorphic 2-groups with isomorphic modular group algebras, J. Reine Angew. Math. 783, 269-274 (2022). ZBL1514.20019.

Breakthroughs in mathematics in 2021

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