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Undoubtedly one of the news that attracted the most attention this year was the result of Yitang Zhang on the Landau–Siegel zeros (see Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros)

Since it is not possible to be attentive to great results in all areas, In general terms, what have been important advances in 2022 in different mathematical disciplines?

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    $\begingroup$ check out Quanta magazine quantamagazine.org they go over many math results. $\endgroup$ Dec 2, 2022 at 4:22
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    $\begingroup$ So, we're just going to ignore December? $\endgroup$
    – Asaf Karagila
    Dec 2, 2022 at 7:21
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    $\begingroup$ Maybe I'm just being grumpy, but aren't the answers necessarily going to be opinion-based? ("Posted this year" is objectively measurable, but "great impact" isn't) $\endgroup$ Dec 2, 2022 at 10:50
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    $\begingroup$ A similar question last year was highly upvoted. I am personally undecided about whether this type of question is appropriate for MO, but it definitely seems inconsistent for the other question to receive 100+ upvotes and for this one to be closed. (Maybe the fact that Yitang Zhang's result was mentioned before it has been properly vetted by experts has something to do with it.) $\endgroup$ Dec 2, 2022 at 16:04
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    $\begingroup$ I hope this become a tradition for every year, it is interesting to know such results from other area that I don't know well $\endgroup$
    – Seewoo Lee
    Dec 15, 2022 at 6:04

6 Answers 6

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One from just a few days ago is Justin Gilmer's breakthrough on the union-closed conjecture, also known as Frankel's conjecture, which says that if one has a finite family of sets which are closed under taking unions, then at least one element appears in at least half the sets. This seems like a really basic combinatorics question and you can see how easily it generates interest by among other things how many Mathoverflow questions there have been about it. It seems like one of those problems where once you hear about it, you have to resist the temptation to just drop things and try to find a simple proof.

But a lot of the obvious things one would try to do for this question fail. We'll say an element is abundant if it appears in at least half of the sets in the family. It isn't too hard to show that if one of your sets is a singleton $\{x\}$ then $x$ must be abundant. And if the smallest size set in your family is two elements, then at least one of its elements in abundant. But the obvious generalization is false. It is possible for none of the abundant elements to appear in the smallest size sets in your family. See discussion at Difficult examples for Frankl's union-closed conjecture.

And there are other things that one might hope for that can break down. For example, the set of elements which appear in at least half the elements need not be itself in our family (see A strengthening of Frankl's union-closed conjecture?).

The strongest results until a few days could not even construct an explicit constant $\delta >0$ where we could prove the weaker union closed conjecture with $\delta$ replacing $\frac{1}{2}$. However,[Justin Gilmer gave a very readable proof at A constant lower bound for the union-closed sets conjecture that such a collection has to have an element which appears in at least 1/100th of all the members in the collection. For more details, see Gil Kalai's discussion at Amazing: Justin Gilmer gave a constant lower bound for the union-closed sets conjecture.

The natural limit to the method of Gilmer is $\frac{3- \sqrt{5}}{2} \approx 0.38$ rather than $\frac{1}{2}$, and there was some speculation that getting up there might be a slog, and some people thought that this might not be a bad idea for a new Polymath project (in a vein similar to the one on prime gaps). However, nearly simultaneously, three different preprints getting the $\frac{3- \sqrt{5}}{2}$ bound appeared, all using slightly different methods (one, An improved lower bound for the union-closed set conjecture, by @WillSawin; two, Approximate union closed conjecture, by a pair of stack exchange users in Mathematics and Theoretical Computer Science, Zachary Chase and Shachar Lovett; and three Improved Lower Bound for Frankl's Union-Closed Sets Conjecture, by @RyanAlweiss, Brice Huang, and Mark Sellke). One of them actually has potential to move beyond that bound.

To some extent the union-closed conjecture is a good example of how there are some really basic things we still don't know. This breakthrough helps alleviate some of that. And it looks plausible that aspects of Gilmer's method may work on some other problems also.

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    $\begingroup$ Maybe these additional follow-up papers are also worth mentioning: arxiv.org/abs/2211.13139, arxiv.org/abs/2212.00658 $\endgroup$ Dec 5, 2022 at 15:00
  • $\begingroup$ @SamHopkins I haven't looked at them. If you want to add them to the answer with some comments by all means feel free. $\endgroup$
    – JoshuaZ
    Dec 5, 2022 at 15:28
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I'd like to mention the resolution of the redshift conjecture for $E_{\infty}$-rings. The latter are a homotopical refinement of usual commutative rings; at least if we restrict to connective ones, we can model them as topological spaces with an addition and multiplication which satisfy the usual ring axioms up to homotopy (but these homotopies have to satisfy higher axioms, resulting in higher homotopies etc.). As for usual commutative rings, we can take the algebraic K-theory of an $E_{\infty}$-ring $R$ -- actually, the output can be viewed as an $E_{\infty}$-ring $K(R)$ again of which the usual algebraic K-groups $K_i(R)$ are just the homotopy groups.

It is a central tenet of chromatic homotopy theory to classify spaces (or $E_{\infty}$-rings) by their height. Any discrete space (like $\mathbb{Z}$) has height at most $0$, topological K-theory (or the space $BU\times \mathbb{Z}$ representing it) has height $1$; higher heights are bit more subtle to define: Technically speaking, the height of an $E_{\infty}$-ring $R$ is the maximum $n$ such that $L_{K(n)}R$ is nonzero (or, if we view $R$ indeed as a space: the maximum $n$ such that the $n$-th Bousfield Kuhn functor of $R$ is nonzero).

It has long been known that $K(\mathbb{Z})$ has height $1$ and actually it almost agrees with its part of pure height $1$ (i.e. the map $K(\mathbb{Z})_p \to L_{K(1)}K(\mathbb{Z})$ is an equivalence above degree $0$ if I recall correctly; this is closely related to the Quillen-Lichtenbaum conjecture, proven by Voevodsky et al.). Motivated by this and (very few) other examples, (one version of) Rognes's redshift conjecture states:

For any $E_{\infty}$-ring, the height of $K(R)$ is always one higher than that of $R$.

This has been proven this year in the monumental paper The Chromatic Nullstellensatz by Burklund, Schlank and Yuan. I want to make a few more remarks:

  • The work of Burklund-Schlank-Yuan is actually at its core about the role which Morava E-theory plays in chromatic stable homotopy theory and proves wonderful results about it. But combined with earlier work of Yuan about the algebraic K-theory of Morava E-theory, it proves the red-shift conjecture in the version above so-to-speak as a by-product.
  • There has been other very important work on the redshift-conjecture and its variants recently. For example, Hahn and Wilson had shown before that the spectra $BP\langle n\rangle$ satisfy redshift (which was the first known example valid at all heights). Moreover, they show also a statement which roughly says that $K(BP\langle n\rangle)$ is well-approximated by its $K(n+1)$-localization. Even more recently, Ausoni, Bayindir and Moulinos have shown redshift for the Morava K-theories $K(n)$ themselves, which are only associative; this implies redshift for many more ring spectra which are not $E_{\infty}$. Note also the recent concrete calculations of Angelini-Knoll, Culver, Höning and Rognes in this context, and a new way to do similar calculations discovered in recent work of Hahn, Raksit and Wilson.
  • There has been already other very interesting work in homotopy theory this year, among which I want to mention two papers by Burklund: one gives the first sublinear bound on the $p$-exponent of the stable homotopy groups of spheres; the other shows that quotients in the stable homotopy theory have much better multiplicative properties than anybody expected before.
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Giles Gardam's paper A counterexample to the unit conjecture for group rings was a big breakthrough in group rings. Giles produces a group $G$ that is torsionfree, such that there are units in $\mathbb{F}_2[G]$ that are not of the trivial form "a nonzero constant from $\mathbb{F}_2$ times an element of $G$". He explicitly names a couple of these nontrivial units.

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    $\begingroup$ It would be nice if you could at least provide a statement of the result from the paper in your answer. $\endgroup$
    – Wojowu
    Dec 5, 2022 at 17:47
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    $\begingroup$ It would also be nice if you could provide a statement of the year the paper appeared, to check that it really was 2022 :-) $\endgroup$ Dec 5, 2022 at 17:56
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    $\begingroup$ @PaceNielsen Well, it already made last year's list! :-). $\endgroup$ Dec 5, 2022 at 20:20
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    $\begingroup$ (Even though, in fairness, it is a nice enough result to be in two lists!) $\endgroup$ Dec 5, 2022 at 20:56
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    $\begingroup$ I think this answer should be deleted since it is not a 2022 result. $\endgroup$ Dec 16, 2022 at 16:37
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Another major result from this year was Rachel Greenfeld and Terry Tao's disproof of the periodic tiling conjecture. The conjecture essentially said that if one has a single tile which aperiodically tiles $\mathbb{R}^n$ then it also periodically tiles $\mathbb{R}^n$. This conjecture was known to be true for $n=2$ when restricted to $\mathbb{Z}^2$ but open for $\mathbb{R}^n$ for all dimensions. The question touches upon issues not just in geometry but also in logic, since a lot of classes of basic tiling problems in their general form turn out to be undecidable (and in fact equivalent to the Halting problem). Greenfeld and Tao constructed a tile in a massive number of dimensions (they don't work it out explicitly, but estimating it seems to involve a tower of exponentials) where there is a tile which aperiodically tiles that space but does not periodically tile it. It is likely that their construction though can be brought down to a much lower dimension, especially given that they disproved the lattice version of the problem and even showed that the lattice version is false in $\mathbb{Z}^2 \times G$ for some finite abelian group $G$.

The preprint is here and there is a good Quanta article explaining some of what went into it.

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    $\begingroup$ The above statement that "this conjecture was known to be true for $n=2$" is slightly garbled, due to confusion between $\mathbb{Z}$ and $\mathbb{R}$. In particular, it is still open whether there is an einstein in $\mathbb{R}^2$. $\endgroup$ Dec 17, 2022 at 16:26
  • $\begingroup$ @TimothyChow Good pointed. Fixed now. $\endgroup$
    – JoshuaZ
    Dec 17, 2022 at 18:17
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To avoid guessing which of arxiv preprints are likely to pass peer review and which are not, I will concentrate on theorems that where peer-reviewed and published in 2022. Most of them appeared in arxiv before.

My favourite theorems in mathematics are the ones that are at the same time great and have easy-to-understand formulation. So, you can just follow the links, read the original papers, and most likely you will be able to understand and appreciate these theorems! Enjoy!

So, the greatest easy-to-understand theorems published in 2022 are:

Finally, you may want to look at my book with the descriptions of all such theorems published from 2001 until 2020.

B. Grechuk, Landscape of 21st Century Mathematics, Springer, Cham, 2021, https://doi.org/10.1007/978-3-030-80627-9

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  • $\begingroup$ It would be better to give a title for your book and a doi link, not a fragile link.springer.com url, just saying :-) $\endgroup$
    – David Roberts
    Dec 24, 2022 at 6:46
  • $\begingroup$ @DavidRoberts I agree, but at least Springer now embeds the DOI in its URL. $\endgroup$ Dec 24, 2022 at 13:17
  • $\begingroup$ Thank you for the suggestion. The link is corrected, and the title is given. $\endgroup$ Dec 27, 2022 at 9:17
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In Measures of maximal entropy for surface diffeomorphisms appeared in the Annals, Buzzi, Crovisier and Sarig proved that (from the Abstract) "$C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case".

It is a huge result in smooth dynamics, and it looks like a major result in general to me, although I'm biased towards dynamical systems.

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    $\begingroup$ -1 "Looks like a major result to me". One might say the same about anything published in the Annals! Surely a good answer to this (controversial) question should at least explain why the result is important. If it doesn't solve a well known open problem, my instinct is that it probably shouldn't make the grade. The wording of your answer suggests that even someone biased towards the area doesn't understand the importance of this result. $\endgroup$
    – HJRW
    Dec 5, 2022 at 10:05
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    $\begingroup$ That's harsh, isn't it? :) I didn't see any special requirement for motivations or explanations in the question, so I just mentioned a paper I judged very important and that I'm reading at the moment. And it's important (also) because it establishes a result that can be understood by anyone just a bit familiar with basic dynamical systems theory. It is also wonderfully written, and almost self-contained. $\endgroup$ Dec 5, 2022 at 12:36
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    $\begingroup$ Understandable usually means fundamental. The converse is obviously not true, as there are very crucial results which are very technical to state. $\endgroup$ Dec 5, 2022 at 14:42
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    $\begingroup$ @AlessandroDellaCorte "Understandable usually means fundamental." Is that true in dynamical systems? At least in number theory and graph theory there's massive amounts of very easy to understand stuff that is not at all fundamental. $\endgroup$
    – JoshuaZ
    Dec 5, 2022 at 14:53
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    $\begingroup$ I don't really get what this result is saying, or why it's important, and I try to be widely-read. I mean, i think i recognise most of the buzzwords, but i have no feeling for why it's important, aside from the facts its a kind of general uniqueness proof that seems nontrivial. So "understandable" is relative. Compare for instance a result like an faster multiplication algorithm, or a model of choiceless set theory where the real numbers are countable. $\endgroup$
    – David Roberts
    Dec 5, 2022 at 20:28

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