Most important results in 2022

Question

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?

Answer 1

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.

Answer 2

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. Note also the recent concrete calculations of Angelini-Knoll, Ausoni, 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.

Answer 3

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.

Answer 4

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:

• Proof that almost all orbits of the Collatz map attain almost bounded values by Tao https://doi.org/10.1017/fmp.2022.8

• Proof that $E_8$ and Leech lattices are universally optimal in dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska https://doi.org/10.4007/annals.2022.196.3.3

• New upper bounds on the minimal density of lattice coverings of ${\mathbb{R}}^n$ by dilates of a convex body, established by Ordentlich, Regev and Weiss https://www.ams.org/journals/jams/2022-35-01/S0894-0347-2021-00984-0/viewer/

• Proof of famous Babai’s conjecture (stating that the diameter of a non-abelian finite simple group is bounded by a polynomial in log(size) of the group) in the case of high-rank classical groups with random generators by Eberhard and Jezernik https://link.springer.com/article/10.1007/s00222-021-01065-x

• Construction of Feigenbaum quadratic-like maps whose Julia set has positive Lebesgue measure by Avila and Lyubich https://annals.math.princeton.edu/2022/195-1/p01

• Proof of the Gaussian Multi-Bubble Conjecture (that is, finding the least Gaussian-weighted perimeter way to decompose ${\mathbb R}^n$ into $q$ cells of prescribed positive Gaussian measure for all $2\leq q \leq n+1$) by Milman and Neeman https://annals.math.princeton.edu/2022/195-1/p02

• Development of the quasi-polynomial (expected) time algorithm for the discrete logarithm problem in finite fields of fixed characteristic by Kleinjung and Wesolowski https://www.ams.org/journals/jams/2022-35-02/S0894-0347-2021-00985-2/

• Proof of effective version of the Oppenheim conjecture by Buterus, Götze, Hille and Margulis https://link.springer.com/article/10.1007/s00222-021-01086-6

• Calculating the probability that a random integral rectangular matrix defines a surjective map by Nguyen and Wood https://link.springer.com/article/10.1007/s00222-021-01082-w

• Establishing the density of the uncovered set in the Erdős covering problem by Balister, Bollobás, Morris, Sahasrabudhe and Tiba https://link.springer.com/article/10.1007/s00222-021-01087-5

• Pointwise ergodic theorems for non-conventional bilinear polynomial averages by by Krause, Mirek and Tao https://annals.math.princeton.edu/2022/195-3/p04

• Construction of body, not a ball, that can float in water in every position, by Ryabogin https://annals.math.princeton.edu/2022/195-3/p05

• Proof that a positive proportion (in fact over 30%) of monic polynomials with integer coefficients have square-free discriminants by Bhargava, Shankar and Wang https://link.springer.com/article/10.1007/s00222-022-01098-w

• Proof of the satisfiability conjecture for all large $k$ by Ding, Sly and Sun https://annals.math.princeton.edu/2022/196-1/p01

• Proof of polynomial analogues of the Chowla and twin primes conjectures by Sawin and Shusterman https://annals.math.princeton.edu/2022/196-2/p01

• Proof that for the equation $x^2+y^2+z^2−xyz=k$ the Hasse Principle holds for almost all $k$'s but fails for infinitely many $k$'s by Ghosh and Sarnak https://link.springer.com/article/10.1007/s00222-022-01114-z

• Proof that the two-colour van der Waerden number $w(3,k)$ grows superpolynomially in $k$ by Green https://doi.org/10.1017/fmp.2022.12

• Proof that one can hear the shape of ellipses of small eccentricity by Hezari and Zelditch https://annals.math.princeton.edu/2022/196-3/p04

• Rigorous proof of scaling relations for planar random-cluster model by Duminil-Copin and Manolescu https://doi.org/10.1017/fmp.2022.16

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

Answer 5

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.

Answer 6

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.