# Breakthroughs in mathematics in 2023

At the end of 2021, Johnny Cage asked about breakthroughs in 2021 in different mathematical disciplines. A similar question has been asked at the end of 2022, so it looks like Johnny Cage originated a nice tradition.

Moderation note: See meta for meta discussion of the appropriateness of this post. Please do not carry on the meta discussion here.

• this question has been asked and closed mathoverflow.net/q/460824/11260, it may be reopened, pending a discussion on Meta meta.mathoverflow.net/q/5818/11260 Commented Dec 26, 2023 at 22:05
• Before asking, I spent some time looking whether this question has already been asked, and found nothing, hence the question. I think such end-of-year discussions are very interesting. Commented Dec 26, 2023 at 22:17
• Also, Quanta, being a math publicizer, naturally has a combinatorics bias. It would be nice to hear of breakthroughs in other areas of math. Commented Dec 27, 2023 at 2:18
• No, sorry, I did not check Meta and closed questions, because I did not think that such an interesting question was closed. I just checked the main list of questions, did not find it, and asked. Commented Dec 27, 2023 at 16:36
• @CarloBeenakker, I for one found this question a useful source of significant results I wasn't aware of. It should be re-opened. Commented Dec 28, 2023 at 14:09

Marcelo Campos, Simon Griffiths, Robert Morris, Julian Sahasrabudhe made a breakthrough in Ramsey theory. The abstract is a good enough summary to just quote it here:

The Ramsey number $$R(k)$$ is the minimum $$n \in N$$ such that every red-blue colouring of the edges of the complete graph $$K_n$$ on $$n$$ vertices contains a monochromatic copy of $$K_k$$. We prove that $$R(k) \leq (4− \varepsilon )^k$$ for some constant $$\varepsilon >0$$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

The solution by D. Smith, J.S. Myers, C. S. Kaplan, C. Goodman-Strauss, posted in arXiv in March 2023, of the einstein problem:

In plane geometry, the einstein problem asks about the existence of a single connected shape ("prototile") that can tessellate the Euclidean plane but only in a nonperiodic way.

(essentially cited from the Wikipedia page). They soon followed up with another paper, giving an aperiodic monotile without reflections. More information, including a video presentation by some of the authors, may be found at the Museum of Mathematics website.

• The formulation in Wikipedia is so badly stated that it shouldn't be quoted (mathematically half incorrect, missing capitals...). I reformulated (but the Wikipedia page should be revised too).
– YCor
Commented Dec 27, 2023 at 10:29
• @YCor I reverted to lowercase because otherwise it looks like the problem has some connection to Albert Einstein, which is does not, except as a joke. Commented Dec 27, 2023 at 12:46
• @TimothyChow Thanks. As a joke on the "Einstein" it would take a capital too. But indeed, the Wikipedia page says it comes from German "ein Stein" (one stone), which justifies the lower case even if confusion with Einstein is intended.
– YCor
Commented Dec 27, 2023 at 12:59
• @YCor it is a "quirk" of German grammar that if you mean "one stone" you write two separate words "ein Stein" but if the two are perceived as a single attribute/property the the two are written together to form the lengthy noun "Einsteinpuzzle" or "Einsteinkachelung" (Kachel means tile and Kachelung is tiling). The confusion with A. Einstein is owed to the resultimg ambiguity. A similar effect can be found in English with e.g. "ugliest dog owner" Commented Dec 27, 2023 at 13:54

Renling Jin found a simplified proof of Szemeredi's theorem using several levels of standardness, a question that Tao has been wondering about for years.

Renling Jin, A Simple Combinatorial Proof of Szemerédi’s Theorem via Three Levels of Infinities, Discrete Analysis, September 2023.

This has now been formalized by Hrbacek conservatively over ZF+ACC in

Hrbacek, Karel. Multi-level nonstandard analysis and the Axiom of Choice. Journal of Logic and Analysis (2024). https://arxiv.org/abs/2405.00621

• I was not aware of this, what a nice result! Commented Jan 16 at 2:44
• @mathworker21, thanks! Hrbacek succeeded in showing that Jin's proof (with three levels of standardness) can be carried out conservatively over ZF+ADC. Commented Jan 16 at 10:29
• @MikhailKatz ADC = axiom of dependent choice? (EDIT: oh, yes, that's it; I see it in the abstract of Hrbacek's paper) Commented May 2 at 12:24
• @DavidRoberts, As I mentioned in my recent edit, Jin's proof can be formalized conservatively over ZF+ACC, rather than ZF+ADC. This is a very recent breakthrough by Hrbacek. Commented May 2 at 13:06
• @MikhailKatz I get it now, I'm just more used to the abbreviations DC and CC. I didn't know if ADC and ACC [which also stands for Ascending Chain Condition!] were something particular to non-standard analysis. Commented May 2 at 23:35

Originally posted by David White:

The biggest result of the year in my area, homotopy theory, was the disproof of the Telescope Conjecture, by Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank. This conjecture was the last of the Ravenel Conjectures from 1984, regarding the structure of chromatic homotopy theory. The telescope conjecture was proved early on for $$n=1$$ and now we know it's false for $$n\geq 2$$, which means that the homotopy groups of spheres are even more complicated than was hoped in the 1980s. Quanta magazine covered this too. What's exciting to me is that the method of disproof continues the development of our computational power in algebraic $$K$$-theory, and suggests we will continue to learn cool things in stable homotopy theory with these new computational tools.

• It's a little odd that the Quanta article was published in August, well before the arxiv posting which didn't go up until (late) October. It seems a bit sensationalistic. Commented Jan 8 at 20:39
• @ZachTeitler The result was actually announced and presented on earlier in the summer at A Panorama of Homotopy Theory. Commented Jan 9 at 5:07

Zander Kelley and Raghu Meka substantially improved the bound in Roth's theorem, showing that any set $$A \subseteq \{1,\dots,N\}$$ with no nontrivial 3-term arithmetic progression must satisfy $$|A| \ll \frac{N}{\exp\left(\log^{1/12} N\right)}.$$ The exponent $$1/12$$ was improved to $$1/9$$ by Bloom and Sisask.

The most notable feature of these upper bounds is that their shape matches that of a classical construction of Behrend, which established the existence of a set $$A \subseteq \{1,\dots,N\}$$ with no nontrivial 3APs and of size $$|A| \gg \frac{N}{\exp\left(\log^{1/2} N\right)}.$$

Just to note, the previous upper bound, also due to breakthrough work of Bloom and Sisask, was $$|A| \ll \frac{N}{\log^{1+c} N}$$ for some absolute $$c > 0$$.

Another paper from Marcelo Campos, Matthew Jenssen, Marcus Michelen and Julian Sahasrabudhe was just posted in ArXiv, which is "the first asymptotically growing improvement to Rogers’ bound from 1947" on the sphere packing problem:

We show there exists a packing of identical spheres in $$\mathbb R^d$$ with density at least $$(1 - o(1))\frac{d\log d}{2^{d+1}}$$ as $$d \rightarrow \infty$$.

From Terence Tao:

A new advance on sphere packing in high dimensions, finding sphere packings that are asymptotically stronger than previous such constructions: https://arxiv.org/abs/2312.10026 . Previous methods relied mostly on taking random lattice packings, with refinements coming mostly from how one defines "random". The approach here is more graph theoretic, starting with a random Poisson process near-packing (in which there are some overlaps), and using a general combinatorial procedure ("the Rodl nibble", a.k.a. the "semi-random method") to eliminate the overlaps; it had previously been applied for other packing problems but this seems to be the first time it was able to improve upon the classical sphere packing problem.

To avoid guessing which of arxiv preprints are likely to pass peer review and which are not, I will talk about theorems that were peer-reviewed and published in 2023. Most of them appeared in arxiv before.

I will list theorems 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 2023 are:

Proof that for every $$\beta>\frac{1}{4}$$, every graph with no $$K_t$$ minor is $$O(t(\log t)^{\beta})$$-colourable, see https://doi.org/10.1016/j.aim.2023.109020 (this has then been improved to $$O(t\log\log t)$$, but this is not published yet, just in arxiv https://arxiv.org/abs/2108.01633). This is a progress towards famous Hadwiger's conjecture predicting that every graph with no $$K_t$$ minor is $$(t-1)$$-colourable.

Proof of the existence of polynomial automorphisms of $${\mathbb C}^2$$ with wandering Fatou components, see https://doi.org/10.1090/jams/1005 This answers a question of Bedford and Smillie from 1991.

Proof that famous Schinzel's hypothesis H is true in 100% cases, in a well-defined sense, see https://doi.org/10.1007/s00222-022-01153-6

Proof of higher uniformity of bounded multiplicative functions in short intervals on average, see https://doi.org/10.1017/fmp.2023.28 and https://doi.org/10.4007/annals.2023.197.2.3

Establishing the rigidity of proper colourings of $${\mathbb Z}^d$$, see https://doi.org/10.1007/s00222-022-01164-3

Proof that if $$n>d \geq 2$$ and $$(n,d)\neq (4,3)$$, then $$100\%$$ of the $$n$$-variable forms of degree $$d$$ with rational coefficients (ordered by height) satisfy the Hasse principle. Also, if $$(n,d)\neq (3,2)$$, then a positive proportion of them has a non-trivial rational zero. See https://doi.org/10.4007/annals.2023.197.3.3

Proof that, in the black-box model of random polynomial systems with prescribed evaluation complexity $$L$$, we can compute an approximate zero of a random structured polynomial system with $$n$$ equations of degree at most $$D$$ in $$n$$ variables with only $$poly(n,D)L$$ operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem, see https://doi.org/10.1017/fmp.2023.7

Proof that for any $$d\geq 2$$, any non-degenerate submanifold in $${\mathbb R}^d$$ is of Khintchine type for convergence, see https://doi.org/10.4310/ACTA.2023.v231.n1.a1 This is a fundamental result in the theory of Diophantine approximation! See also https://doi.org/10.1007/s00222-022-01171-4 for another 2023 result in this area.

A new lower https://doi.org/10.1007/s00222-022-01177-y and upper https://doi.org/10.1007/s11425-022-2193-8 bounds for Erdos--Hooley delta-function that measures the maximal concentration of divisors of $$n$$ in short intervals

Proof of the Erdos primitive set conjecture https://doi.org/10.1017/fmp.2023.16

Proof that any multiple polylogarithm of weight $$n\geq 2$$ can be expressed as a linear combination of multiple polylogarithms of depth at most $$n/2$$ and products of polylogarithms of lower weight, see https://doi.org/10.1090/jams/1011

Proof of sharp isoperimetric inequalities for affine quermassintegrals, see https://doi.org/10.1090/jams/1013

A solution to Erdos and Hajnal's odd cycle problem https://doi.org/10.1090/jams/1018

Resolution of the Erdos-Sauer problem on regular subgraphs https://doi.org/10.1017/fmp.2023.19

Proof that random polynomials are irreducible with high probability https://doi.org/10.1007/s00222-023-01193-6

Resolution of Banach's isometric subspace problem in dimension four https://doi.org/10.1007/s00222-023-01197-2

Proof of the Erdos-Faber-Lovasz conjecture for all large $$n$$: there is a constant $$n_0$$ such that every linear hypergraph on $$n\geq n_0$$ vertices has chromatic index at most $$n$$, see https://doi.org/10.4007/annals.2023.198.2.2

Proof of the Erdos-McKay conjecture, which predicted that every $$n$$-vertex $$C\log n$$-Ramsey graph contains an induced subgraph with exactly $$x$$ edges for every not-too-large integer $$x$$, see https://doi.org/10.1017/fmp.2023.17

Verification of Polya's conjecture for Euclidean balls https://doi.org/10.1007/s00222-023-01198-1

Analysis of a multi-parameter variant of the Bellow-Furstenberg problem https://doi.org/10.1017/fmp.2023.21

Description of the equality cases in the Alexandrov-Fenchel inequality for convex polytopes https://dx.doi.org/10.4310/ACTA.2023.v231.n1.a3

Proof that any finitely generated field of characteristic $$\text{char}(F) \neq 2$$ can be characterized by a first-order sentence in the language of rings, see https://doi.org/10.4007/annals.2023.198.3.4

Proof that for every smooth Jordan curve $$\Gamma$$ and every cyclic quadrilateral $$Q$$ in the Euclidean plane, there exists a quadrilateral similar to $$Q$$ whose vertices lie on $$\Gamma$$, see https://doi.org/10.1007/s00222-023-01212-6

• May I ask: Are you preparing a sequel to Theorems of the 21st Century: Volume I? Commented Jan 13 at 14:00
• I have a document with the collection of all suitable theorems with short descriptions, and I keep updating it by including new theorems "on the way", but I do not know when I will have time to convert this to Theorems of the 21st Century: Volume II... Commented Jan 14 at 11:27

The Polynomial Freiman-Ruzsa Conjecture was solved by Gowers, Green, Manners, and Tao in this preprint.

The result states that if $$A$$ is a subset of $$\mathbb{F}_2^n$$, and $$|A+A|\leq K|A|$$, then $$A$$ can be covered by $$2K^C$$ cosets of a subgroup $$H\leq\mathbb{F}_2^n$$ with $$|H|\leq |A|$$ (where $$C$$ is an absolute constant).

Many regarded this as one of the biggest open problems in additive combinatorics. The result has a number of striking equivalences and applications. Further information can be found in the preprint. This survey by Ben Green is also a valuable resource (see page 20 in particular).

• One notable feature of this work is that the main result was fully formalized in Lean surprisingly quickly (before the paper was even submitted, as a matter of fact). See the Lean Zulip chat for more details. Commented Jan 10 at 3:03
• The generalization to finite abelian groups of bounded torsion (by the same authors) is also now available: arxiv.org/abs/2404.02244 Commented May 2 at 12:47

Jared Duker Lichtman and Alexandru Pascadi set a new world record for reaching beyond "the square root barrier" in the smooth numbers problem. The details are discussed in this blog post and also this numberphile video.