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.
Continuing this, let me ask exactly the same question about 2023.
Moderation note: See meta for meta discussion of the appropriateness of this post. Please do not carry on the meta discussion here.
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.
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
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.
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 $c > 0$.
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
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.
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).
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.