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
