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 analogousanalogues 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
