Primes in large arithmetic progressions & smooth shifted primes

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{0.5312}$. This result extends moduli beyond $x^{0.5233}$ due to Zhang and Polymath, and
$x^{0.5238}$ recently obtained by Maynard. All these results build on well-known 1986 work of Bombieri, Friedlander, and Iwaniec.