Pointwise convergence of polynomial multiple ergodic averages along the primes

We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages 1/N ∑ₙ₌₁N (n) f₁(TP₁(n) x)⋯ fₖ(TPₖ(n) x) as N→ ∞, where is the von Mangoldt function, T X → X is an invertible measure-preserving transformation of a probability space (X,ν), P₁,…, Pₖ are polynomials with integer coefficients and distinct degrees, and f₁,…,fₖ∈ L^∞(X). This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. Building on the foundational work of Krause--Mirek--Tao (Ann. of Math., 2022), Kosz--Mirek--Peluse--Wright (arXiv: 2411.09478, 2024), and Krause--Mousavi--Tao--Ter\"{a}v\"{a}inen (arXiv: 2409.10510, 2024), we develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages. This method combines harmonic analysis techniques across multiple groups with the newest inverse theorem from additive combinatorics. In particular, the principal innovations of this framework include: (i) an inverse theorem and a Weyl-type inequality for multilinear Cram\'{e}r-weighted averages; (ii) a multilinear Rademacher-Menshov inequality; and (iii) an arithmetic multilinear estimate.