The original proof of the PCP Theorem composes a Reed-Muller-based PCP with itself, and then composes the resulting PCP with a Hadamard-based PCP [Arora, Lund, Motwani, Sudan and Szegedy ({\em JACM}, 1998)].
Hence, that proof applies a (general) proof composition result twice.
(Dinur's alternative proof consists of logarithmically many gap ... more >>>

One of the most fundamental problems in the field of hypothesis testing is the identity testing problem: whether samples from some unknown distribution $\mathcal{G}$ are actually from some explicit distribution $\mathcal{D}$. It is known that when the distribution $\mathcal{D}$ has support $[N]$, the optimal sample complexity for the identity testing ... more >>>

We study the Kaufman--Oppenheim coset complexes (STOC 2018, Eur. J. Comb. 2023), which have an elementary and strongly explicit description. Answering an open question of Kaufman, Oppenheim, and Weinberger (STOC 2025), we show that they support sparse direct-product testers in the low soundness regime. Our proof relies on the HDX ... more >>>