The main result is a new PCP theorem in which, for each positive $\delta$, the verifier makes two queries to an alledged proof of almost-linear length over a constant-size alphabet and has soundness error $\delta$. By almost-linear we mean linear up to polylogarithmic factors and the constants (both in the degree of this polynomial and the alphabet size) depends on $\delta$ only. This improves over Dinur's celebrated result, which holds for $\delta$ that is close to 1 (e.g., $\delta=0.99$).
The result builds on analogous resuls for dearndomized direct product testing obatined (independently) by Dikstein, Dinur and Lubotzky and by Bafna, Lifshitz, and Dor Minzer. This is combined with a novel routing scheme that uses HDXs (and with an alphabet reduction technique (specifically of Dinur and Harsha)). The routing scheme bridges the gap between unstructured sets of random queries and structured ones (i.e., the structure arises from the original PCP being parallelized).
We will discuss recent PCP constructions based on high-dimensional expanders that achieve small soundness and quasi-linear size, which are two key properties of PCPs. To do so we discuss the idea of "derandomized hardness amplification", which is a soundness amplifying procedure that only incurs a mild size blow-up, and show how to achieve it (in the context of PCPs) via high-dimensional expanders.