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.

No special background will be assumed.

Based on joint works with Mitali Bafna, Noam Lifshitz, Nikhil Vyas and Zhiwei Yun.