This project is focused on probabilistically checkable proofs (PCPs) and high dimensional expanders. The central objective of this project is to connect between the two terms: PCPs and HDXs, that come from very different worlds.
The first term, PCPs, is a well studied object in theoretical computer science and has to do with methods of proof that allow for interaction and randomness. It turns out that with randomness one can provide proofs that can be checked extremely efficiently. The verifier doesn't need to read the proof, only probe a tiny random part of it. PCPs are useful in various contexts, one important one is that of showing lower bounds for approximation. Certain problems are hard to compute, and PCPs show that they are even hard to approximate.
The second term, HDX, is a construction of a combinatorial object which is like a graph but with higher dimensions, namely a simplicial complex. Such objects are known to exist via deep number theoretic constructions, and more recently also more elementary group theoretic constructions have been described. The high dimensional aspect of the object means that it is a certain generalization of expander graphs which themselves are very important and central objects in multiple fields including computer science and group theory.
The goal of the project is to explore ways to bring HDXs into the study of PCPs, and to bring PCP insight and questions into the study of HDXs. One of the most promising directions is to construct new error correcting codes based on HDXs that have properties similar to codes that come from PCPs. Error correcting codes are ubiquitous in the information age, and the extra "local-testability" properties have applications for example in blockchains.
This has been achieved in our work on C^3 LTCs. We constructed new combinatorial high dimensional expanders of a new tyoe (cubical rather than simplicial) and used them to build error correcting codes that are localy testable yet have constant rate and distance. This is considered a breakthrough and has been open or even sometimes conjectured to not exist.
Another goal that has been achieved is that of proving low-soundness agreement tests, and these have lead in follow up work (by another team) to PCPs. Thereby demonstrating that indeed better PCPs can come of HDX!
