Local-to-global Expansion and PCPs

Introduction

This proposal aims to delve further into the study of Probabilistically Checkable Proofs (PCPs), a cornerstone of modern theoretical computer science, that exhibits some of the most powerful local to global behavior. Any NP proof can be written in a format that is locally testable, meaning that local pieces of the proof imply very rich global structure. This theory is strongly tied to hardness of approximation and has significant applications in cryptography. A major goal is to develop simpler and more efficient PCP constructions, with better parameters, as well as to deepen our understanding of robust encodings with local to global features, such as PCPs.

Methodology

The main methodology of this exploration will be through harnessing the power of the beautiful emerging theory of high-dimensional expansion (HDX). Generalizing the concept of expander graphs to higher dimensions, HDX emerges as an exciting cross-disciplinary theory with roots in:

• Group theory
• Number theory
• Algebraic topology
• Combinatorics
• Theoretical computer science

The HDX theory is characterized by a local-to-global principle, which allows inferences about the global structure based on local link properties. This principle bears significant potential for areas like property testing and it has already shown its power with the construction of C^3 locally testable codes. As these are very closely connected to PCPs, we seek to harness HDX towards advancing our understanding of more general local-to-global encodings, and potentially paving the way for novel PCP constructions.

Research Directions

The research directions outlined in this proposal cover a wide array of goals, including:

1. The exploration of the HDX theory
2. The construction of new PCPs
3. Inapproximability
4. The creation of novel error-correcting codes

These directions seek to bridge various areas of mathematics and computer science, promising to contribute significantly to the field and open up new horizons for further research and applications.