The project sits in computational complexity and the analysis of algorithms, focusing on when algorithmic problems are provably infeasible to solve or even to approximate efficiently. Classic results show that many NP-hard optimization problems can’t be approximated efficiently unless P = NP. The PCPABF pushes this frontier further by defining the following objectives:
• Objective: advance Khot’s Unique Games Conjecture (UGC) using probabilistically checkable proofs (PCPs) and the analysis of Boolean functions.
Technical focus: study expansion and structure in high-dimensional combinatorial objects (like Grassmann and Johnson graphs) and develop new tools in Boolean function analysis, sharp thresholds, and hardness of approximation.
• Objective: analyzing computational problems via linear-algebraic or lattice-based frameworks.
Technical focus: Formalizing computational problems in algebraic rather than purely combinatorial ones. Reducing diverse computational or decision/optimization problems to standard lattice problems (SVP, CVP, lattice-basis problems, integer-kernel/image problems, etc.) or to problems in integer linear algebra. Studying the complexity (hardness, approximability, algorithmic solvability) of these lattice / linear-algebra derived problems, often in high dimension, and understanding how hardness scales with dimension, approximation factor, norm choice, and basis quality. Exploring what this algebraic/lattice viewpoint reveals about the nature of computational hardness, potentially providing evidence for hardness beyond NP (e.g. showing “intermediate complexity”), or building cryptographic primitives (or proofs) based on that hardness. Developing or harnessing algebraic / geometric methods (basis reduction, normal forms, integer matrix theory, dual lattices / duality, geometry-of-numbers, lattice-based reductions) to tackle problems that traditional combinatorial methods struggle with.
