From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE

TL;DR

This paper surveys the deep connections between combinatorics, analysis, and PDEs, highlighting how geometric and arithmetic combinatorics relate to oscillatory integrals and wave equations.

Contribution

It provides a comprehensive overview of emerging links between combinatorial problems and analytical PDE techniques, unifying diverse mathematical areas.

Findings

Identifies connections between Kakeya problem and wave stability

Links arithmetic progressions to oscillatory integral estimates

Highlights the role of combinatorics in PDE well-posedness

Abstract

We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic progressions), oscillatory integrals (such as the Bochner-Riesz, restriction, and local smoothing problems), and the local and global well-posedness theory for non-linear dispersive and wave equations.