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

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.

32 pages, 9 figures, submitted, Notices Amer. Math. Soc. (The published version is likely to be substantially shorter)

Cited by in corpus (16)

• On the size of Kakeya sets in finite fields
• A sharp bilinear restriction estimate for paraboloids
• Incidences between points and lines in R^4
• Finite field models in additive combinatorics
• Highly incidental patterns on a quadratic hypersurface in ${\mathbb{R}}^4$
• Incidences with curves in R^d
• Exponentially sparse representations of Fourier integral operators
• Additive combinatorics with a view towards computer science and cryptography: An exposition
• Improved restriction estimate for hyperbolic surfaces in R3
• Improving dimension estimates for Furstenberg-type sets
• A Generalization of the Convex Kakeya Problem
• Sylvester-Gallai type theorems for approximate collinearity
• On the arithmetic Kakeya conjecture of Katz and Tao
• The Inverse Kakeya Problem
• Construction of a one-dimensional set which asymptotically and omnidirectionally contains arithmetic progressions
• Small Furstenberg sets