Higher dimenisonal Elekes-Szabó problem

• Applicant: Dr. Tingxiang Zou
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 574217313

This project investigates the higher-dimensional Elekes--Szabó problem, which seeks to classify algebraic varieties whose intersections with Cartesian products of finite sets do not exhibit the expected combinatorial growth. While the classical Elekes--Szabó Theorem over the field of complex numbers shows that such behaviour is only possible for varieties closely related to abelian algebraic groups, new developments suggest a richer structure in higher dimensions, potentially involving nilpotent group actions. Our goal is to extend the theory beyond its current limitations, focusing in particular on the genuinely higher-dimensional case. We also seek to understand this phenomenon over a broader class of ambient varieties and in positive characteristic. Achieving this requires synthesizing techniques from several mathematical disciplines. The project lies at the intersection of incidence geometry, algebraic group theory, and additive combinatorics, and builds on recent progress in model theory, approximate subgroup theory, and algebraic geometry to develop a unified framework for expansion phenomena in higher dimensions.

• DFG Programme: Emmy Noether Independent Junior Research Groups