The past few years have seen a flurry of new incidence results in the wake of Guth and Katz's seminal paper on the ErdÅ‘s distinct distance problem. Many of these incidence theorems only apply to points and algebraic varieties in (real) Euclidean space. In many cases it is reasonable to expect that analogous theorems should hold over ℂ, but limited progress has been made in this direction. In this talk, I will discuss a new point-curve incidence theorem in the complex plane. The proof of this theorem introduces some ideas that are new to incidence geometry, including Frobenius' theorem on integrable distributions, a quantitative version of Noether's ascending chain condition, and Chevalley's upper semi-continuity theorem. This is joint work with Adam Sheffer. |