Sets with few ordinary lines

Given $n$ points in the plane, an ordinary line is a line that contains exactly two of these points, and a $3$-rich line is a line that contains exactly three of these points. An old problem of Dirac and Motzkin seeks to determine the minimum number of ordinary lines
spanned by n noncollinear points, and an even older problem of Sylvester (the orchard planting problem) seeks to determine the maximum number of $3$-rich lines. In recent work with Ben Green, both these problems were solved for sufficiently large $n$, by combining
tools from topology (Euler’s formula), algebraic geometry (the Cayley-Bacharach theorem, and the classification of cubic curves), additive combinatorics (via the group structure of said cubic curves), and even some Galois theory (through the theorem of Poonen and
Rubinstein that a non-central interior point in the unit disk can pass through at most seven chords connecting roots of unity). We will discuss how these ingredients enter into the solution to these problems in this talk.