Extremal problems in combinatorial geometry

Combinatorial geometry is a field that studies combinatorial problems that involve some simple geometric objects/notions, such as: lines, points, distances, collinearity. While such problems are often easy to state, some of them are very difficult, have a deep underlying theory, and remain (or have remained) open for many decades. Extremal problems in the field ask how large or small a certain quantity, defined over a finite set of geometric objects, can be under certain restrictions. One is also interested in characterizing the configurations that attain the extremum.

In the talk I will describe few classical examples to such problems (distinct distances, unit distance, ordinary lines, orchard-planting problem), and will review (briefly) recent progress in some of them. I will then switch to a simpler variant of these questions, in which the points are assumed to lie on a constant-degree algebraic curve. I will introduce a general unified approach to attack problems of the latter kind.