Abstract: Given a set L of n lines in three dimensions, a joint is a point incident to at least three non-coplanar lines of L. What is the maximum possible number of joints that such a set of lines can have? A simple grid construction shows that this number can be at least of the order of magnitude of n^{3/2}, and it has been conjectured, about 20 years ago, that this is also the upper bound for any set of n lines in 3D.

The conjecture has been affirmatively solved only about two years ago, by Larry Guth and Nets Hawk Katz, using a proof technique based on fairly simple tools from algebraic geometry, a totally new and exciting approach to combinatorial problems of this kind in discrete geometry.

The joints problem, interesting but not that central, is a special case of a large class of incidence problems in combinatorial geometry, in which one seeks sharp bounds on the maximum possible number of incidences between points and lines or curves in the plane and in higher dimensions. I will also briefly review this broader and major topic in the talk.

I will present in the talk a simplified version of the new algebraic machinery, and show how it solves an extended version of the joints problem, in arbitrary dimensions, as well as several other related incidence problems.

Joint work with Haim Kaplan, Eugenii Shustin, and (the late) Gyorgy Elekes.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006