Abstract: Suppose you take n points in the plane, not all on a line. A celebrated result, the Sylvester-Gallai theorem, states that there is an “ordinary line”, that is to say some line passing through precisely two of the points. I will talk about some recent work with Terence Tao in which we have established that there are at least n/2 ordinary lines for large n. This bound is tight. Although this is a result in combinatorics the proof requires some simple ideas from a variety of areas including plane algebraic geometry and additive number theory. I will try to give an account of these connections at a level suitable for a general mathematical audience.
Professor Green is the Waynflete Professor of Pure Mathematics at Oxford University. Further information can be found on his website.
The talk will be followed by the reception in the Huxley Common Room.