Nov. 17, 3:30pm, Wean 8220

Abdul Basit, Rutgers

On the number of ordinary lines determined by sets in complex space

Abdul Basit, Rutgers

On the number of ordinary lines determined by sets in complex space

Abstract:

The classical Sylvester-Gallai theorem states the following:
Given a finite set of points in the *2*-dimensional Euclidean plane, not
all collinear, there must exist a line containing exactly *2* points
(referred to as an ordinary line). In a recent result, Green and Tao
were able to give optimal lower bounds on the number of ordinary lines
for large finite point sets.

In this talk we will consider the situation over the complex numbers.
While the Sylvester-Gallai theorem as stated is false in the complex
plane, Kelly's theorem states that if a finite point set in
*3*-dimensional complex space is not contained in a plane, then there
must exist an ordinary line. Using techniques developed for bounding
ranks of design matrices, we will show that either such a point set
must determine at least *3n*/*2* ordinary lines or at least *n*-*1* of the points are contained in a plane. (Joint work with Z. Dvir, S.
Saraf and C.Wolf.)

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.