Geometry and Topology Seminar: The polynomial method in incidence geometry and harmonic analysis.
23 June 2016 13:00 in CM221.
When we want to understand a geometric picture, finding the zero set of a polynomial hiding in it can be very helpful: it can reveal structure and allow computations. This technique is known as the polynomial method, and was first used to count point-line incidences in 2008 by Dvir, for the solution of the Kakeya problem in finite fields. Since then, the polynomial method has revolutionised discrete incidence geometry, largely thanks to the fact that interaction of lines with varieties is, to an extent, well-understood. Recently, Guth discovered agreeable interaction between varieties and tubes as well, opening up the exciting possibility that many problems of point-tube incidence flavour could also have algebraic structure; and such problems are of interest in harmonic analysis. In this talk, we will present the polynomial method via simple discrete analogues of the Kakeya problem, and discuss its potential to be extensively used in harmonic analysis.