The polynomial method in combinatorial incidence geometry
Seminar Room 1, Newton Institute
Combinatorial incidence geometry is concerned with controlling the number of possible incidences between a finite number of geometric objects such as points, lines, and circles, in various domains. Recently, a number of breakthroughs in the subject (such as Dvir's solution of the finite field Kakeya conjecture, or Guth and Katz's near-complete solution of the Erdos distance problem) have been obtained by applying the _polynomial method_, in which one uses linear algebra (or algebraic topology) to efficiently captures many of these objects inside an algebraic variety of controlled degree, and then uses tools from algebraic geometry to understand how this variety interacts with the other objects being studied. In this talk we give an introduction to these methods.