Algebraic Techniques in Geometry: The 10th Anniversary

Abstract

This year we are celebrating the 10th anniversary of a dramatic revolution in combinatorial geometry, fueled by the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were thought to be unreachable with more traditional techniques. The new era has begun with two groundbreaking papers of Guth and Katz, the second of which has (almost completely) solved the celebrated distinct distances problem of Paul Erdös, open since 1946. In this talk I will survey, as time permits, some of the progress that has been made since then, including a variety of problems on distinct and repeated distances and other configurations, on incidences between points and lines, curves, and surfaces in two, three, and higher dimensions, on polynomials vanishing on Cartesian products with applications, and on cycle elimination for lines and triangles in three dimensions.