Algebraic Techniques in Geometry: The New Revolution
I will survey the immense progress in combinatorial and computational geometry in the past seven years, triggered by the infusion of techniques from algebraic geometry and algebra, as introduced by Guth and Katz and further developed by the community at large.
This has led to solutions of very hard problems, with the crown jewel being a nearly complete solution to Erdos's 1946 problem on distinct distances in the plane.
In this talk I will survey the recent developments. They include new bounds for other variants of the distinct distances problem, new bounds for incidences between points and lines or other geometric entites in various contexts, and re-examination of the theory of Elekes, Ronyai, and Szabo on polynomials vanishing on grids, and numerous applications thereof.
In the (short) time that I will have I will only highlight some of the key developments, and demonstrate the new approach by a few examples.
The talk might assume some basic knowledge in algebra and geometry. Passion for geometric problems is always a plus.