Jonathan Tidor, Stanford University

Abstract

** Please note day change to Tuesday
Many problems in discrete geometry can be naturally encoded by a graph. Using tools from graph theory then gives information about the original geometric problem. In recent years, this paradigm has been strengthened by noting that, in many cases of interest, the corresponding graph is in the class of so-called "semialgebraic graphs". Proving strong results on the structure of semialgebraic graphs then immediately has many consequences back in the geometric setting. Semialgebraic graphs are useful for studying a number of problems including the Erdős unit distance problem and many of its variants, point-line incidence problems studied by Szemerédi–Trotter and by Guth–Katz, general problems about incidences of varieties, and many more examples.

In this talk, I will introduce some problems in discrete geometry and define semialgebraic graphs. Then I will discuss a number of new structural and extremal results about semialgebraic graphs and the geometric consequences of these results. These include a regularity lemma with asymptotically optimal bounds and an improvement on the Zarankiewicz problem for semialgebraic graphs. These results are proved via a novel extension of the polynomial method, building upon the polynomial partitioning machinery of Guth–Katz and Walsh.

Based on joint work with Hung-Hsun Hans Yu.

See: https://sites.google.com/view/rutgersdmseminar