Many problems in discrete geometry can be conveniently encoded by a structure known as a semialgebraic graph. Some of these problems include the Erdős unit distance problem and its variants, the point-line incidence problems studied by Szemerédi–Trotter and by Guth–Katz, general problems about incidences of varieties, and many more examples.
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, as well as results 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.