UBC Math Department Colloquium: Jonathan Tidor

Many problems in discrete geometry can be conveniently encoded by a structure known as a semialgebraic graph. These problems include 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.

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 improved bound 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.