Discrete geometry, semialgebraic graphs, and the polynomial method

Many problems in discrete geometry can naturally be encoded by a structure known as a semialgebraic graph. These include the Erdős unit distance problem, incidence problems involving algebraic objects, and many more. In this talk, I will discuss several new structural and extremal results about semialgebraic graphs. These include a very strong regularity lemma with optimal quantitative bounds as well as progress 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 of Walsh. Based on joint work with Hung-Hsun Hans Yu.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/