The ACO Seminar (2025–2026)

Many problems in discrete geometry can be encoded combinatorially via a structure known as a semialgebraic graph -- a graph whose edge relations are defined by polynomial equations and inequalities. These include the Erdős unit distance problem, incidence results of Szemerédi--Trotter, and numerous other problems. In this talk, I will introduce these ideas and discuss what is known about the class of semialgebraic graphs. This includes new structural and extremal results: a very strong (and quantitatively optimal) regularity lemma as well as improved Turán-type bounds for semialgebraic graphs. The proofs of these results are inspired by an idea of Bukh and Vasileuski and use novel polynomial method tools, which generalize the polynomial partitioning machinery of Guth--Katz and of Walsh.