Geometry Seminar

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Semialgebraic graphs and polynomial partitioning

Speaker: Jonathan Tidor, Princeton U

Location: Warren Weaver Hall 1314

Videoconference link: https://youtu.be/blybaHFf0w4

Date: Tuesday, November 11, 2025, 6 p.m.

Synopsis:

A semialgebraic graph is a graph whose vertices are points in Euclidean space and whose edge relation is defined by polynomial inequalities on the vertices. Numerous problems in discrete geometry can be encoded by a semialgebraic graph. These include the Erdős unit distance problem and its variants, incidence problems involving algebraic and semialgebraic objects, and many more. I will discuss a number of new structural and extremal results for semialgebraic graphs and some geometric consequences of these results. These include a very strong regularity lemma with optimal quantitative bounds as well as progress on the semialgebraic Zarankiewicz problem. These results are proved using a novel extension of the polynomial partitioning machinery of Guth–Katz and of Walsh.

Based on joint work with Hung-Hsun Hans Yu.