Real polynomial partitioning

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The Guth--Katz polynomial partitioning was first introduced in their ground-breaking result on the Erdős distinct distances problem, and it has found many applications in incidence geometry since then. In this talk, I will discuss two recent works that apply the tool. The first is a joint work with Gabriel Currier and József Solymosi, which shows that near optimizers of the Szemerédi–Trotter theorem must be "rigid" in some sense. The second is a joint work with Jonathan Tidor, which concerns hypergraphs that are called semialgraic. A semialgebraic hypergraph is a hypergraph with vertices being points in some Euclidean space and edges defined by some semialgebraic relations. I will talk about how polynomial partitioning is useful to prove a strong regularity lemma and a Zarankiewciz-type result for semialgebraic hypergraphs.

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