Geometry Seminar

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The Szemerédi-Trotter Theorem Over Arbitrary Field Of Characteristic Zero

Speaker: Jiahe Shen, Columbia University

Location: Warren Weaver Hall 1314 and on Zoom

Date: Tuesday, February 10, 2026, 6 p.m.

Synopsis:

Let \(P\) be a set of \(m\) points and \(L\) a set of \(n\) lines in \(K^2\)), where \(K\) is a field with \(\mathrm{char}(K)=0\). We prove the incidence bound \(I(P,L)=O(m^{2/3}n^{2/3}+m+n).\) Moreover, this bound is sharp and cannot be improved. This resolves the Szemerédi-Trotter incidence problem for arbitrary field of characteristic zero. The key tool of our proof is the Baby Lefschetz principle, which allows us to reduce the problem to the complex case. Based on this observation, we further derive several related results over (K), including Beck’s theorem, the Erdős-Szemerédi sum–product estimate, and incidence theorems involving more general algebraic objects.

If time allows, we can also discuss some related open problems.

Notes:

In person and on Zoom.  Plz contact Boris Aronov if you are not NYU-affiliated and want to attend in person and/or to be put on the mailing list and get the Zoom information.