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Seminars

Special Colloquium

Counting Joints and Triangles

2026/10/15 (Thu.) 15:30~16:30
  • Date : 2026/10/15 (Thu.) 15:30~16:30
  • Location : Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)
  • Speaker : Ting-Wei Chao (Massachusetts Institute of Technology)
  • Abstract : In this talk, we will go through problems in incidence geometry and extremal graph theory, and show some interesting connection between these problems. The joints problem asks for the maximum number of joints formed by $n$ lines in $\mathbb{R}^3$, where a joint is a common intersection of three non-coplanar lines. Using the polynomial method, Guth and Katz showed that there are at most $O(n^{3/2})$ joints. Together with Hung-Hsun Hans Yu, we refined their method and proved a tight bound on this problem. When restricting to specific configurations, the joints problem becomes a generalization of the Kruskal—Katona theorem, and our theorem resolves a conjecture of Bollobás and Eccles. By a similar reduction, the restricted version of the multijoints problem in $\mathbb{R}^3$ becomes a colorful version of the Kruskal—Katona theorem on counting rainbow triangles in graphs. I will discuss several sharp results in this direction from joint work with Maya Sankar and Hung-Hsun Hans Yu.
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