Ting-Wei Chao, Carnegie Mellon University

Hung-Hsun (Hans) Yu, Princeton University

Abstract

The joints problem asks to determine the maximum number of joints N lines can form, where a joint in a d-dimensional space is a point on d lines in linearly independent directions. Recently, we determined the maximum exactly for k choose d-1 lines in d-dimensional space, namely k choose d. What is more important is that we are able to prove a structural result determining all optimal configurations, and this is the first success of the polynomial method in this direction. It turns out that our result implies a conjecture of Bollobás and Eccles as an immediate corollary regarding a generalization of the Kruskal–Katona theorem. In this talk, we will talk about the connection to that conjecture and also give a high-level overview of the key ideas. 

See: https://sites.google.com/view/rutgersdmseminar