Speaker:  Matt Larson  (Stanford)

Title:  Cross-ratio degrees

Abstract:  

Given n-3 subsets of {1, …, n} of size 4, the cross-ratio degree counts the number of ways to place n marked points on the Riemann sphere such that the n-3 cross-ratios are prescribed generic complex numbers. If more than k-3 of the sets involve just k of the points, then those k points are overdetermined and the cross-ratio degree vanishes. We show that this is the only reason why a cross-ratio degree can vanish: if no subset of the points is overdetermined, then the cross-ratio degree is positive. This gives a new proof of Laman’s theorem characterizing graphs whose generic embedding in the plane is rigid. Joint with Joshua Brakensiek, Christopher Eur, and Shiyue Li.