Combinatorics Seminar: The tropical Cayley-Menger variety

Seminar | November 25 | 12:10-1 p.m. | 939 Evans Hall

 Robert Krone, UC Davis

 Department of Mathematics

The Cayley-Menger variety is the Zariski closure of the set of vectors specifying the pairwise squared distances between n points in $R^d$. For a graph on n vertices, a coordinate projection of the Cayley-Menger variety gives the possible edge lengths of the embeddings of the graph into $R^d$. Tropicalization converts an algebraic set into a polyhedral complex, the "combinatorial shadow" of the original variety. When $d=2$, the tropical Cayley-Menger variety is the set of sums of two ultrametrics on n leaves. We can describe its polyhedral structure in terms of pairs of rooted trees. This description leads to a new, tropical, proof of Laman's theorem, which is a characterization of the minimal generically rigid graphs in $R^2$. This is joint work with Daniel Bernstein.