Abstract:

In tropical geometry one works over the tropical semiring $(\mathbb{R},\min,+)$. There are several natural ways to define collinearity of points in tropical space. One is to demand that their tropical convex hull is one-dimensional. A more restrictive version is to require that the tropical convex hull is generated by two of the points.

Using the former definition, the space $T_{d,n}$ of $n$ points on a line in tropical $d$-space is relevant in mathematical biology since it encodes sets of $n$ species on phylogenetic trees with $d$ leaves. Confirming a conjecture of Develin, Markwig and Yu recently showed that (a subdivision of) $T_{n,d}$ is shellable.

The latter definition leads to a space $B_{n,d}$. Its elements are the $d \times n$ matrices of Barvinok rank 2 that play a role in certain combinatorial optimization problems. Develin conjectured that $B_{n,d}$ is a manifold. I will sketch a simple proof of the conjecture and discuss the topology of this space.