Presented by: 
Ngoc Tran, University of Texas, Austin
Tue 20 May, 3:00 pm - 4:00 pm
Building 67, Room 442

Polytropes are tropical polytopes which are also ordinary polytopes. They
arise in many places: as the set of feasible solutions to the all-pair
shortest path linear program; as cells in affine Coxeter arrangement; as
building blocks of tropical polytopes. One natural goal is to classify them
up to combinatorial equivalence.

The combinatorial types of polytropes are in bijection with cones in a
certain Grobner fan GF. In 2010, Joswig and Kutlas computed and
classified polytropes up to dimension 4. Higher dimensions seem
computationally infeasible, and the fan structure was not well-understood.
We show that while the fan GF is rather large, its structure is
essentially captured in one subfan induced by a specific hyperplane
arrangement. We give an explicit description of this arrangement, and
thereby successfully classify polytropes in dimension 5.

In this talk, I sketch the proof and discuss the open problem of classifying
polytropes in dimension 6 and higher.