Tropical Mathematics and its Applications

Lunch will be at the Botanical Garden: either meet at the Mathematical Sciences building at 11:30 to head over, or join at the Gardens when you arrive.

Dinner will be near the train station to allow people to make quick departures.

To register, please send an e-mail to b.smith9@lancaster.ac.uk and/or roan.s.talbut@durham.ac.uk.

Tropical Fermat-Weber points (Ruriko Yoshida)

In this talk, we discuss on the geometry of tropical Fermat--Weber points, that is, optimal solutions to a location problem over a projective space using a dissimilarity measure derived from the tropical metric. It is well-known that for a given sample, such points are not necessarily unique, and we show that the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem, and we describe the polytrope of optimal locations in the terminology of tropical geometry. Here we also consider an extension of  the previous work to a more general setting by considering a quasi-metric on the tropical projective torus that, in essence, provides a weighted interpolation between the symmetric and asymmetric tropical distances.  We revisit the tropical Fermat--Weber problem over the tropical projective torus using this measure.

From posets to polytopes: with a view towards tropical statistics (Kamillo Ferry)

A poset with weights on its relations satisfying the triangle inequality along chains defines a polytrope, a polyhedron that is both tropically and classically convex. We show how the combinatorial types of such polytropes correspond to regular central subdivisions of the root polytope of the poset, and that the weighted Hasse diagram gives their minimal facet description. This correspondence is used to enumerate generic types for up to seven nodes. We end with the statistical origin of the problem: polytropes are the supports of max-linear Bayesian networks, and the weighted Hasse diagram determines which parameters are identifiable. Joint work with Carlos Améndola.

Automorphism of column spaces and maximal subgroups of tropical matrix semigroups (Thomas Aird)

The set of n-by-n matrices over the tropical semiring forms a semigroup under tropical matrix multiplication. In this talk, I will explore the relationship between the subgroup structure of this semigroup and the automorphisms of tropical column spaces. I will begin by showing that the matrices sharing the same row and column spaces as a given idempotent matrix form a group, and that these groups are precisely the maximal subgroups of the semigroup. Each such group is naturally isomorphic to the group of tropical linear automorphisms of its common column space.