Speaker: Ben Smith (Manchester)
Title: Face Posets of Tropical Polyhedra
Abstract: Tropical polyhedra can be viewed as the logarithmic limit of families of ordinary polyhedra. As members of these families may have differing face structures, it is difficult to define a coherent notion of face for tropical polyhedra. In this talk, we offer a possible solution by introducing a notion of tropical faces for a special class of tropical polyhedra arising as tropicalisations of blocking polyhedra. We then show how this face structure may be extended to all tropical polyhedra. Furthermore, we show how this notion of tropical face is intimately related to ideas from commutative algebra and order theory. This is joint work with Georg Loho.

Speaker: Marta Panizzut (TU Berlin)
Title: Local Dressians of matroids
Abstract: Tropical Grassmannians are rational polyhedral fans parametrizing realizable tropical linear spaces in tropical projective spaces. These are contractible polyhedral complexes arising from the tropicalization of linear spaces. Herrmann, Jensen, Joswig and Sturmfels introduced the Dressians, outer approximations of the tropical Grassmannians which parametrize all tropical linear spaces in tropical projective spaces. Moreover, they remarked that a stratification based on matroids can be described on Dressians, motivating definition local of a Tropical Grassmannians, Dressians and local Dressians will be the main characters of the talk. After introducing the main concepts, I will focus on the fan structures these objects coming from the Plücker relations and as subfans of the secondary fans of matroid polytopes. This is based on joint work with Jorge Alberto Olarte and Benjamin Schröter.

Speaker: Ebrahim Patel (Oxford)
Title: Modelling networks with Tropical Mathematics
Abstract: I will present some work carried out with Masters students on the popular application of optimal railway timetabling; we show that max-plus algebra can be used to optimise the network structure as well. Generalising to a max-min-plus system allows the modelling of threshold dynamics on networks. Thus, I propose a max-min-plus model of social and biological processes on networks. Asymptotically, the max-min-plus system reduces to a max-plus system; how this happens is an open question.