3:00pm - 5:00pmComputational tropical geometry
Chair(s): Kalina Mincheva (Yale University), Yue Ren (Max Planck Institute for Mathematics in the Sciences, Germany)
This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Connectivity of tropical varieties
Diane Maclagan1, Josephine Yu2
1Warwick University, 2Georgia Tech
The standard algorithm to compute tropical varieties makes crucial use of the fact that the tropicalization of an irreducible variety is connected. I will discuss joint work with Josephine Yu showing that the tropicalization of a d-dimensional irreducible variety satisfies a stronger d-connectedness property.
Tropical convex hull of polytopes
Cvetelina Hill1, Sara Lamboglia2, Faye Pasley Simon3
1Georgia Tech, 2Goethe Universität Frankfurt, 3North Carolina State University
Tropical convexity has been mostly focused on tropical convex hull of finitely many points, i.e., tropical polytopes. Moreover there has been some work on polytropes which are convex tropical polytopes. In this talk I will consider the tropical convex hull of polytopes and polyhedra. I will show that these are convex sets and that in some cases tconv(conv(S))=conv(tconv(S)) and tconv(pos (S))=pos(tconv(0,S)) for a finite set S. This will lead the way to compute the tropical convex hull of a tropical variety.
Algorithmic questions around tropical Carathéodory
Georg Peter Loho
London School of Economics
Since Imre Bárány found the colourful version of Carathéodory's theorem in 1982, many combinatorial generalizations and algorithmic variations have been considered. This ranges from variations of the colour classes to different notions of convexity. We take a closer look at the tropical convexity version of this theorem. We provide new insights on colourful linear programming and matroid generalizations from a tropical point of view, by considering additional sign informations. We focus on explicit constructions for 'colourful simplices'. The difficulty of the arising algorithmic questions ranges from greedily solvable to NP-hard.
Convergent Puiseux series and tropical geometry of higher rank
Ben Smith
Queen Mary University of London
Tropical hypersurfaces arising from polynomials over the Puiseux series are well studied and well understood objects. The picture becomes less clear when considering Puiseux series in multiple indeterminates. Unlike their rank one counterparts, these higher rank tropical hypersurfaces are not ordinary polyhedral complexes, but we shall see they still have a large amount of structure. Moreover, by restricting to convergent Puiseux series we show how one can describe them via the rank one tropical hypersurfaces arising from substitution of indeterminates. We will also consider a couple of applications of this framework, including a new viewpoint for stable intersection in the vein of symbolic perturbation.