Tropical · Tropical methods in MAthematics and Computer Science

This proposal joins three themes around tropical arithmetics: WP1. Tropical methods in game theory.Mean payoff games form an interesting class in complexity theory since they are known to be in NP, but it is not known whether they can be solved in polynomial time. Our objective is to use tropical operators for the development of new and fast algorithms to solve mean payoff games. In addition, we search for strategies to establish a polynomial time algorithm.WP2. Tropical structures for matroids. In a recent paper, we have introduced a novel approach to study matroid representation in terms of a new algebraic structure: the representation theory of the matroid is completely controlled by its ""foundation"". Our objective is to continue this powerful theory by broadening the foundations and developing computational tools to determine the foundation of a matroid. Additionally, we aim for an understanding of foundations of 3-connected matroid, which conjecturally reveals a deep connectivity property for the foundation.(3) Tropical Riemann-Roch.The tropical Riemann-Roch theorem has found important applications in Brill-Noether theory. Up to date, this theorem is a purely combinatorial statement about graphs. Our objective is to use the richer structure of tropical scheme to develop a cohomological understanding and proof of the Riemann-Roch theorem. This involves the development of sheaf cohomology and etale morphisms for tropical schemes and an understanding of Berkovich skeleta as tropical schemes.Due to the interdisciplinary nature of this proposal (game theory and matroids form a part of computer science, our methods stem from a mathematical background), we chose Groningen as a basis to perform this proposal. The Bernoulli Institute in Groningen merges Mathematics and Computer Science in one departent, with three additional centers AI, CDSS and CogniGron. Moreover the BI hosts virtually all tropical geometers of the Netherlands.""