(1) In collaboration with Marianne Akian, Stephane Gaubert and Matthias Mnich, we develop a new method to algorithmically solve mean payoff games that improves on previous methods and provides a linear time algorithm for a subclass of games. Ideas that build up on this algorithm have been investigated by Robert Modderman in his master thesis under my supervision at the University of Groningen.
(2) In collaboration with Matthew Baker, we have exploited our new methods to study matroid representations. As a first (completed) project, we have categorized and generalized lift constructions of matroid presentations, which lead to new concrete results in matroid theory and algebraic geometry. A follow-up paper on computational methods for the central invariant that is used in our method is nearly completed. A computer program developed by Chen and Zhang, with help by Baker and myself, has served to probe into computational complexity that we unreachable before. This led to new structural insight, which we are actively investigating.
(3) The most fundamental insight towards the tropical Riemann Roch Theorem is that tropical curves in the context of this result behave in fact like surfaces. This requires a much more sophisticated setup in the sense of a Grothendieck-Riemann-Roch theorem. We are working actively (in different constellations of collaborators) on the foundational theory that is necessary to achieve this goal. One first important stepping stone was achieved jointly with Manoel Jarra: we develop a general theory of flag matroids with coefficients, which is the backhold for sheaf cohomology.
