Periodic Reporting for period 1 - Tropical (Tropical methods in MAthematics and Computer Science)
Reporting period: 2021-04-01 to 2023-03-31
Tropical arithmetic has been introduced by the computer scientist Imre Simon, and its name is inspired by Simon’s residency in Brazil.
The basic idea is simple: optimization problems involve naturally the minimization (sometimes maximization) operation. Replacing usual addition and multiplication of real numbers by minimum and addition, respectively, leads to tropical
matrix calculus and tropical varieties. The advantage of tropical arithmetic is that it is easy to solve (number theoretic problems become combinatorial problems) and that the operations are fast to compute (minimum and addition are linear in the size of the input, multiplication is quadratic).
A striking example of an every day’s life application are tropical algorithms for the scheduling of trains, which incidentally found a pioneering model for the Dutch railroad system.
Our scientific proposal touches three themes around tropical arithmetic: algorithm development, structural insights into matroids and computational tools for algebraic geometry. In more detail, these are:
(1) Tropical methods in game theory. The objective is the development of new and fast algorithms to solve mean payoff games. We investigate strategies to solve the “P versus NP”-problem for mean payoff games.
(2) Tropical structures for matroids. The objective is to continue a recent approach to matroid representation that is based on a novel type of algebraic object, which is called a pasture.
(3) Tropical Riemann-Roch (D3.1–D3.4). The objective is to develop a cohomological understanding of the tropical Riemann-Roch theorem that is based on tropical scheme theory.
The basic idea is simple: optimization problems involve naturally the minimization (sometimes maximization) operation. Replacing usual addition and multiplication of real numbers by minimum and addition, respectively, leads to tropical
matrix calculus and tropical varieties. The advantage of tropical arithmetic is that it is easy to solve (number theoretic problems become combinatorial problems) and that the operations are fast to compute (minimum and addition are linear in the size of the input, multiplication is quadratic).
A striking example of an every day’s life application are tropical algorithms for the scheduling of trains, which incidentally found a pioneering model for the Dutch railroad system.
Our scientific proposal touches three themes around tropical arithmetic: algorithm development, structural insights into matroids and computational tools for algebraic geometry. In more detail, these are:
(1) Tropical methods in game theory. The objective is the development of new and fast algorithms to solve mean payoff games. We investigate strategies to solve the “P versus NP”-problem for mean payoff games.
(2) Tropical structures for matroids. The objective is to continue a recent approach to matroid representation that is based on a novel type of algebraic object, which is called a pasture.
(3) Tropical Riemann-Roch (D3.1–D3.4). The objective is to develop a cohomological understanding of the tropical Riemann-Roch theorem that is based on tropical scheme theory.
(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.
(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.
The progress of this progress pushed forward the knowledge within the community of tropical geometry and matroid theory. In particular, they form one of the pillars of the recent developments in geometric matroid theory, a new flourishing field, not at least thanks to the field medal award to June Huh. These new developments clarify and solve long-standing problems in combinatorics.