Matroids in tropical geometry

Many problems in mathematics and its applications are concerned with understanding sets of solutions of polynomial equations. These sets are called "algebraic varieties", and their study forms a very rich field of mathematics called "algebraic geometry". Over the last few years, a new approach to studying algebraic varieties, called "tropical geometry", has gained a lot of interest. Tropical geometry aims to study algebraic varieties by transforming them into "polyhedral" objects, which can be studied using discrete methods. This approach has, in many cases, given rise to novel insights and new computational techniques. This project aimed to discover and develop connections between tropical geometry and the field of "matroid theory", which are useful for the further development of tropical geometry and its applications. The main results obtained include the development of a rich algebraic theory relevant to tropical geometry, in which matroid theory plays a central role. Also, new geometric objects that broaden the link between tropical geometry and matroid theory were discovered and studied.

The main outcome has been to further understand essential objects in tropical geometry, such as tropical linear spaces and tropical homogeneous spaces. Also, to use new combinatorial insights to study tropical schemes, and develop in this way a useful algebraic foundation for tropical geometry. The applicant has introduced new mathematical objects that have gotten significant interest by the research community, and that are expected to play an important role in future. As part of this project, a total of 7 research papers were produced (some of which are still in preparation). As secondary goals, the project managed to strengthen the combinatorial community in Norway by organizing yearly meetings on discrete mathematics, and to contribute to the internationalization of the community by actively participating in conferences, collaborating with several international researchers, and organizing yearly workshops with leading international scholars as invited speakers.

This is a proposal for a 3-year postdoctoral research project in pure mathematics, with the goal of strengthening and expanding the connections between the fields of tropical geometry and matroid theory. Tropical geometry is a relatively new area of mathematics that offers a way to study algebraic varieties and other structures by transforming them into combinatorially defined polyhedral objects, allowing the use of combinatorial and matroidal methods for their study. This approach has, in many cases, given rise to novel insights and new computational techniques. The project aims to develop old and new connections between tropical geometry and matroid theory, by investigating fundamental structures like tropical linear spaces, tropical homogeneous spaces, tropical schemes, and tropical discriminantal varieties. The resulting combinatorial insights will be useful for the development of tropical geometry and its applications, and are expected to be of great interest to other researchers.