Matroids in tropical geometry

Mange problemer i matematikk og dens anvendelser dreier seg om å forstå mengder av løsninger på polynomielle likninger. Disse mengdene kalles "algebraiske varieteter", og studiet av disse er en rik gren av matematikken som kalles "algebraisk geometri". De siste årene har en ny tilnærming til studiet av algebraiske varieteter, "tropisk geometri", vekket stor interesse. I tropisk geometri forsøker man å studere algebraiske varieteter ved å gjøre dem om til "polyhedrale" objekter, som kan studeres med diskrete metoder. Denne nye tilnærmingen har i flere tilfeller ført til ny innsikt og nye beregningsteknikker. Dette prosjektet søkte å utforske og utvikle koblinger mellom tropisk geometri og feltet "matroideteori", noe som er nyttig for den videre utviklingen av tropisk geometri og dens anvendelser. De viktigste resultatene som ble fått inkluderer utviklingen av en rik algebraisk teori som er relevant for tropisk geometri. I denne teorien spiller matroideteori en sentral rolle. I tillegg ble nye geometriske objekter oppdaget og studert, som utvider koblingen mellom tropisk geometri og matroideteori.

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.