# Matroids in tropical geometry

Lead Research Organisation:
Queen Mary University of London

Department Name: Sch of Mathematical Sciences

### Abstract

Tropical geometry is the geometry obtained when the operations of addition and multiplication on the real numbers are replaced by the operations of minimum and addition, respectively. Tropical mathematics have been studied in many different contexts, but a deep connection to algebraic geometry has only been established in the last few decades. This development has led to numerous applications in many different areas, such as enumerative algebraic geometry, mirror symmetry, optimisation, and computational biology.

Matroids are mathematical objects that abstract many different notions of independence throughout mathematics. They are essential in tropical geometry, as they play the same role as linear subspaces in classical mathematics. The connection between tropical geometry and matroid theory is deep and strong, and has been very beneficial to both fields.

Recently, the PI and his collaborators have introduced two new notions in tropical geometry that promise to be very useful for the field: tropical ideals and tropical CSM classes. Tropical ideals serve as algebraic and combinatorial objects that keep track of the equations that define a tropical variety. Tropical CSM classes are tropical objects that carry combinatorial and topological information about any smooth tropical variety. Matroids are essential in the construction of both of these objects.

The aim of this project is to continue to develop the strong connections between matroid theory and tropical geometry, by pushing the study of these two novel tropical notions: tropical ideals and tropical CSM classes. Investigating these promising objects will push the reach of tropical geometry further, opening the door to numerous applications such as a tropical study of Hilbert schemes, a deeper exploration of realisability questions in tropical geometry, and new approaches to enumerative algebro-geometric problems.

Matroids are mathematical objects that abstract many different notions of independence throughout mathematics. They are essential in tropical geometry, as they play the same role as linear subspaces in classical mathematics. The connection between tropical geometry and matroid theory is deep and strong, and has been very beneficial to both fields.

Recently, the PI and his collaborators have introduced two new notions in tropical geometry that promise to be very useful for the field: tropical ideals and tropical CSM classes. Tropical ideals serve as algebraic and combinatorial objects that keep track of the equations that define a tropical variety. Tropical CSM classes are tropical objects that carry combinatorial and topological information about any smooth tropical variety. Matroids are essential in the construction of both of these objects.

The aim of this project is to continue to develop the strong connections between matroid theory and tropical geometry, by pushing the study of these two novel tropical notions: tropical ideals and tropical CSM classes. Investigating these promising objects will push the reach of tropical geometry further, opening the door to numerous applications such as a tropical study of Hilbert schemes, a deeper exploration of realisability questions in tropical geometry, and new approaches to enumerative algebro-geometric problems.

### Planned Impact

This project will develop fundamental research in tropical geometry, strengthening its foundations and expanding its reach and applications. This will have an immediate impact for other researchers in the field, which in turn will lead to impact for academics in nearby fields such as algebraic geometry and combinatorics, and ultimately for end-users outside of mathematics.

Tropical geometry has already had numerous applications in areas outside of mathematics. For instance, Paul Klemperer used tropical geometry to design a new type of auction called "product-mix auction", which was used by the Bank of England to arrange loan terms during the financial crisis. Other applications to economics have been developed since then, such as Yoshinori Shiozawa's application to Ricardian theory of international trade.

Another example comes from the study of deep neural networks in the field of artificial intelligence. As described by Zhang, Naitzat, and Lim, tropical geometry provides the perfect language to understand the class of functions that can be computed by a deep neural network with rectilinear activation functions. This close connection provides an algebro-geometric insight into important aspects of this very popular machine-learning technique, which has just begun to be exploited.

To maximise the opportunities of impact outside of academia, the PI will stay up to date on the different applications of tropical geometry in other areas, including machine learning and economics. He will continue to give colloquium talks and presentations to non-specialists in more distant fields, with the aim of introducing them to tropical geometry and helping them realise some of its possible applications. In addition, the PI will engage in public outreach activities, such as delivering public lectures on tropical geometry. This will help foster interest in science and mathematics, and communicate to the public some of the recent research in this field.

Tropical geometry has already had numerous applications in areas outside of mathematics. For instance, Paul Klemperer used tropical geometry to design a new type of auction called "product-mix auction", which was used by the Bank of England to arrange loan terms during the financial crisis. Other applications to economics have been developed since then, such as Yoshinori Shiozawa's application to Ricardian theory of international trade.

Another example comes from the study of deep neural networks in the field of artificial intelligence. As described by Zhang, Naitzat, and Lim, tropical geometry provides the perfect language to understand the class of functions that can be computed by a deep neural network with rectilinear activation functions. This close connection provides an algebro-geometric insight into important aspects of this very popular machine-learning technique, which has just begun to be exploited.

To maximise the opportunities of impact outside of academia, the PI will stay up to date on the different applications of tropical geometry in other areas, including machine learning and economics. He will continue to give colloquium talks and presentations to non-specialists in more distant fields, with the aim of introducing them to tropical geometry and helping them realise some of its possible applications. In addition, the PI will engage in public outreach activities, such as delivering public lectures on tropical geometry. This will help foster interest in science and mathematics, and communicate to the public some of the recent research in this field.

## People |
## ORCID iD |

Felipe Rincon (Principal Investigator) |

### Publications

Anderson N
(2022)

*Paving tropical ideals*in Journal of Algebraic Combinatorics
De Medrano L
(2023)

*Chern Classes of Tropical Manifolds*
Maclagan D
(2022)

*Varieties of tropical ideals are balanced*in Advances in Mathematics
Rincón F
(2023)

*Tropical Combinatorics*in Notices of the American Mathematical SocietyDescription | The connection between matroids and tropical geometry has been developed much further as a result of this work. The project focussed on the study of two fundamental objects in tropical geometry: tropical ideals and CSM classes of matroids. On the one hand, tropical ideals have been much better understood, and the relation between their algebra and their geometry has been developed much further. This includes deep results such as the fact that the variety associated to a tropical ideal is a balanced polyhedral complex of the same dimension as the tropical ideal, an elimination theorem for tropical ideals, and the strong Nullstellensatz for tropical ideals. Applications to the study of Newton polytopes of zero-dimensional ideals, and closures of tropical varieties in tropical toric varieties were developed as well. On the other hand, the theory of CSM classes of matroids has also been explored further, and generalised to the much broader context of tropical manifolds. Of particular interest are powerful results that prove that CSM classes are preserved under the tropicalisation process, satisfy a certain adjunction formula, and comply with Noether's Formula in the case of tropical surfaces. Additionally, a new presentation of CSM classes in the Chow ring of a matroid has been obtained. The project has resulted in a total of six top-quality academic publications. At the moment, two have already been published, one has been submitted for publication, and three more are in preprint status and will soon be submitted. |

Exploitation Route | The foundational results on tropical ideals and CSM classes of matroids developed in this project provide other researchers in the area with powerful tools for exploring novel research questions in tropical geometry, its algebraic foundations, and the geometric study of matroid theory. |

Sectors | Other |

URL | https://sites.google.com/site/feliper84/publications |

Description | CSM cycles with Kris Shaw at Oslo |

Organisation | University of Oslo |

Country | Norway |

Sector | Academic/University |

PI Contribution | Expertise on matroids and polyhedral geometry. |

Collaborator Contribution | Expertise on real algebraic geometry and tropical geometry. |

Impact | In preparation: "Chern classes of tropical manifolds". |

Start Year | 2021 |

Description | Tropical Ideals with Diane Maclagan at Warwick |

Organisation | University of Warwick |

Department | Warwick Mathematics Institute |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Expertise in matroids and polyhedral geometry. |

Collaborator Contribution | Expertise in algebraic geometry, tropical geometry, and commutative algebra. |

Impact | Preprint: "Varieties of tropical ideals are balanced". In preparation: "Strong Nullstellensatz for Tropical Ideals". |

Start Year | 2021 |

Description | Workshop on Matroids and Tropical Combinatorics |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Postgraduate students |

Results and Impact | I organised a workshop with 70 participants, mostly PhD students and young researchers, on Matroids and Tropical Geometry. The workshop consisted mainly of 3 minicourses given by international world experts, and of 15 participant talks. |

Year(s) Of Engagement Activity | 2023 |

URL | https://sites.google.com/view/matrocomlondon/home |