# LMS Workshop on Tropical Mathematics, February 2023

- Location
- Arts Building - Lecture Room 7 (2nd Floor) (R16 on campus map)

- Dates
- Friday 17 February 2023 (13:00-18:00)

- Contact

The School of Mathematics hosted the LMS Workshop on Tropical Mathematics on 17 February 2023. This is a hybrid meeting. Links to the lectures are given below with the abstracts.

This workshop aimed to foster future collaborations between specialists in various branches of tropical mathematics. As such, the talks were varied in their content. Talks were delivered in a hybrid mode (two talks on Campus and three talks via Zoom), with the intention of widening the scope of researchers who were able to deliver a talk and take part in the workshop.

## Confirmed speakers

- Victoria Schleis (University of Tübingen)
- Alheydis Geiger (Max Planck Institute for Mathematics in the Sciences)
- Abhimanyu Gupta (Technical University of Delft)
- Georg Loho (University of Twente)
- Michel van Garrel (University of Birmingham)

## Schedule

12:00-13:00: Lunch (Staff house).

Those of you who arrive early please find Sergey Sergeev in the School of Mathematics (Watson Building) or, alternatively, meet Sergey and other participants in the foyer of the building shortly before 12:00.

13:15-14:00: Victoria Schleis (University of Tübingen)

Title: Linear degenerate tropical flag varieties

Abstract: Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules. As linear degenerations of flag varieties are difficult to analyse algebraically, we describe them in a combinatorial setting and further investigate their tropical counterparts.

In this talk, I will introduce matroidal, polyhedral and tropical analogues and descriptions of linear degenerate flags and their varieties obtained in joint work with Alessio Borzì. To this end, we introduce and study morphisms of valuated matroids. Using techniques from matroid theory, polyhedral geometry and linear tropical geometry, we use the correspondences between the different descriptions to gain insight on the structure of linear degeneration. Further, we analyse the structure of linear degenerate flag varieties in all three settings, and provide some cover relations on the poset of degenerations. For small examples, we relate the observations on cover relations to the flat irreducible locus studied in representation theory.

Victoria Schleis's talk on YouTube.

14:05-14:50: Alheydis Geiger (Max Planck Institut)

Title: Self-duality in tropical geometry

Abstract: In classical algebraic geometry, self-dual point configurations arise, for example, as the hyperplane section of a canonical curve. Self-duality of generic point configurations translates to interesting geometric properties: 6 distinct points are self-dual if and only if they lie on a conic, 8 distinct self-dual points in generic position form a Cayley Octad. Self-dual point configurations are parametrised by a subvariety of the Grassmannian Gr(n,2n) and its tropicalisation. This talk will focus particularly on the tropical aspects. We will analyse the tropical self-dual Grassmannian trop(SGr(3,6)) and make first steps towards trop(SGr(4,8)).

This is joint work with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.

Alheydis Geiger's talk on YouTube.

15:00-15:30: Break

15:35-16:20: Abhimanyu Gupta (Technical University of Delft)

Title: Stability of max-plus algebraic discrete-event systems

Abstract: The modelling, analysis, and control of the system evolution defined over events results in the discrete-event systems framework. Examples include manufacturing and transportation networks. Max-plus algebra, with maximisation and addition as its basic operations,(and associated algebraic structures) conveniently handle the timing aspects of discrete-event systems when the schedule of operation of different tasks, such as order of trains, is made deterministic. We model discrete-event systems using a hybrid dynamical systems approach where purely (max-plus) algebraic models, derived from timing constraints among events, are enriched with automata-theoretic conflict resolution schemes to treat variable schedules. In this talk, I will present a Lyapunov-theoretic framework for studying stability of hybrid models in max-plus algebra. Then I present the application of the proposed framework to study boundedness of trajectories generated by a semigroup of matrices in the max-plus algebra. The resulting decision problems are then formulated as mixed-integer programs. I close the presentation with a brief discussion on the complexity of the decision problems.

Abhimanyu Gupta's talk on YouTube.

16:25 - 17:10: Georg Loho (University of Twente)

Title: Signed tropical convexity

Abstract: Tropical convexity inherently has only non-negative objects. Resolving this restriction leads to the introduction of signed tropical numbers. They allow us to formulate signed tropical convexity, which arises in different flavours.

I will survey basic properties of two notions of signed tropical convexity, TO-convexity and TC-convexity, which are based on open and closed signed tropical halfspaces, respectively. On the technical level, tropical hemispaces and signed tropicalisation play a crucial role.

While this sheds new light on the correspondence with mean payoff games and linear programming, many (complexity) questions remain open.

This talk is based on joint work with Mateusz Skomra.

17:15 - 18:00: Michel van Garrel (University of Birmingham)

Title: Tropical Enumerative Geometry

Abstract: Consider the complex projective plane with its 3 axes (x-axis, y-axis and line at infinity). Fix two points away from the axes. How many conics are there that pass through the two points and intersect each axis in 1 point only? This is the type of question that tropical enumerative geometry answers. The answer is 4, which corresponds to 1 tropical curve with multiplicity 4. I will explain the correspondence between algebraic curve counts and tropical curve counts in this setting and others. This is based on joint work with Brini and Bousseau.