Tropical Mathematics and its Applications

To register, please send an e-mail to b.smith9@lancaster.ac.uk, yue.ren2@durham.ac.uk or oliver.clarke@durham.ac.uk.

Alejandro Vargas (Warwick)

On tropical covers and embeddings

A rank-1 divisor on a smooth algebraic curve C corresponds to a non-trivial ramified cover φ : C → ℙ¹. The minimum degree of such φ is an intrinsic invariant, i.e. the gonality of C, which by Brill-Noether theory equals ⌈g/2 + 1⌉ if C is generic. When the genus g of C is even, the number of maps witnessing the gonality is finite and equals the (g/2+1)-th Catalan number. Tropically, rank-1 divisors do not necessarily correspond to maps, namely the theory contains some combinatorial artifacts, i.e. non-realizable divisors. Nonetheless, in joint work with Draisma, the expected Catalan number was recovered in the analogous count of maps φ witnessing the gonality of a metric graph Γ, by using certain multiplicity for the φ. Despite this and other several success stories, there are several outstanding open questions around tropical covers and the correspondence with divisors, both in the curve and in the higher dimensional setting.

This talk surveys some known results and lists some open questions.

Cat Rust (QMUL)

Tropicalising the Moduli Space of Collapsed Stable Maps to a Toric Surface

Fixing a proper toric surface X, one can consider the moduli space of rational curves in X which satisfy given tangency conditions to its toric boundary. These tangency conditions are specified by a vector x ∈ (Z2)n, and as x varies, the geometry of the moduli space changes. This allows us to consider geometric invariants of the space to be a function of x. Naturally, we can ask how the invariants change as x changes and aim to find regions of  (Z2)n  where a given invariant is constant. These moduli spaces have a stratification by equivalence classes of tropical maps called combinatorial types. Using this tropical stratification, we show how to find regions of (Z2)n such that the class of the moduli space in the Grothendieck ring of varieties is constant.

Thibault Poiret (St. Andrews)

Tropicalizing some natural covers of the space of curves.

Tropical geometry has proved extremely useful to study smooth projective curves and their deformation theory. The reason is that the moduli space of stable curves, which is proper, admits a tropicalization map to the space of stable graphs, and the fibres of this map are spaces of smooth curves. This relates a space we care about (smooth curves) with a space we are better equipped to study (stable curves) via the combinatorics of tropical objects. We can do something similar to spaces of roots, some natural covers of

the space of curves. Space of roots show up naturally in the study of curves themselves, but also in Hurwitz theory and enumerative geometry. I will talk about what they are, how to compactify them, and how to explicitly construct the tropicalization map on the compactification. 

Violeta Lopez (St. Andrews)

A new Brill–Noether general tropical curve

A classical result in the study of algebraic curves is the Brill–Noether theorem. In 2012, Cools, Draisma, Payne and Robeva provided a new proof of this theorem using tropical methods. They did it by showing that the tropical curve known as the chain of loops is “Brill–Noether general”. In this talk we will review some concepts and results related to the tropical Brill–Noether theory. After this, we will present a new tropical curve which is Brill–Noether general. This is my current research work.

Lena Weis (TU Berlin)

Shellings of tropical hypersurfaces

The shellability of the boundary complex of an unbounded polyhedron is investigated. For this concept to make sense, it is necessary to pass to a suitable compactification, e.g., by one point. This can be exploited to prove that any tropical hypersurface is shellable. Under the hood there is a subtle interplay between the duality of polyhedral complexes and shellability. Translated into discrete Morse theory, that interplay entails that the tight span of an arbitrary regular subdivision is collapsible, not shellable in general. This talk is based on joint work with George Balla and Michael Joswig.

Roan Talbut (Imperial)

Tropical Geometry for Phylogenetic Statistics

The identification of the tropical Grassmannian and the space of phylogenetic trees has inspired a range of research on the use of tropical geometry for phylogenetic statistics. We will review these connections and the various avenues of research which have followed. In particular, we consider the problem of comparing probability distributions on spaces of differing dimensions. We construct a Wasserstein distance between measures on different tropical projective tori via tropical projections of probability measures.  We prove that this distance is symmetric, whether mapping from a low dimensional space to a high dimensional space or vice versa.

Linxuan Li (QMUL)

Tropical fans supporting a reduced 0-dimensional complete intersection

A tropical fan is called regular if it supports a reduced 0-dimensional complete intersection, and for some cases the classification of regular fans is already complete. In this talk, I will recall the construction of the tropical intersection product via two equivalent ways, introduce the existing classifications from Fink and Esterov-Gusev, and give some unified answers of the classification problem to lower dimensional regular tropical fans.

Patience Ablett (Warwick)

Multigraded Hilbert schemes of toric varieties

Hilbert schemes are a useful tool allowing us to understand subschemes living in a fixed ambient space. The Hilbert scheme of P^n is well understood; in particular results of Hartshorne and Gotzmann allow us to identify and explicitly describe the connected components. Multigraded Hilbert schemes give a nice generalisation from this setting to any smooth projective toric variety. In this talk we will introduce these multigraded Hilbert schemes and discuss the status of attempts to generalise the results of Hartshorne and Gotzmann to this setting.