Tropical mathematics in Swansea
 7-9 July, 2014
Swansea, UK


Abstracts

Alex Fink (QMUL)
   
    Stiefel linear spaces

Unlike the classical situation, not all tropical linear spaces are row spaces of matrices, i.e., stable sums of points.  We call the ones which are Stiefel tropical linear spaces.  We discuss their combinatorial structure viewed through the lenses of transversal matroids or of tropical parametrised images, and a nice atlas of local coordinate systems for their moduli space.

Stiefel spaces appear to be of relevance to tropical foundations, in giving the right contruction of, at least, principal ideals.  We explain this connexion as well, into which investigation is ongoing.

This talk is based on joint work with Felipe Rincon and with Jeffrey and Noah Giansiracusa.


Mark Gross (Cambridge)

1. Cluster algebras and mirror symmetry
2. Logarithmic Gromov-Witten invariants


Rod Halburd (UCL)

    Tropical Nevanlinna theory

Nevanlinna theory studies the value distribution of meromorphic functions in the complex plane. In this talk we will describe a tropical version of some key results in the theory that apply to tropical meromorphic functions, which are piecewise linear functions on the real line. Applications to so-called ultra-discrete equations will be discussed.


James Hook (Manchester)

    Max-plus singular values and the Hungarian scaling

The modulus of the roots of a polynomial can be approximated by the max-plus roots of an associated max-plus polynomial. Likewise the eigenvalues/singular values of a matrix cab be approximated by the max-plus eigenvalues/singular values of an associated max-plus matrix. The advantage of using max-plus algebra is that the 'max-plus version' of a problem (finding roots, eigenvalues, singular values etc...) is typically much easier to solve than the original classical problem. Information gained by solving the max-plus problem can then be used to help solve the original classical algebra problem.

In my talk I will give an introduction to max-plus eigenvalues and singular values and discuss some interesting connections between max-plus singular values and Hungarian scaling, which is a classical algebra diagonal scaling strategy based on optimal assignments (a.k.a maximal matchings).


Marianne Johnson (Manchester)

    Idempotent tropical matrices

This talk concerns an exact characterisation of those tropical polytopes which arise as the column (or row) space of a (multiplicatively) idempotent tropical matrix (joint work with Kambites and Izhakian). A particularly  interesting class of tropical idempotents arises from finite (semi)-metric spaces. We garner some further information about the geometric structure of the corresponding column/row spaces in these cases.


Elisa Postinghel (KU Leuven)

    Faithful tropicalisations and torus actions

Given a closed subvariety X of the affine n-space, there is a surjective map from the analytification of X in the sense of Berkovich to its tropicalisation. It is natural to ask whether this map has a continuous section. Affirmative answers are given in recent work by Baker, Payne, and Rabinoff in the case of curves, and by Cueto, Haebich, and Werner in the case of Grassmannians of 2-spaces. In this talk we show how one can construct such a section when X is a linear space or is obtained from a linear space smeared around by a torus action. In particular, this gives a new, more geometric proof for the Grassmannian of 2-spaces and it also applies to some examples of determinantal varieties.  Joint work with J. Draisma.


Felipe Rincon (Warwick)

    Tropical ideals

In a recent preprint, Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties. In this talk I will describe how the structure of these tropical schemes is closely related to valuated matroids, and how it naturally leads to an interesting and more general notion of tropical ideals. This is joint work with Diane Maclagan.

Tony Yue Yu (Paris 7)

    Gromov compactness in non-archimedean analytic geometry

I will begin by explaining how the theory of Berkovich spaces gives a natural framework of global tropicalization. It enables us to apply techniques of tropical geometry in a wider context than toric varieties. The first step of doing enumerative geometry in the setting of Berkovich spaces is the non-archimedean analog of Gromov’s compactness theorem. I will introduce a notion of Kähler structures in non-archimedean geometry using metrizations of virtual line bundles. Then I will explain the construction of the k-analytic stack of stable maps and the proof of the compactness theorem. If time permits, I will discuss the tropicalization of the moduli space of stable maps.  The reference is arXiv:1401.6452.