Tropical Mathematics and its Applications, November 2017

A joint research group in tropical mathematics has been formed by researchers in UK mathematics departments at universities including Manchester, Birmingham, Warwick, Queen Mary and Swansea, with financial support from the London Mathematical Society.

This page gives details of the next meeting, to be held in Birmingham. Funds are available to support the attendance of UK-based postgraduate students. For more information, please contact Dr Sergey Sergeev.

Venue

All talks will take place in Lecture Theatre B, Watson Building (School of Mathematics), University of Birmingham.

Speakers

• Marvin Anas Hahn (University of Tübingen, Germany)
• Daniel Jones (INRIA and École Polytechnique, Paris, France)
• Laure Daviaud (University of Warwick, UK)
• Nikolai Krivulin (University of St Petersburg, Russia)

Agenda

Please note: where academics have agreed to share their presentations, these are included as PDF-format downloads.

13:00 Marvin Anas Hahn: A tropical approach to the piecewise polynomiality of monotone Hurwitz numbers

Hurwitz numbers are important enumerative objects connecting various areas of mathematics. These objects can be defined in terms of factorisations in the symmetric group. Double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone double Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem and it was proved that these objects are piecewise polynomial in a certain sense. The aim of this talk is to use a connection between monotone double Hurwitz numbers and tropical covers in order to give algorithms to compute the polynomials for monotone double Hurwitz numbers using Erhart theory.

13:45 Daniel Jones: A discrete geometry model of fire propagation in urban areas (PDF, 2.9 MB)

We explore the phenomenon of fire propagation in urban areas using a discrete geometry model. Fires of this kind are fundamentally different to, say, wildland fires. In the latter, Hamilton-Jacobi PDEs have been used and the fire-front is found to be elliptical according to some parameters. Fire propagation in cities, however, is inherently discrete and we discuss the geometry of the cellular automata model here. The main theoretical result states that the long term fire front is a polyhedral ball. The theory can be viewed as an extension of tropical spectral theory to lattices. We finally give some suggestions for incorporating the theory into a working model for fire simulation and show some videos demonstrating the early stages of such a model. It is also interesting to look at famous fires from history to see the theory in action.

14:30 Tea/Coffee in School of Mathematics

15:00 Laure Daviaud: Max-plus automata and tropical identities (PDF, 373 kB)

In this talk, I will (re)introduce the strong connection between the model of max-plus automata and the finitely generated semigroups of tropical matrices. I will in particular discuss the question of the existence of identities in the semigroup of square matrices of dimension n. This question is open in the general case, but using the automata point of view, I will give some results about triangular matrices and matrices of dimension 2. This talk is based on joint works with Marianne Johnson and Mark Kambites.

15:45 Nikolai Krivulin: Tropical Optimization Framework for Analytical Hierarchy Process (PDF, 376 kB)

We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytical Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. The framework involves the Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. We reduce the log-Chebyshev approximation to multidimensional tropical optimization problems, and offer complete direct solutions to the problems in the framework of tropical mathematics. The results obtained provide a closed-form solution to the rating problem of interest as either a unique score vector (up to a positive factor) or as a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques to find those vectors from the solution set, which least and most differentiate between the alternatives with the highest and lowest scores, and thus can be well representative of the entire solution.

17:00 Dinner
We plan to go for an early dinner, somewhere in Selly Oak. Please inform Sergey Sergeev whether you are going to join the organisers and the speakers for dinner.

Financial support

Financial support for UK-based postgraduate students is awarded on a first come, first served basis; please give an estimate of your travel costs when confirming your attendance.

Directions from Central Birmingham Stations

(Birmingham New Street, Birmingham Moor Street)

The University of Birmingham (Edgbaston Campus) is located near University train station (to our knowledge, surprisingly, this is a unique station with this name in the UK).

If you arrive at Birmingham Moor Street then walk to Birmingham New Street station: there is no direct train from Birmingham Moor Street to University.

When you are at Birmingham New Street, it is safe to find platform 12 and take a train to Redditch or to Longbridge. Then leave the train at the second stop: University. There are several other trains which stop at University: check the electronic screens.

From the train station, descend downhill towards the clock tower, and keep walking straight until you see a bridge. The main entrance to Watson Building is on the left of that bridge (if you are walking from the train station). You will find the conference venue on the First floor (not on the Ground floor). Pointers to the venue will be attached in appropriate places.

Contacts

To request financial support and for more information, please contact Dr Sergey Sergeev.