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Tropical Mathematics & its Applications Supported by the LMS 15th January 2020 School of Mathematics, University of Manchester Local Organisers: Marianne Johnson, Mark Kambites, Ben Smith |
* Zeinab Toghani (QMUL)
* László Végh (LSE)
with graduate student talks from:
* Arthur Kennedy-Cochran-Patrick (Birmingham)
* Stefano Mereta (Swansea)
* Kayleigh Ward (Swansea)
All talks will talk place in Frank Adams Room 1, Alan Turing Building, University of Manchester. (See below for advice on how to get here.)
13:00 - 14:15: Graduate Student Talks
13:00 - 13:20: Bounds for the Factor Rank Transient of Inhomogenous Matrix Products - Arthur Kennedy-Cochran-Patrick (University of Birmingham)
We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length have factor rank equal to the cyclicity of the matrices associated digraph. The first case was shown in [6][L. Shue, B.D.O. Anderson, S. Dey: On steady state properties of certain max-plus products. Proceedings of the American Control Conference, Philadelphia, Pensylvania, (June 1998), 1909 1913.] and we establish a bound on the transient after which this starts to happen for any product of matrices whose length exceeds that bound. The second, third and fourth cases will look to generalise the digraph structure to develop bounds on the transients of a wider range of digraphs.
13:25 - 13:45: Tropical Ideals and Discriminants - Kayleigh Ward (University of Swansea)
13:50 - 14:10: Scheme theoretical approach to differential tropical geometry - Stefano Mereta (University of Swansea)
Starting from the presentation of tropical differential equations as in the work of Grigoriev and the following paper by Aroca, Toghani and Garay, in which the fundamental theorem of differential tropical geometry has been proven, I’ll try to put their work in the framework of semiring theory. Using it, I’ll prove a result going towards the definition of a differential scheme over a semiring whose points are the solutions of a differential tropical equation.
The long term goal will be to define a universal differential semiring whose spectrum is the moduli space of what we call differential enhancements of a given valuation.
14:15 - 15:00: Coffee
15:00 - 15:50: Tropical Differential Algebra - Zeinab Toghani (Queen Mary university of London)
16:00 - 16:50: Signed tropical convexity - László Végh (London School of Economics)
Convexity for the max-plus algebra has been studied from different directions including discrete geometric, scheduling, computational complexity. As there is no inverse for the max-operation, this used to rely on an implicit non-negativity assumption. We remove this restriction by introducing 'signed tropical convexity’. Our notion has several natural formulations in terms of balance relations, polytopes over Puiseux series and hyperoperations. We obtain several structural theorems including a new Farkas lemma and a Minkowski-Weyl theorem for polytopes over the signed tropical numbers. This is joint work with Georg Loho.
17:00 Dinner
We plan to go for an early dinner, somewhere near to the train station. It would be helpful if you could let Marianne know if you intend to join us for dinner.
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 Manchester Piccadilly stationWalking from Manchester Piccadilly to the Alan Turing Building should take around 20 mins. Leave the station by the Fairfield Street exit (head down the escalators or lift from the main concourse) which brings you out at a big road junction. Cross both main roads, and go along a smaller road (Granby Row) to the left of the Bull's Head pub. Keep straight on, as the road becomes a pedestrian walk and then a road again, and at the phoneboxes turn left onto Sackville Street. Go under the railway bridge and continue past some constructions works - where the road bends off to the right, follow the left-hand pavement which becomes a footpath and goes through an underpass. Afterwards, keep left under the motorway flyover (avoiding a deeper underpass ahead) before bearing right (avoiding yet another underpass to the left). After very carefully crossing the motorway sliproad, you find yourself on Brook Street. Walk down this (away from the flyover). At the intersection with Grosvenor Street, cross both roads and then continue along (now Upper) Brook Street on the opposite side. Cross the next side-road (Booth Street East, carefully again!), continue past the Aquatics Centre car park and then the Alan Turing Building is on your right. Walk past what looks like an entrance (glass wall with fire doors) and turn right into a partially covered walk-way. The actual entrance to the Alan Turing building should now be on your right.