## Tropical Algebra and Railway Optimization

Simon Willerton pointed out a wonderful workshop, which unfortunately neither he nor I can attend… nor Jamie Vicary, who is at Birmingham:

Tropical Mathematics & Optimisation for Railways, University of Birmingham, School of Engineering, Monday 18 June 2018.

If you can go, please do—and report back!

Tropical algebra involves the numbers $(-\infty, \infty]$ made into a rig with minimization as the addition and addition as the multiplication. It’s called a rig because it’s a “ring without negatives”.

Tropical algebra is important in algebraic geometry, because if you take some polynomial equations and rewrite them replacing + with min and × with +, you get equations that describe shapes with flat pieces replacing curved surfaces, like this:

These simplified shapes are easier to deal with, but they shed light on the original curved ones! Click the picture for more on the subject from Johannes Rau.

Tropical algebra is also important for quantization, since classical mechanics chooses the path with minimum action while quantum mechanics sums over paths. But it’s also important for creating efficient railway time-tables, where you’re trying to minimize the total time it takes to get from one place to another. Finally these worlds are meeting!

Here’s the abstract, which shows that the reference to railway optimization is not just a joke:

Abstract. The main purpose of this workshop is to bring together specialists in tropical mathematics and mathematical optimisation applied in railway engineering and to foster further collaboration between them. It is inspired by some applications of tropical mathematics to the analysis of railway timetables. The most elementary of them is based on a controlled tropically linear dynamic system, which allows for a stability analysis of a regular timetable and can model the delay propagation. Tropical (max-plus) switching systems are one of the extensions of this elementary model. Tropical mathematics also provides appropriate mathematical language and tools for various other applications which willbe presented at the workshop.

The talks on mathematical optimisation in railway engineering will be given by Professor Clive Roberts and other prominent specialists working at the Birmingham Centre for Railway Research and Education (BCRRE). They will inform the workshop participants about the problems that are of actual interest for railways, and suggest efficient and practical methods of their solution.

For a glimpse of some of the category theory lurking in this subject, see:

• Simon Willerton, Project scheduling and copresheaves, The n-Category Café.

### 6 Responses to Tropical Algebra and Railway Optimization

1. Tim Porter says:

John has cross posted this on the n-Categroy Café and I have added a comment there that might be of use. This is lovely stuff to teach and my comment includes some links to notes that I used to use when I taught this in Bangor years ago. It is highly relevant to people interested in a categorical approach to problems and is great fun!

2. amarashiki says:

The time for tropical geometry and non-archimedean geometry into physics is to come sooner than expected…

3. Ebrahim says:

I am currently trying to apply max-plus (tropical) methods for the scheduling of airport activities. I am using similar techniques to that used to model railway timetables, so I am gutted to have only learned about the Birmingham workshop a day too late! If anyone is interested, please do get in touch – I am particularly keen on understanding how tropical mathematics may provide a more efficient tool than traditional methods for developing critical path methods for project completion.

• John Baez says:

I am gutted to have only learned about the Birmingham workshop a day too late!

Ouch! Sorry. You should read this blog more often.

If anyone reading this went to that workshop, I hope you tell us a bit about what happened!

• Ebrahim, are you on Marianne Johnson’s mailing list for “Tropical Mathematics and its Applications meetings”? If not, you should be! That’s where I found out about the meeting.

• Ebrahim says:

Hi Simon – no I am not, but I know Marianne from my Manchester days! Thanks.

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