I'm trying to finish a bunch of papers. I usually get started writing around noon or 1 pm, and when I get into it it's hard for me switch gears and write a lecture, especially since I've been trying to go to the gym almost every day at 6.

It's getting harder to write the lectures as the book proceeds and the 'sketches' get more sketchy, leaving me to fill in more details.

I have the feeling that many students have fallen behind the rather quick pace of the lectures, leaving only a small band of energetic followers.

My energy is slowly running out.

As for 2, I don't know if I *should* be filling in so many details. Maybe people would be happier if I gave more of an overview. This will be even more of an issue soon. Fong and Spivak give just a rough definition of 'monoidal category' in Section 4.4.3. The definition is a bit complicated, but it's a fundamental concept in category theory. Should I spend time to fill in the details or not? This is just one example of the decisions we face.

As for 3, it would be great to hear from people who *aren't* in the small band of energetic students who are leaving lots of comments on the lectures and solving lots of puzzles.

As for 4, that's mainly my problem, but I should warn you: I'm considering finishing at the end of chapter 4, after a good explanation of monoidal categories, compact closed categories, and PERT charts as another application of \(\mathcal{V}\)-enriched profunctors. I've put a lot of energy into this course and hate the idea of quitting before its done, but it's also tough to wake up each morning and know I need to spend an hour or two writing lectures along with papers. This will get a bit tougher on Wednesday when I go to Singapore.

]]>It's free to read, free to publish in, and it's about building big things from smaller parts. Here's the top of the journal's home page right now:

Here's the official announcement:

]]>We are pleased to announce the launch of

Compositionality, a new diamond open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition. To learn more about the scope and editorial policies of the journal, please visit our website at www.compositionality-journal.org.

Compositionalityis the culmination of a long-running discussion by many members of the extended category theory community, and the editorial policies, look, and mission of the journal have yet to be finalized. We would love to get your feedback about our ideas on the forum we have established for this purpose:http://reddit.com/r/compositionality

Lastly, the journal is currently receiving applications to serve on the editorial board; submissions are due May 31 and will be evaluated by the members of our steering board: John Baez, Bob Coecke, Kathryn Hess, Steve Lack, and Valeria de Paiva.

https://tinyurl.com/call-for-editors

We will announce a call for submissions in mid-June.

We're looking forward to your ideas and submissions!

Best regards,

Brendan Fong, Nina Otter, and Joshua Tan

Eventually there will be a way to apply for this school, which I'll announce on the Azimuth blog.

]]>I have an innate need to learn by coding small tools and I also have to apply it to practical software problems. And I am overwhelmed by the number of different threads discussing topics too advanced for me.

Where should I start actually ?

]]>- Applied Category Theory 2019, July 15-19, 2019, Oxford, UK.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our members is as varied as the systems being studied. The goal of the ACT2019 Conference is to bring the majority of researchers in the field together and provide a platform for exposing the progress in the area. Both original research papers as well as extended abstracts of work submitted/accepted/published elsewhere will be considered.

There will be best paper award(s) and selected contributions will be awarded extended keynote slots.

The conference will include a business showcase and tutorials, and there also will be an adjoint school, the following week (see webpage).

- Submission of contributed papers: 3 May
- Acceptance/Rejection notification: 7 June

Prospective speakers are invited to submit one (or more) of the following:

Original contributions of high quality work consisting of a 5-12 page extended abstract that provides sufficient evidence of results of genuine interest and enough detail to allow the program committee to assess the merits of the work. Submissions of works in progress are encouraged but must be more substantial than a research proposal.

Extended abstracts describing high quality work submitted/published elsewhere will also be considered, provided the work is recent and relevant to the conference. These consist of a maximum 3 page description and should include a link to a separate published paper or preprint.

The conference proceedings will be published in a dedicated Proceedings issue of the new *Compositionality* journal:

http://www.compositionality-journal.org

Only original contributions are eligible to be published in the proceedings.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Use of the Compositionality style is encouraged. Submission is done via EasyChair:

https://easychair.org/conferences/?conf=act2019

- John Baez (U.C. Riverside)
- Bob Coecke (University of Oxford)

- Bob Coecke (chair)
- John Baez (chair)
- Christina Vasilakopoulou
- David Moore
- Josh Tan
- Stefano Gogioso
- Brendan Fong
- Steve Lack
- Simona Paoli
- Joachim Kock
- Kathryn Hess Bellwald
- Tobias Fritz
- David I. Spivak
- Ross Duncan
- Dan Ghica
- Valeria de Paiva
- Jeremy Gibbons
- Samuel Mimram
- Aleks Kissinger
- Jamie Vicary
- Martha Lewis
- Nick Gurski
- Dusko Pavlovic
- Chris Heunen
- Corina Cirstea
- Helle Hvid Hansen
- Dan Marsden
- Simon Willerton
- Pawel Sobocinski
- Dominic Horsman
- Nina Otter
- Miriam Backens

- John Baez (U.C. Riverside)
- Bob Coecke (University of Oxford)
- David Spivak (M.I.T.)
- Christina Vasilakopoulou (U.C. Riverside)

Is there perhaps a subtle difference between these two? Seven sketches doesn't seem to mention quotient categories at any point

]]>Their book is free here:

- Brendan Fong and David Spivak,
*Seven Sketches in Compositionality: An Invitation to Applied Category Theory*.

If you're in Boston you can actually go to the course. It's at MIT January 14 - Feb 1, Monday-Friday, 14:00-15:00 in room 4-237.

They taught it last year too, and last year's YouTube videos are on the same YouTube channel.

]]>We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 School. It will begin February 18, 2019 and culminate in a meeting in Oxford, July 22–26. Applications are due January 30th! For more details, go here:

]]>Hoping to discuss once I've read and digested it some!

]]>If you are unaware of the game, here's a brief summary. Factorio is an open-ended game where you build factories to harvest raw resources and convert them into a manufactured goods. You do this by building structures which convert some resources to other resources (at a certain rate). For example, a stone furnace smelts iron ore to iron plates (and requires coal for power). [\mathrm{StoneFurnace}: \ \mathrm{IronOre} + \mathrm{Coal} \rightarrow \mathrm{IronPlate} ] The wiki page has explanations for all game mechanics. You can probably already see the similarities to resources theories (Chapter 2) and codesign diagrams (Chapter 4).

Like real factories, we want to design efficient/effective factories that produce an end product at a given rate (or produce a given amount, etc.). A simple version of this question is this: how many factories do we need to make iron plates from iron ore and coal if we want to produce iron plates at a given rate \(r\)? As you can find here, the furnace makes 0.28 plates/sec so we would need \(n = \lceil \frac{r}{0.28} \rceil\) furnaces, and supply them with iron ore at a rate \(r\) and coal at a rate \(0.0225 n\) at least. This question is easy to answer with a single building, but gets harder as the factory gets more complicated. So the goal is to come up with a way to describe a factory using category theory, where

- objects are resources/reagents/ingredients: iron ore, plates, etc.
- morphisms are processes which convert resources/combinations of resources to other resources.

But with that category, we also want to be able to ultimately answer questions such as "How many furnaces do I need?", "At what rate do I need to produce a certain raw material to produce an end resource?"

I think there's several routes to an answer (as is often in science/math). Initially, I thought about this problem using feasibility relations (Chapter 4), but found it tricky to keep track of the resources. After reading Tai-Danae Bradley's booklet, I realized that the theme of functorial semantics is useful here. Instead of creating one category to capture both how recipes can be combined and the corresponding rates, we can split the problem into two slightly separate problems:

- Construct a category, call it \(\mathbf{Fact}\), which describes how recipes can be combined to produce bigger recipes.
- Construct a second category, call it \(\mathbf{Rate}\), which describes production, consumption, supply and demand rates for the resources and processes. And along with it, a functor \(F\) which maps recipe diagrams in \(\mathbf{Fact}\) to \(\mathbf{Rate}\) which tells us what the rates of production/consumption are.

Hopefully, this is enough info to get started. While I have some results for both problems, I'd rather not spoil the fun of discovering them for anyone else (at least until others have a chance to try it).

]]>In other words, how would I draw the following diagram:

using the notation where morphisms are arrows? Sure, I can always do this:

where I just say the function accepts a product as the input, but I feel this is just raising another question: how did I end up with \( A \times B \) ? A possible answer could be that we can just specify the product using the universal property and we somehow just "have" it.

But I feel this doesn't get to the gist of the answer. To translate a monoidal product to usual notation, we'd need an arrow to accept two things as input. Arrows are inherently one-dimensional objects and have as inputs two-dimensional objects, points. I suspect that using two-dimensional shapes as arrows instead of one-dimensional could help alleviate the problem. Which is exactly what string diagrams are, in the end!

Is this sort of reasoning valid? Where can I read more about this? Are there higher-dimensional generalizations of string diagrams?

This seems like an important thing to know but I haven't been able to find good resources. CT is usually introduced as points and arrows between them, but does this mean there's an inherent limitation to arrow notation? It took me quite a while trying to draw products using arrows before I realized this might not be possible.

]]>- Tai-Danae Bradley,
*What Is Applied Category Theory?*

Abstract.This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.

Check it out!

]]>But it looks like I'm doing it in a strange way. Later tonight I'll post Lecture 77, titled "The End". But then *after that* I'll post Lecture 76, part two of "The Grand Synthesis".

Huh?

It just worked out that way. I started writing "The Grand Synthesis" but got distracted by listing various references for further study, and realized they should go in a separate post, which I'm almost done writing. Since it's been way too long since I've posted *anything*, I'll post that first, and then go back to "The Grand Synthesis".

I was an art student and, like all art students, I was encouraged to believe that there were a few great figures like Picasso and Kandinsky, Rembrandt and Giotto and so on who sort-of appeared out of nowhere and produced artistic revolution.

As I looked at art more and more, I discovered that that wasn’t really a true picture.

What really happened was that there was sometimes very fertile scenes involving lots and lots of people – some of them artists, some of them collectors, some of them curators, thinkers, theorists, people who were fashionable and knew what the hip things were – all sorts of people who created a kind of ecology of talent. And out of that ecology arose some wonderful work.

The period that I was particularly interested in, ’round about the Russian revolution, shows this extremely well. So I thought that originally those few individuals who’d survived in history – in the sort-of “Great Man” theory of history – they were called “geniuses”. But what I thought was interesting was the fact that they all came out of a scene that was very fertile and very intelligent.

So I came up with this word “scenius” – and scenius is the intelligence of a whole... operation or group of people. And I think that’s a more useful way to think about culture, actually. I think that – let’s forget the idea of “genius” for a little while, let’s think about the whole ecology of ideas that give rise to good new thoughts and good new work.

Maybe we can speak also of a categorical "scene" too.

]]>I'm writing this partially to remind myself to make sure to tell all of you about this book when it comes out!

]]>I *almost* overcame my resistance today, but I had to write a blog article about the work I'm doing with Metron on the Complex Adaptive System Composition and Design Environment project, to help convince the people at DARPA that we're doing good work, and that seems to have sucked out today's supply of energy.

I will try again tomorrow morning.

Anyway, here is that blog article:

It's about some new work John Foley and are doing... a continuation of the earlier work described here:

Part 1. CASCADE: the Complex Adaptive System Composition and Design Environment.

Part 2. Metron’s software for system design.

Part 3. Operads: the basic idea.

Part 4. Network operads: an easy example.

Part 5. Algebras of network operads: some easy examples.

Part 6. Network models.

Part 7. Step-by-step compositional design and tasking using commitment networks.

There, that should give you something to read about applied category theory until tomorrow!

]]>• John Baez and Jade Master, Open Petri nets.

Abstract.The reachability semantics for Petri nets can be studied using open Petri nets. For us an 'open' Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category \(\mathbb{O}\mathbf{pen}(\mathrm{Petri})\). Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of \(\mathbb{O}\mathbf{pen}(\mathrm{Petri})\). Here we describe the reachability semantics, which assigns to each open Petri net the relation saying which markings of the outputs can be obtained from a given marking of the inputs via a sequence of transitions. We show this semantics gives a symmetric monoidal lax double functor from \(\mathbb{O}\mathbf{pen}(\mathrm{Petri})\) to the double category of relations. A key step in the proof is to treat Petri nets as presentations of symmetric monoidal categories; for this we use the work of Meseguer, Montanari, Sassone and others.

I've started blogging about it here:

There's a lot of applied category theory in here. We build a symmetric monoidal category where the morphisms are open Petri nets, like this:

But actually do much more: we build a symmetric monoidal double category. A double category has two types of arrows, which go horizontally and vertically, and also squares:

We use the squares to describe maps between open Petri nets.

]]>The basic laws of arithmetic, like \(a \times (b + c) = a\times b + a \times c\), are secretly laws of set theory. But they apply not only to sets, but to many other structures!

Emily Riehl explained this in Eugenia Cheng's recent "Categories for All" session at a meeting of the Mathematical Association of America. Check out her slides:

- Emily Riehl, Categorifying cardinal arithmetic, MAA MathFest, August 4, 2018

To hear her discuss this argument, and math in general, listen to her episode of *My Favorite Theorem*:

- Evelyn Lamb, Emily Riehl's favorite theorem,
*Scientific American*, May 24, 2018.

Her proof that \(a \times (b + c) = a\times b + a\times c\) involves three facts about the category of sets:

It has

**binary coproducts**(the disjoint union \(A + B\) of two sets).It has

**binary products**(the cartesian product \(A \times B\) of two sets).It is

**Cartesian closed**(there's a set \(\mathrm{Fun}(A,B)\) of functions from \(A\) to \(B\)).

The defining property of the coproduct \(A + B\) is that a function from \(A+B\) to any set \(X\) is 'the same as' a function from \(A\) to \(X\) together with a function from \(B\) to \(X\).

The defining property of the product \(A \times B\) is that a function from any set \(X\) to \(A+B\) is the same as a function from \(X\) to \(A\) together with a function from \(X\) to \(B\).

The defining property of \(\mathrm{Fun}(A,B)\) is that a function from any set \(X\) to \(\mathrm{Fun}(A,B)\) is the same as a function from \(A \times X\) to \(B\). This change of viewpoint is called **currying**, and Riehl uses it twice in her proof.

She calls the defining property of \(+\) 'pairing', and she uses that twice too. I would call this **copairing**, but she wisely decided that sounds too technical.

Her proof actually never uses the defining property of \(\times\), which I would call **pairing**.

Her argument works for any category that has the necessary properties! For example, the category of topological spaces and continuous maps. But it's nice to see that lurking inside grade-school arithmetic there is an introduction to category theory.

]]>A bunch of you have probably not had time to digest the material on Galois connections. The next lecture will be on Monday. I will spend more time explaining those, with more examples and applications.

But right now I'd like to pose a little puzzle, completely unrelated to the course material, to keep you entertained over the weekend.

Here's an interesting function:

$$ f(x) = \frac{1}{2\lfloor x \rfloor - x + 1} $$

\(\lfloor x \rfloor\) is the floor function: it's the largest integer less than or equal to \(x\). For example,

$$ \lfloor 1.999 \rfloor = 1, \qquad \lfloor 2 \rfloor = 2 $$

We're starting to see the floor function in our discussion of Galois connections, and we'll see more of it, but that's not the point here. Here I want you to take the number \(0\) and keep hitting it with this function \(f\). I'll start you off:

$$ 0 $$

$$ f(0) = \frac{1}{2\lfloor 0 \rfloor - 0 + 1} = 1 $$

$$ f(1) = \frac{1}{2\lfloor 1 \rfloor - 1 + 1} = \frac{1}{2} $$

$$ f(1/2) = \frac{1}{2\lfloor \frac{1}{2} \rfloor - \frac{1}{2} + 1} = 2 $$

$$ f(2) = \frac{1}{2\lfloor 2 \rfloor - 2 + 1} = \frac{1}{3} $$

$$ f(1/3) = \frac{1}{2\lfloor \frac{1}{3} \rfloor - \frac{1}{3} + 1} = \frac{3}{2} $$

$$ f(3/2) = \frac{1}{2\lfloor \frac{3}{2} \rfloor - \frac{3}{2} + 1} = \frac{2}{3} $$

$$ f(2/3) = \frac{1}{2\lfloor \frac{2}{3} \rfloor - \frac{2}{3}+ 1} = 3 $$

So far we're getting this sequence:

$$ \frac{0}{1}, \frac{1}{2}, \frac{2}{1} , \frac{1}{3}, \frac{3}{2}, \frac{2}{3}, \frac{3}{1} , \dots $$

Here I'm writing all the numbers as fractions in lowest terms, because this will help you see some patterns. And my puzzle is quite open-ended:

**Puzzle:** What are the interesting properties of this sequence? What patterns can you find?

If you already know from your studies of math, please *don't* answer! Let others have the fun of discovering everything for themselves! One can discover a lot of amazing things by pondering this sequence.

Since a lot of you are programmers, the first step might be to compute the first hundred terms and show them to us! I urge you to show them in the way I just did, including writing numbers like \(3\) as \(\frac{3}{1}\).

]]>- Brendan Fong and David Spivak,
*Seven Sketches in Compositionality*. See also the website with videos.

Chapter 3 is about databases - and now we will finally meet categories, functors, and universal constructions.

Lectures will start on Monday 28 May 2018. But you can start reading the book and doing exercises now!

[[Fredrick Eisele]] is putting some exercises onto this forum. Please try some and put up your answers! If one of them is not up yet at the following location, you can put it there.

- Exercise 1 - Chapter 3
- Exercise 3 - Chapter 3
- Exercise 4 - Chapter 3
- Exercise 5 - Chapter 3
- Exercise 7 - Chapter 3
- Exercise 8 - Chapter 3
- Exercise 9 - Chapter 3
- Exercise 11 - Chapter 3
- Exercise 13 - Chapter 3
- Exercise 14 - Chapter 3
- Exercise 17 - Chapter 3
- Exercise 22 - Chapter 3
- Exercise 23 - Chapter 3
- Exercise 24 - Chapter 3
- Exercise 25 - Chapter 3
- Exercise 29 - Chapter 3
- Exercise 31 - Chapter 3
- Exercise 32 - Chapter 3
- Exercise 35 - Chapter 3
- Exercise 37 - Chapter 3
- Exercise 39 - Chapter 3
- Exercise 45 - Chapter 3
- Exercise 48 - Chapter 3
- Exercise 51 - Chapter 3
- Exercise 53 - Chapter 3
- Exercise 54 - Chapter 3
- Exercise 59 - Chapter 3
- Exercise 61 - Chapter 3
- Exercise 62 - Chapter 3
- Exercise 65 - Chapter 3
- Exercise 66 - Chapter 3
- Exercise 67 - Chapter 3
- Exercise 72 - Chapter 3
- Exercise 74 - Chapter 3
- Exercise 75 - Chapter 3
- Exercise 81 - Chapter 3
- Exercise 82 - Chapter 3
- Exercise 84 - Chapter 3

Chapter 4 is largely about profunctors. I just ran into a nice explanation of how profunctors show up in Haskell:

- Dan Piponi, Profunctors in Haskell.

We've also got some threads going where you can discuss exercises in Chapter 4:

]]>I drew a part of the Hasse diagram of such a preorder:

It mainly consists of things talked about in the first two and a half chapters (things I know so far). First of all, does this look correct?
I'm asking because this closely corresponds to my internal view of the topic - when I'm learning what these abstract structures are - I'm internally arranging them in a preorder! It's only after so many years I realized that this *thing* I'm internally arranging in my head corresponds to a preorder. I guess this is one of the wonders of CT - it allows me to more clearly communicate what I have in my head.

Now, some of this structure is captured linguistically, by adding various prefixes to a word. But I feel that language doesn't properly capture all the intricacies, as is for the example of the unital commutative quantale. Just knowing the word quantale doesn't tell me how it's connected to a preorder.

But seeing it as a part of a larger diagram immediately tells me many things! At a first glance, I can tell that it's a symmetric monoidal closed preorder with *some extra structure*. I can even fill in the blanks in some cases with minimal effort just by realizing some part of a square is missing. I think some famous mathematician said (Tao, perhaps?) that most of his papers were just completing the hypercube in this sort of way (although I can't find the reference).

So my question is, is this sort of diagrammatic reasoning useful? Does this scale as I add more objects?
I assume it would be like Fredrick commented to my question; perhaps it'd become unmanageable, where each structure could be factorized by *all* the axioms that apply to it.
Googling indeed yields many results that are very cluttered!

But then again, all these pictures present a sort of a flat, unnested hierarchy.
Could *smart nesting* somehow alleviate the problem? What if we tried to *define* our structures in such that the resulting diagram becomes nested and self-referential? And to try to answer that question, aren't we trying to do *exactly that* with category theory?
As far as I understand, with all the self-referential objects in CT (Category of categories, preorder of preorder structures...) one of the things we're trying to do is to find the 'best' way to define things such that there are no leaky abstractions and that theorems naturally follow.

To illustrate what I mean: I realized I have some redundancy in my previous graph and this is the improved version:

This notation follows closely Seven sketches, where the inside of the boxes represents objects of certain categories and dotted arrows represent functors, or in this case, monoidal maps. But those categories themselves can be arranged in a preorder! So this shows us that we have several preorders (one of which contains the object preorder itself) between which there is a preorder structure! We can say that \( \mathcal{F} \leq \mathcal{M} \) (the entire F is less defined than entire M) Now, I'm aware that this diagram can be further improved, but I think I've managed to get the point across.

I'd love to be able to see more of this diagram; to have a 'world map' of sorts where I can locate myself and observe both the nearby landscape as well as the distant mountain ranges.

Is this sort of approach feasible?

]]>Eugenia Cheng did her Ph.D. with Martin Hyland in Cambridge, developing and fixing some ideas James Dolan and I had about \(n\)-categories. She's become a quite famous explainer of mathematics. She's giving a public talk here, explaining that category theory is "the logical study of the logical study of how logical things work". On May 28th she's giving a talk in Brisbane:

Abstract.Category theory can be thought of as being “very abstract algebra”. It is thought of as “too abstract” by some people, and as “abstract nonsense” by some others. In this talk I will show that while it is abstract, it is far from being nonsense. I will argue that the abstraction has a purpose and that broad applicability is one of the powerful consequences. To demonstrate this, I will show how I apply concepts of category theory to important questions of life such as prejudice, privilege, blame and responsibility. I will introduce the category theory concepts from scratch so no prior knowledge is needed. These concepts will include objects and morphisms, isomorphisms and universal properties.

Here's a talk she gave at "Lambda World" last year:

]]>**Puzzle MD 1**:

From the New York Times (2013)

There are 10 steps in front of you that you are about to climb. At any point, you can either take one step up, or you can jump two steps up. In how many unique ways can you climb the 10 steps?

**Puzzle MD 2**: Suppose there were \(10^7\) steps in front of you. Let \(N\) be the number of unique ways you can go up these steps. What are the last 10 digits of \(N\)?

(\(\star\)) What about the last 10 digits after \(10^{100}\) steps?

**Puzzle MD 3**: Suppose you could either go up the stairs 3 different ways: using your left foot to go up one, your right foot to go up one, or hop up two at a time. There's still \(10^7\) steps in front of you, what are the last 10 digits of the count now?

(\(\star\)) What about the last 10 digits after \(10^{100}\) steps?

**Puzzle MD 4**: Now suppose you can go either 1 step, 2 steps, or 3 steps at a time. Once again, what are the last 10 digits of the count for \(10^7\) steps?

(\(\star\)) What about the last 10 digits after \(10^{100}\) steps?

**Puzzle MD 5**: Finally, assume you can go up either 1 step, 2 steps, or two different ways of going 5 steps, or three different ways of going 9 steps. What are the last 10 digits for the count for \(10^7\) steps?

(\(\star\)) What about the last 10 digits after \(10^{100}\) steps?

]]>- 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 "ri**n**g without **n**egatives".

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*.