## A Small Exercise in Categorification

To celebrate the fact that I’ve finally got latex working on the blog, I’m going to bore you with a post about combinatorics and categories…

After some nice student questions in the Foundations course the other day, David and I were playing around with the number of surjections that exist between finite sets. And somehow this led down the road of writing the number of maps between two finite sets as a sum over the number of pre-image sets. There’s a very nice formula for this, actually. The number of surjections ${rm Sur}^n_k$ from set of $n$ things to a set of $k$ things is given by $k!S(n,k)$, where $S(n,k)$ is the Sterling number of the second kind, which counts the ways of partioning an n-element set into k subsets. (And it’s pretty easy to backwards reason why the number of surjections is this number!) The number of injections ${rm Inj}^n_k$ is the falling factorial, $(k)_n=k(k-1)cdots (k-n)$.

## Because it’s there.

Today I climbed up the biggest hill readily visible from Maseno with Aryan, a San Dieagan who’s living at the hospital.  Basic strategy: Pick the largest hill visible from Maseno.  Climb it.  Repeat.  (Well, the repeat will be for another day, but we have some candidates picked out.)  We didn’t really know what to expect in terms of paths; essentially, there’s beautiful hills all around with these giant granite boulders sticking out of them, but we have little idea what the local interaction with them is.  The one we climbed today had farms situated on it, probably to about half way up, including some really steep little plots that we passed through on the way back down. There are cows wandering around everywhere, occasionally exactly where you want to pass by, but that was the most ferocious animal we met.

At the road side we were approached by a guy who was intent on being our guide up the hill; he tried to scare us into taking him on by telling us that the hill was full of vipers and panthers.  This wasn’t terribly persuasive, though: I think we would have been in just as terrible a position if he’d been with us as not.  He didn’t look quite sturdy enough to take on a panther in a fist fight.

## Teaching, teaching, teaching….

Yesterday I spent five hours in front of a board!  Getting some good honest work in…  Two hours of normal class,doing the Foundations of Pure Math course, two hours of seminar, and an hour making a video lecture for the algebraic structures class.

Here are the videos for the algebraic structures class; it seems like a nice deliverable! It’s for the first section of the notes, which gives an introduction to the definition of a group, along with a bunch of examples.
Group Definition
Integers modulo n
Permutation Group
Dihedral Group
General Linear Group
The videos were shot in David’s house; the clip-on microphone that I picked up last year all but eliminated the terrible echo in the room. On the other hand, there’s a good bit of noise in the audio, which it would be great to figure out how to eliminate.

## Groups, Groups, Groups…

Lots of teaching things going on right now!  David and I were tasked (with extremely short notice) with developing an online abstract algebra course (entitled ‘algebraic structures’) for second-year undergraduates, and have been hashing out a good direction for the course to take.

One of the main things we realized (in a few hours of back and forth) is that the usual first course in abstract algebra probably isn’t really locally appropriate.  This is largely because we can’t be assured that the later maths classes — which hook abstract algebra into a wide variety of different contexts — will ever be available to the students.  The other big idea we came up with is that it would be great if we got students to the point where they could look at a mathematical problem and find the influence of algebra, and (in a perfect world) craft solutions to the problem appropriately.

## Amagoro Mini-Maths Camps

This weekend we ran four mini-maths camps! These are half-day events held at high schools, which give a sort of introduction to the kinds of things that we do at the main maths camp in August at Maseno.  On this particular trip, we were partnered with the Elewana Project, which works directly with a number of schools in the area; they mainly bring in students from the US during the summer to work with students, though they are beginning to do some two-week camps aimed at improving educational outcomes for secondary students.

The upshot on this particular trip was that the schools we visited had decently-maintained computer labs, thanks to Elewana’s ongoing efforts with the schools.  The schools also have student computer clubs, which allows students to have regular access to their computers.  As such, our program for the camps were focused on:

1. Getting across the basic idea that mathematics is about understanding and applying a system of rules, and
2. Getting a little bit of familiarity with the software, so that the students can explore and learn more after we’ve left.

## Akirachix MobileGarage

Today is the kick-off of the Maseno MobileGarage, a two-day boot-camp hosted by the Akirachix, a group promoting ITC development for women and more generally.  The idea is to give trainings on mobile application development to the local students.  I gave a short 15-minute talk for the kick-off emphasizing the importance of developing open-source tools in addition to focus on winning the lottery in the app store.

With luck, I’ll be able to poach an enterprising undergraduate to help develop an Android version of the photo-uploading script I wrote; I think it would be a great app for getting student work up on a website quickly and easily, which facilitates peer-learning and peer-review of student submissions.

Good times!

## Sage for Specht Modules

As the strike drags on, I’ve had some time to actually do some maths.  In particular, I’m preparing to run a weekly seminar on using representation theory for certain statistical problems.  The plan is to work from some old lecture notes by Persi Diaconis entitled ‘Group Representations in Probability and Statistics.’ These deal with, for example, how long one should apply a random shuffling process before the thing which is being shuffled is well-mixed.  Particular examples include the question of how many times one should shuffle a deck of cards, and how long one should let a random walk on Z_n run before we can be reasonably sure that every point has been reached.  There are numerous real-world applications of the results, and it uses a lot of first-rate representation theory along the way!

## More Strike Down-Time

There’s still no foreseeable end to the teacher and lecturer’s strike (in fact, the doctors are joining in, too, now), though I’m assured that there is to be a meeting on Monday to try to negotiate an end.  As such, we’ve been running lots of errands in the downtime…  And in-between work, we’ve been eating healthy amounts of ice-cream and playing a lot of Dominion!

## Three Strikes

The over-riding event of the last week and a half has been the teacher’s and lecturer’s strikes going on across the country.  As an outsider who’s just shown up in-country, I can’t really speak to the nuances of what’s going on.  Essentially the teachers struck to try for a pay increase, and recently the lecturers followed suit.  On Monday, a meeting was held by the Maseno Vice-Chancellor to try and end the strike: the attendees decided to go back to work the next day.  But the actual lecturer’s union was having a separate concurrent meeting on the other side of campus and thus wasn’t considered in the vote; on Tuesday morning they went department to department shutting things down and locking up offices.  In fact, before the action on Tuesday, someone told me that the ‘Vice-Chancellor had called off the strike.’  I heard that and thought, hmmm, that’s not usually how these things work…

As it stands, we don’t really know how long the strike will last.  It’s given us plenty of time to work on non-class related things, like processing work permits and thinking about what to do for our research seminar(s).  (Which I’ll post more on later, of course!)

## Maths Camp Song

I worked a couple summers at Camp Winnebago, near Augusta, Maine.  (Founded 1919, long before the car was a thing!)  One of the great things about Winnebago was that they had some thirty camp songs, one for every occasion, mostly written in the 40’s I think.  The most sticky of the bunch was ‘Goodnight Winnebago,’ which the entire camp would sing every night before bed; it’s a kind of theme song for the camp.  (And a good lullabye for getting a pack of kids ready to chill out for the night!)

We’ve been thinking for most of the last year that it would be great to have a theme for the maths camp.  And after nothing happening on it for many months, I wrote a bunch of lyrics the last night of camp and performed it (with some predictable hiccups) at the closing ceremony and then again at the final assembly, before we sent all the students home.

Yesterday we managed to round up a bunch of undergrads in the Maseno music program and work the song up a bit more.  The students were really creative and great to work with; I absolutely cannot imagine getting something like this together in just a day back in Toronto.  Here’s the last recording of the day!