Combinatorics and Optimization, Day 1

I wrote this last night at my grandparents’ house, which has no Internet connection I can feasibly use (dial-up does not count), so I had to wait until today to post it from the University of Waterloo campus. All references to “today” refer to Monday, July 5.

This week, Rachael, Aaron, and I have travelled to Waterloo, Ontario for two math conferences. The first is the Combinatorics & Optimization Summer School, a two-day event consisting of several talks and, yes, food! The second is the Canadian Undergraduate Math Conference, which also entails much talking and eating. I was reluctant to attend at first, because I dislike travelling. However, my grandparents live in Waterloo, so this was a convenient way to visit them for a week while still getting paid. With that incentive, I managed to convince myself that these conferences would be interesting and probably even useful to my research. This was only the first day, but so far I remain convinced in those respects.

I’ve been up since 4:30 in the morning. Let me take a moment to reflect on the fact that we flew from Thunder Bay to Toronto in an hour and a half, traversing—or rather, bypassing—the largest freshwater lake in the world. And we did it in a metal behemoth that harnesses complex physics and engineering to work miracles.

Flight is awesome.

OK, science-geeky moment over: back to math.

Today there were four talks. We arrived late to the first talk, by about fifteen minutes, but it was still very interesting. It concerned the colouring of graphs on surfaces.

Following a short break, the second talk discussed the Borsuk conjecture, which asks a question about the existence of a certain partition of any set in d dimensions. This was my favourite talk of the day, for several reasons. Firstly, I learned a lot about the diameter of sets, a topic with which I was not familiar. Topology involves a lot of geometry, something for which I lack proper intuition. Yet still it interests me, probably because of its ability to formalize that geometry. I like abstraction. Secondly, the presenter told the story of how Kahn and Kalai proved the Borsuk conjecture false. They took a problem that had been open for nearly seventy years, solved it in a week, and wrote a short, about one-paragraph proof. It’s a wonderful example of how unpredictable and exciting mathematics can be: sure, sometimes math research involves long, boring days reading papers and staring at a problem on a chalkboard. Sometimes, just sometimes, it leads to the most interesting results.

After lunch, we listened to a talk about cutting cake—specifically, how to divide a cake into sections such that no one person would complain that he or she received a worse section. It was by the far the most accessible of the three talks, and the presenter had a very engaging manner. Unfortunately, my fatigue caught up with me during this talk, and I found myself nodding off during the most interesting parts. We learned a little about hypergraphs, which, as the name implies, are like regular graphs but on crack.

The last talk was on symmetric groups and their combinatoric properties. Last week, my prof showed me how we may be able to make use of the symmetric groups to solve the problem on which I‘m working this summer. The talk was more of a review of things I had already learned in group theory two years ago, which was still useful considering the gap in time.

The day began winding down as we went to a pub-like house for dinner. Then we trekked across campus to the residence where Rachael and Aaron are staying. We got lost in the process, of course, but eventually found our way thanks to a map and, moreso, a helpful student.

More to come on Tuesday’s schenanigans tonight or tomorrow morning!