Mathematicians (like all other people) come in different forms. Some work in the abstract and are most happy there, whilst others feel the urge to ground themselves with intuitive examples. I’m definitely a member of the latter camp, and so seeing mathematics excites me far more than being force-fed notation.
As such, whenever I ask somebody what exciting bit of maths they are working on, I steadily brace myself to witness a fumbling of the highest order. After some time I have to admit my cluelessness, which is often met by silence and a forged excuse by the explainer to leave the conversation.
Despite these regular occurrences, it is surely true that any piece of abstract mathematics can be traced back through its history to an interesting intuitive problem or idea. Now it’s this, the thing that started it all, that you should be telling the people outside of your field, rather than starting your spiel with Wojciechowski’s nineteenth lemma on the quasi-regularisation of modular semi-primitive topological bananas.
The lecture on Billiards and Moduli Spaces given yesterday by Fields Medallist Curtis T. McMullen was a perfect example of how maths should be delivered to a non-specialist audience. I found the talk to be delightfully down-to-earth whilst still maintaining the exactness of his material. The mathematics was developed to the extent that it could still be accompanied by insightful images.
Let’s have a peak at the content of the talk; the theory of dynamic billiards. Say you have a billiards table with no pockets and you set a ball in motion. If we suppose that there are no impeding forces then the ball will roll around on the table forever, changing direction only when it rebounds off of an edge. It’s known by a theorem of Veech, that on a standard rectangular billiards table, the motion of the ball will either be periodic (that is, it repeats itself after some time) or it will continue to cover new area, never repeating, and will eventually reach every point on the table. It all depends on the starting slope of the ball’s motion.
Actually, Veech showed that this is fact also holds for an entire class of billiards table, not just the rectangular ones. Now this theorem had obviously had some effect on McMullen, and by sharing it he showed us an interesting entry point to his research. McMullen has been studying billiards on trickier types of spaces, in particular, moduli spaces, using complex analysis
There’s nothing too scary about these spaces; think of the classic Super Mario Brothers game where you walk off the screen to the left and reappear on the right. This game is built on a moduli space – two edges are glued together so as to be thought of as the same edge!
Billiards on a moduli space works as follows. You have some polygon on which you wish to play billiards. However, we define a gluing on the edges of this polygon, so that rather than rebounding off an edge, the ball goes through the wall and comes back in elsewhere. McMullen then talked about problems involving these spaces for some time, before showing us a video made by his PhD student Diana Davis.
You should check out it out here; it’s a fantastic example of an abstract theorem made accessible to the general public.