This is an excerpt from a conversation between me and my-other-self in a parallel universe who did not decide to major in mathematics.
Naught: Hello, Shunsuke. I know you’re doing research with a Ph.D candidate here at McGill in Math…but what do you actually do? What is research in Math anyways?
Prime: Hello there. Yes, I’ve been fortunate enough to be researching and discovering the elegance of mathematics.
Naught: But what’s there left to discover in Math? Don’t we know all of Math?
Prime: Wow, hold your horses. Mathematics isn’t dead. It’s growing just like you and me, just like my universe and perhaps yours too! Some mathematicians prove propositions, extend ideas that already exist, or characterize things that we know a little bit about. What I do is I’m trying to find characterizations in a branch of geometry.
Naught: So like characterizing right triangles and equilaterals?
Prime: That’s the right idea of characterization: having three equal angles characterises an equilateral, and similarly, having two equal angles characterizes an isosceles. But geometry doesn’t necessarily mean Euclidean geometry, and right triangles and equilaterals only “live” in Euclidean geometry. Poincaré once said that “one geometry cannot be more true than another; it can only be more convenient.”
Naught: Umm… I don’t really get it.
Prime: So let’s say we’re trying to get to a party. How would you figure out how to get there?
Naught: Probably by using the address?
Prime: Right, but what if it’s a very minor street up in the mountains and your GPS doesn’t recognize the address?
Naught: That’s never happened to me before.
Prime: But let’s say it does, then what’s an alternative to find the location of the party?
Prime: Perhaps you can put the latitude and longitude into your GPS.
Naught: Okay, so what’s your point?
Prime: My point is that both the address and the coordinates uniquely define the location. You would prefer the address, but there might be situations where address isn’t the most convenient. A satellite would prefer the coordinates. In addition to that, the two system work in a completely different way!
Naught: So you would use different kinds of geometry in different situations?
Prime: Exactly. As Euclid explored the world of Euclidean geometry, and stated theorems and properties of this way of looking at things, you can imagine that the same can be done for the other branches of geometry too.
Naught: But isn’t that like using chopsticks when you have fork and knife?
Prime: What do you mean?
Naught: I just don’t see the point of using an alternative when I’m used to using one.
Prime: Remember the address example, there might be situations where your proverbial fork wouldn’t be the most convenient — or at least, would be pretty unwieldy. It just happens that we are frequently dealing with flat surfaces, like on a desk, and Euclidean geometry is great for that. But what if you want to figure out the area of a triangle drawn with vertices on the North Pole, Montreal, and Tokyo.
Prime: Then the sum of all angles ends up larger than 180°, not exactly what we were taught in high school, right? But this is a physically existing triangle! And as you might imagine, (½ × base × height) will not give you the area for this triangle.
Prime: In this case, theorems and tools developed in non-Euclidean geometry come in handy. You can use a similar technique used in calculus and chop up the curved triangles into much smaller triangles so that the curvature is negligible and sum up all the area of the subdivided triangles calculated using our good old friend (½ × base × height).
Naught: So you can still use Euclidean geometry.
Prime: But why bother when we have more convenient tools! Why not use chopsticks to pick up sushi when it’s easier! At least it is for me.
Naught: Okay. I think I get it. But how about you? What do you do specifically?
Prime: I explore the world of algebraic geometry and try different things to see if I can identify nice patterns that I can classify into or unfold a general law that governs a particular situation. This sounds more like research that you’re used to hearing about, right? My supervisor and I have been looking at quadrilaterals and we have gathered some information but whole picture yet to be cleared up. And we’re almost out of time so I’ll let you know more about what I do next time.