by Jonathan Kujawa
As we know from the Law of Small Numbers, coincidences happen. Indeed, Ramsey's Theorem tells us they are downright unavoidable. Unfortunately, not all can be happy coincidences. In the first two weeks of July we lost three remarkable women of mathematics: Maryam Mirzakhani, Marina Ratner, and Marjorie Rice.
The most famous was Maryam Mirzakhani. She passed away on July 14th. This was reported in the New York Times, the Economist, and across the internet (including, of course, here at 3QD). Her widespread fame was in large part due to the fact that she was both the first woman and the first Iranianian to earn the Fields Medal. As we discussed at the time, Mirzakhani worked in geometry. More specifically, she worked with moduli spaces: these are geometric spaces where each individual point is itself a rich geometric object.
That is, imagine you are interested in studying geometric objects of a certain type (say, circles). Rather than study them one by one, you could study them as a collection. If you say two circles are "close" to each other when their radii are close in size, then you have a way of measuring distances in the space of circles and this lets you unleash 2000 years of geometric tools on the problem of circles. Of course, Mirzakhani studied moduli spaces of much more complicated gadgets than circles, but the principle remains the same.
Loyal 3QD readers saw this approach in action a few months ago. Thanks to the work of Canterella, Needham, Shonkweiler, and Stewart, we know it is possible to match the points on the sphere with triangles in a way where nearby points on the sphere match up to nearby triangles in Triangle Space. If we wanted to be fancy (say, at a geometers' cocktail party), we could have said that they showed that the moduli space of triangles can be identified with the sphere.
Their primary motivation was to give an answer to one of Lewis Carroll's bedtime problems [1]. But thinking geometrically also gives us entirely new insights. A big one is that with geometry we have the ability to talk about the shortest path between two points. But the shortest path depends on the ambient geometry. As we all know, the shortest distance between two points in the plane is given by a straight line. However, on a sphere, the shortest path is given by a "great circle".

