## Moduli Spaces

What do you think about when you see a circle?

Depending on the context, a mathematician might think of a circle in many ways. One option is to think about a circle as a collection of  infinitely many points, rather than to think of it as a line. It is the collection of points that lie a fixed distance away from a central point.

This perspective is important because we can think of each point on the circle as representing something. For example, in the picture below the blue point represents the red arrow (or ray) that starts at the centre of the circle and passes through that blue point:

We can do this for all points and arrows: each point on corresponds to the arrow starting at the centre of the circle and passing through that point, and each arrow corresponds to the point where it meets the circle.

This gives a bijection between points on the circle and arrows from the centre. But our correspondence between points and lines is better than a bijection: it seems quite natural. The reason for this is if two arrows are very close together then the corresponding points on the circle are also close together, and vice versa – our bijection has captured information about the topology of the lines.

In fact, we can find how similar two arrows are by finding the distance between their associated points on the circle.

Why is this useful? It’s much easier to think about a circle than it is to think about a whole collection of arrows. We can see from the fact that a circle is 1-dimensional that all such arrows can be described using one parameter – the angle of the arrow is enough information to define which arrow we’re talking about. We can stratify  the space of arrows by subdividing the circle, for example on the picture below the green region corresponds to arrows that point “upwards”:

We say that the circle is a moduli space for our arrows – each point on the circle represents an arrow in the right kind of “natural” way.

What if, instead of arrows starting at a given point, we were interested in lines passing through a point? What shape could we use to parametrize these – i.e. what could our new moduli space be? We have a problem because each line passes through two points on the circle instead of one so we no longer have a bijection:

Can we still use a circle as the moduli space? (Answer: yes!)

In the above examples we are looking for ways to classify arrows and lines. It is an important problem in maths to classify more complicated objects for example algebraic curves, or fly wings!

Ezra Miller is interested in understanding the moduli space of fly wings – a space where each point corresponds to a different kind of fly wing, where we say two fly wings are different if the veins make different shapes.

Clearly something much more complicated space than a circle will be needed to describe the different kinds of vein shapes of fly wings.  In fact, there are lots of different spaces that we could use. Once we have selected a shape, there are interesting and useful questions we can ask:

1. Distance: in our previous example, we had a clear notion of how far apart two arrows were – we could measure the distance travelled to get from one point to another by travelling along the circle. It’s harder to say how far apart two fly wings are, and we want a measure of distance that agrees with our biological intuition.
2. Stratifying the space: before, we could stratify our space of arrows by selecting some region of the circle (for example, the upper half). What about for the fly wing – are there ares of our moduli space that correspond to biologically significant subsets of flies?

There are also many areas of pure maths where people study moduli spaces, which I will talk about in a future blog post. They are a (fairly) simple but important intuitive concept that plays a role in both pure and applied maths.