Riemann surfaces involve calculus, geometry, and complex numbers. This blog post goes through the main ideas related to these topics and describes how they can be combined together. This builds up to an introduction to Riemann surfaces at the end of the article, as well as a brief introduction to the Riemann—Roch theorem, which I studied in my AMSI project.

My AMSI project involved something called a Riemann surface, but I’d like to start with calculus. Calculus is the computation of continuous processes and how they change, usually thought of as changing in time or in space. A simple example is velocity. For a geometric example, imagine a complicated curve. We can use calculus to find the straight line which most closely matches the curve near a specific point—this is called a tangent line. It also allows us to solve more sophisticated problems, like determining how the tangent line varies when we approximate the curve near different points—this is related to curvature.

To get from calculus to Riemann surfaces, we need to add in two ingredients. The first ingredient to add is geometry. Mixing geometry and calculus gives calculus on curved spaces, which is called differential geometry. A famous application of differential geometry is that it allows for analysis of the curvature of space-time in Einstein’s theory of general relativity. Curved spaces, called surfaces, look flat when viewed when close-up, but look more interesting when viewed further away. For example, the surface of the Earth appears like a flat plane in everyday life to someone standing on it. But viewing it from space reveals that it is globe.

The second ingredient is complex variables. The complex numbers come from introducing a number i which squares to equal negative one. They allow any polynomial equation to be solved, and have many applications to physics and engineering, such as for electromagnetic waves and signal processing. A complex number has two components, for example 7 + 5i. The first component is an ordinary number, and the second component is a multiple of i. Since there are two components this can be viewed as a two-dimensional space, which is why the complex numbers are also called the complex plane. However, a complex number is more than just two coordinates, because we have a natural way of multiplying and dividing complex numbers.

We can also do calculus with complex variables. Functions which can be differentiated using complex numbers satisfy remarkably strict conditions—for example, in the situation studied in my report the only complex differentiable functions are constant functions, which are basically just numbers. Therefore, we loosen our study to consider meromorphic functions, which are complex differentiable at most points, but they are (intuitively speaking) allowed to explode towards infinity at the rest.

Now we can combine everything to understand what a Riemann surface is. This is a surface which when viewed closely looks like the complex plane, not just the usual flat plane. So with a Riemann surface we can look at curvature, do calculus, and have a connection to complex numbers. In my project I studied the Riemann—Roch theorem. This is a formula which, given a Riemann surface, tells you how many meromorphic functions there are on it. (Actually, there are infinitely many meromorphic functions, so what it tells you is the dimension of the space of these functions.)

Tiernan Cartwright
University of Sydney

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