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I took my first class on Complex Analysis last semester and I wanted to continue learning more about it this semester doing independent readings. I was advised to read up on Riemann Surfaces but I'm having a hard time grasping what the motivation for them is and finding an intuitive explanation for what exactly they are. Any help would be greatly appreciated!

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I have two different answers regarding the motivation for studying Riemann surfaces. Before I begin, I should say that, intuitively, a Riemann surface is just a surface that's made up of the same "stuff" as the complex plane. That is, it's a surface on which you can define things like holomorphic (analytic) functions, meromorphic functions, contour integrals, and so forth.

Geometrically, the primary feature of the complex plane is that you can measure the angle between two curves (since holomorphic functions are conformal). As a result, a Riemann surface is really the same thing as a conformal surface, i.e. a surface on which it is possible to measure angles.

Now, part of the motivation for considering such surfaces comes from complex analysis itself, as outlined by Hurkyl. However, there are at least two other important sources of motivation.

1. Algebraic Geometry

One important motivation for studying Riemann surfaces comes from algebraic geometry, which includes the study curves and surfaces defined by systems of polynomial equations. It turns out that the complexified version of an algebraic curve is often a Riemann surface.

For example, recall that a conic section is a curve in $\mathbb{R}^2$ defined by an equation of the form $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0. $$ where $B^2 \ne 4AC$. If you allow $x$ and $y$ to take complex values, then the same equation defines a Riemann surface in $\mathbb{C}^2$ (this is a two-dimensional surface sitting inside of four-dimensional space), of which the original conic section is a cross-section.

If we move from quadratic equations to cubic equations involving $x$ and $y$, then the resulting algebraic curve is called an elliptic curve. As with conic sections, elliptic curves can also be thought of in $\mathbb{C}^2$, where they become Riemann surfaces. Elliptic curves are among the most important objects in modern mathematics, and are related to the classical theory of elliptic functions and elliptic integrals

It turns out that the theory of algebraic curves (such as conic sections and elliptic curves) becomes much nicer if you allow complex coordinates, i.e. if we make the curve into a Riemann surface. Mostly this is because of the Fundamental Theorem of Algebra, which you can use to predict the number of solutions to any system of polynomial equations. For example, a pair of quadratic equations will usually have four solutions, and therefore two conic sections will usually intersect at four points (though some of these points may be complex). This is a special case of something called Bézout's theorem.

2. Topology

Another important motivation for studying Riemann surfaces comes from topology. One of the most important themes in modern topology is that topological objects are sometimes best understood by endowing them with geometric structure. For example, a torus can be obtained by gluing together opposite sides of a Euclidean square, and the topology of a torus is often best understood using Euclidean geometry on the square.

The classification of surfaces states that every closed, orientable surface is homeomorphic to either a sphere, a torus, or a multi-holed torus. Spheres have their own geometry, and the geometry of a torus is Euclidean, but it turns out that all the other closed, orientable surfaces are best understood using hyperbolic geometry. For example, a two-holed torus can be obtained by gluing together the opposite sides of a certain regular octagon in the hyperbolic plane, and this description endows the two-holed torus with something called a "hyperbolic structure". A surface with a hyperbolic structure is called a hyperbolic surface.

Now, there is a close connection between complex analysis and hyperbolic geometry. For example, the Poincaré disk model of the hyperbolic plane imagines the hyperbolic plane as an open disk endowed with a certain non-Euclidean metric. Using this model, the isometries of the hyperbolic plane turn out to be Möbius transformations, and indeed a hyperbolic isometry is the same thing as a holomorphic bijection of the unit disk.

Using this connection, one can show that there is essentially a one-to-one correspondence between complete hyperbolic surfaces and Riemann surfaces. Thus, any serious study of the topology of surfaces inevitably leads to Riemann surfaces. For example, one of the best ways to understand the self-homeomorphisms of a closed surface is to consider how a homeomorphism affects the hyperbolic structure, i.e. to investigate the action of the mapping class group on Teichmüller space. This idea was part of the foundation for Thurston's investigation of surface automorphisms, leading to the theory of pseudo-Anosov maps.

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Consider the function $\sqrt{z}$.

This really isn't a function, per se, because at most points, it has two different values, rather than the single value a function is supposed to have. e.g. both $\mathbf{i}$ and $\mathbf{-i}$ are square roots of $-1$.

While you can adopt various conventions like selecting a "branch" of the square root to be a principal value at all points -- e.g. like how we decide when working purely with the reals that square roots of positive numbers should be positive numbers -- this loses some information.

You can think of the Riemann surface for $\sqrt{z}$ as addition an extra little bit of information to specify which branch of the square root you want. A point on the surface is not simply a complex point, but it also includes some extra information to specify which branch of the square root you want.

e.g. one way to express a point on this surface is as a pair $z_\spadesuit$ or $z_\heartsuit$, where $z$ is a nonzero value. We define the square root function so that $\sqrt{z_\spadesuit}$ is the principal value of $\sqrt{z}$, and $\sqrt{z_\heartsuit} = -\sqrt{z_\spadesuit}$. e.g. $\sqrt{1_\heartsuit} = -1$.

Another way to think of the Riemann surface is as being the analog of the graph of a function, but more suitable for multi-valued functions like $\sqrt{z}$. In this case, it would be the set of all points $(z,w)$ in two complex dimensions that satisfy the equation $w^2 = z$ and $z \neq 0$. To compare with the above, the point $(1, -1)$ would be the point $1_\heartsuit$ by the alternative notation in the paragraph above.

(I haven't used Riemann surfaces much -- there are probably other useful reasons to study them beyond being a very useful way to work with multi-valued functions, but that's all I've used them for)

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