Room 501, Department of Mathematics,
UCL, Gower Street, London, WC1E 6BT
j.d.evans • ucl.ac.uk
I have compiled a list of books I think are good for keen undergraduates. Some of these I enjoyed as an undergraduate. Others I have discovered since and would have enjoyed as an undergraduate.
Complex analysis is a natural way into topology.
Idea from multivariable calculus: If you have a map $F$ from $\mathbf{R}^n$ to $\mathbf{R}^m$ you say it's differentiable if, for every point $x$, there is a linear map $d_xF$ from $\mathbf{R}^n$ to $\mathbf{R}^m$ such that $F(x+\epsilon)=F(x)+d_xF(\epsilon)+|\epsilon|(\eta(\epsilon))$ for some $\eta$ which goes to zero as $|\epsilon|$ goes to zero. This linear map, in coordinates, is the matrix of first partial derivatives of the components of $F$.
In the complex world... If you have a map $F$ from $\mathbf{C}^n$ to $\mathbf{C}^m$ you say it's complex differentiable if, for every point $x$ there is a *complex* linear map $d_xF$ from $\mathbf{C}^n$ to $\mathbf{C}^m$ such that the same thing holds. Incredibly that's a much stronger condition than just having a real linear map as your derivative. It's equivalent to the Cauchy-Riemann equations. For example it implies that $F$ is infinitely complex differentiable and that the Taylor series converges. This is the first and simplest case of "elliptic regularity": the Cauchy-Riemann equations are a system of elliptic PDE (equivalent to the real and imaginary parts satisfying Laplace's equation) and solutions to elliptic PDEs are very rigid objects. They're so rigid that the whole subject of complex analysis on compact manifolds reduces to algebra: for example, holomorphic functions on the Riemann sphere are necessarily constant (Liouville's theorem); if you start allowing poles then you get Laurent polynomials, but nothing more. This is the beginning of a deep and fruitful relationship between complex analysis and algebraic geometry which you will be able to sense from the books I recommended.
(Note that the word elliptic is heavily overused, so elliptic PDE has little to do with elliptic curves/functions. The book on elliptic functions I mentioned is about how complex analysis/algebraic geometry relates to number theory.)
Riemann surfaces: One of the confusing things that can happen in complex analysis is that a function is multivalued - for example $z\mapsto\sqrt{z}$ is a 2-valued function (every number has two square roots, except zero which has one). The way one fixes this is to make a cut in the complex plane, in such a way that there is a single-valued branch of the function defined away from this cut. Let $B\subset\mathbf{C}$ denote the cut (for example, in the case $\sqrt{z}$ you might take $B=\{z\in\mathbf{R}\ :\ z>0\}$). You can now take several copies of $\mathbf{C}\setminus B$, each with a different branch of the function, and glue them together by matching up the two sides of the cut. The result is a weird-looking surface which has a single-valued holomorphic function defined on it. This is called a Riemann surface. It can be a topologically interesting surface. In the case $\sqrt{z}$ it's just $\mathbf{C}$ again, but in the case $\sqrt{z^3-z}$ it's a punctured torus, for example.
So even if you start off just being interested in functions on $\mathbf{C}$, you end up being interested in functions on surfaces with interesting topology. This is how the subject of topology starts to force itself upon you.
Once you realise this, you start to see it everywhere. For example, a complex function $F$ satisfying $F(x+1)=F(x)$ and $F(x+i)=F(x)$ can be thought of as a complex function defined on the quotient space $\mathbf{C}/(\mathbf{Z}\oplus i\mathbf{Z})$. This quotient space is a torus! (Imagine bending the square $\{x+iy\ :\ x\in[0,1],\ y\in[0,1]\}$ over so that the top and bottom edges match, then bending the resulting tube around so that the left and right edges match). Complex functions with more interesting periodicities (symmetries) can be considered as holomorphic functions on more complicated surfaces.
Having established that geometry on curved/topologically nontrivial surfaces is interesting, here are some places you can read about it, starting in dimension 2 and gradually getting higher. The highlight in dimension 2 is probably the Gauss-Bonnet theorem (which tells you that, on any given topological surface, the averaged curvature is independent of the geometry!) - Madsen and Tornehave explain how this generalises in higher dimensions by introducing Chern-Weil theory and characteristic classes.
Why is Rolfsen's "Knots and Links" in the list? I think it's a good book to help you start to visualise complicated topological things (which I think is important for intuition). Here are some fun things you can try and visualise:
Another reason to be interested in curved geometry is the fact that it underpins Einstein's general theory of relativity. Some good books include:
and my all-time favourite physics book is
The following are more advanced books, for those wanting a more advanced (but still accessible) look at geometry, topology and related subjects.
Last updated 27th February 2015.