I am a recent PhD specializing in algebraic geometry. But I also want to do research in some other areas of math (e.g. quantum computation, compressed sensing, and PDE's). What would be a good way of learning these so that they can become a research interest?
I have very little background in physics. My plan/goal is to begin research in one of these areas by next February. I do not want to be limited in the areas in which I do my research (e.g. only doing research in algebraic geometry).
It may not be relevant to your situation, but a bit of advice I was once given, which I think is a good one, is that moving fields is an excellent idea but it is even better to make the path continuous. Given the interconnectedness of mathematics, this advice is easier to follow than it at first seems.
I think our questioner is aware of the difficulties of switching fields and if not he or she soon will be, so let me be naive and try to be constructive.
For Quantum Computation, Isaac Chuang and Michael Nielsen's "Quantum Computation and Quantum Information" has become a standard introduction to the subject, suitable for a graduate student in either mathematics, physics or computer science.
Since I have no idea what background you have in PDEs (you could be a specialist in D-modules for all I know and find these suggestions childish), here are some texts I've become acquainted with:
-V.I. Arnold's "Lectures on Partial Differential Equations" gives a beautifully geometric and intuitive understanding of PDEs, introducing and weaving together contact and symplectic geometry. The table of contents looks quite basic, but it contains the depth you should expect from Arnold.
-Lawrence C. Evans "Partial Differential Equations" is nice and contains the basic notions from Functional Analysis, Sobolev Spaces, Weak Theory and Regularity Theory. It does a good job of being self-contained and trying to give physical interpretations of various PDEs.
-Gilbarg and Trudinger have the classic "Elliptic PDEs of Second Order", which is dense, but a classic nonetheless.
As a mathematician you don't need to learn how physicists think in the next year. Physicists have different ways of looking at problems and are constrained to their own paradigms just as mathematicians are. It is often quicker to pick up advanced physics if you know advanced mathematics, with many excellent bridge texts by world-class mathematicians. Examples that come to mind are Bott's "Morse Theory Indomitable" which includes an exposition of some of Witten's ideas for a mathematician. Atiyah's "Geometry and Physics of Knots" is also an excellent example of this. Feynman's Lectures are great, but won't advance you to research. It's more like a Caltech undergraduate degree bound in 3 volumes.
Finally, as a note of inspiration, I have heard of at least two new faculty who self-studied PDEs in their post-doctoral years. One was supplanting a thesis in deformation theory and integrable systems, the other in knot theory and Floer homology. It is definitely a hard path to follow, but is sometimes necessary for growth. Also, bear in mind that Ed Witten was a history major as an undergrad, dropped out of economics grad school before applying for Princeton applied math and then switching to physics. Raoul Bott switched from electrical engineering to mathematics after his PhD (a much harder path, one might argue). Finally, my personal hero, Douglas Hofstadter, after quitting his Berkeley math PhD and finishing a 7+ year physics PhD in Oregon, then lived at home for a few years re-tooling himself as an AI researcher. Now he has a Pulitzer and tenure at a university -- not too shabby.
Good luck!
Although I encourage you to pursue your strengths and passions, even if it means changing fields, I also advise some caution if you do not yet have tenure. If you feel you can still do more work in your current field that will help build a strong tenure case, you should make sure you continue to devote the bulk of your efforts on that. And I would also be wary of telling people about your plans to change fields (of course, it's a little too late for that advice). Stuff like that can really damage an otherwise strong letter of recommendation: "So-and-so has done brilliant work in .... but has told me that he intends to shift his research away from that and to .... Unfortunately, I cannot comment on the latter."
But if you feel certain you have reached a dead end in your current direction of research, for whatever reason, then you should of course shift full force into something new. Even then, I encourage you to make the initial shift into a direction that can be somehow sold as a natural outgrowth of your current research. Later, when you're a tenured full professor, you can do whatever the heck you want.
I merely quote Halmos:
A creative thinker is alive only so long as he grows; you have to keep learning new things to understand the old. You don't really have to change fields--but you must stoke the furnace, branch out, make a strenuous effort to keep from being locked in.
While it seems a bit late for the OP, I'd like to add some remarks that might be helpful to the subsequent readers.
Having completed my PhD in Combinatorics, I seem to be inevitably converting to Computational Biology. Let me point out some of the most significant aspects that I have encountered:
Obstacles:
The letters of recommendation. Although mentioned by Deane Yang, I feel this was not mentioned strongly enough. I find this to be a major obstacle to my future in computational biology. I have lots of people who would be willing to vouch for me for my expertise in combinatorics, but very few who would vouch for my expertise in computational biology. Those who can vouch for me are only able to make limited comments due to only working in the area a short time. [PS. One tip -- make sure the referees in your previous field have some idea of the significance of your work in the new field (thereby reducing the problem raised by Deane Yang)]
Lack of publications. And moreover, the overall lack of relevant brownie points -- e.g. I've refereed papers in combinatorics, I'm a member of the AMS and other societies, and so on, which are not very relevant.
Advantages:
It is multidisciplinary. So most people who enter this area have a PhD in Biology, Computer Science, Mathematics, Statistics, etc., but not in Computational Biology itself. So most people are in the same boat.
These are neighbouring fields. The advantages of this have already been discussed.
There is significantly more funding in computational biology. This is sheer numbers -- there are more jobs available, so they are easier to get.
There are real world applications. It makes it much easier to argue that this research is worth funding (thinking research grants).
Finally, one tip: try not to "switch" fields, but gradually change from one to another.
The transition from mathematics to physics can be very hard. Most importantly, don't let yourself get stuck because of the lack of rigor and of the lack of proper motivation for doing things in a certain way. Very often the motivation becomes clear only in hindsight (and this hindsight at times may only come as a result of first doing the required calculation even without the complete understanding of what is going on), so it is important to try to move forward and (provisionally) ignore the bit that got you stuck. See also this text by S.P. Novikov about his personal experience of learning physics; some of the above suggestions were taken from there. Also, whenever possible, try to pick several books or sets of lecture notes on the subject rather than sticking to a single one. As for the actual learning of physics, you can try to look into the Feynman Lectures on Physics to get a general feeling of the way the physicists think.
With respect to gowers' comment, I have a few recommendations about how you might move in a continuous path from algebraic geometry to compressed sensing. I know very little about compressed sensing myself, but some of my colleagues work in the field.
I believe Venkat Chandrasekaran's work with Pablo Parrilo et al. on rank-sparsity incoherence (check Venkat's website) uses some tools from AG. I think this may also be true of Ben Recht's work with Pablo et al. on nuclear norm minimization as a surrogate for rank minimization.
In general, it seems reasonable that problems of finding low-rank solutions to equations are much more interesting from an AG perspective than those of "classical" compressed sensing in which one merely wishes to find sparse solutions. Surely there are many settings in which rank-minimization problems arise beside the two I mentioned above.
Quantum computations: there are some interesting books; when I was an undergrad I was quite happy about Classical and Quantum Computation by three wonderful mathematicians with big teaching talent.
You'll also be much more knowledgeable in that area if you learn basic quantum mechanics and statistical mechanics; all of this probably is a topic big enough for a separate question.
Generally, though, if you are interested in any field on a research level you should probably take a look at what people are doing: in many cities you can find a good seminar on at least one of those topics nearby. And (to me) looking at what kinds of questions people solve on arXiv or MathOverflow seems like a good way to "see where the wind is blowing".
What is your intended career path? If a research university, I would say: don't change fields until after you are tenured (as Deane said).
If you are interested in developing quantum algorithms, there is a nice up-to-date survey by Childs and van Dam called Quantum algorithms for algebraic problems, available at http://arxiv.org/abs/0812.0380. For PDE's, you might look into hyperbolic polynomials and hyperbolicity cones. The latter are characterized algebraically, there are lots of open questions about them, and their study is also practially relevant to engineering applications like convex optimization (in case practical applications, or perhaps hot funding areas, are what you're looking for).
I like your style. Right now I'm distracting myself from solving a PDE arising in quantum mechanics using techniques from compressed sensing.
Of the 3, compressed sensing is the easiest to pick up as it's more of a new signal processing trick than a fully developed research area. Checking out Candes' ICM talk (http://dsp.rice.edu/cs) as well as Igor's obsessive blog (http://nuit-blanche.blogspot.com/) does not require that much of a time commitment and can be a lot of fun. Finding research problems is another story.