Making progress in learning hard math? I'm not a mathematician, but a computer scientist, but the following problem has puzzled me and I'm wondering how people counter it. In the last 5-6 years, I've probably read the first 5 sections of chapter 1 of Hartshorne's Algebraic Geometry 10 times or more and also each time done a fair amount of the problems.
My problem tends to be that to actually reach a level of understanding, so that I can retain some knowledge, I need to venture much further in an area. However, the problem always seems to be finding enough time within a reasonably short timeframe in order to do this.
I'm sure that I can't be the only one who has tackled this issue? How are other people coping with this? How do you make sure that you're making progress and not relearning the same old material over and over again?
 A: Finding the time to work on a problem, and finding the most productive way to use your valuable time are fundamental to pretty much everything you want to learn.  There are loads of self-help books devoted to this topic ranging over management, sport, music, etc.
I find the best approach is first to try to define your goals and then try to create a path that will get you there with appropriate testing/measurement points along the way.  Some subgoals may need to be modified or redefined or replaced along the way, but the main goals should be relatively fixed.  Work out how much time you actually have to work on your goals and then allocate time appropriately.  Don't try to do too many things at once, and don't make your subgoals too difficult.  Lots of little successes will give you the motivation to keep going.
More specifically for learning mathematics it is important to do lots of problems and, especially if you are ever planning to do research, to try to create and solve your own problems.  As you read a proof for example, ask yourself where each of the assumptions of the theorem used in the proof, what happens in the case where they are not all fulfilled, and so on.  Always try to find examples and counterexamples for any definition.  Use these examples in the general theorems to see what happens in these particular cases to develop your intuition and cement your understanding.
I'm also trying to learn algebraic geometry and have discovered Ravi Vakil's lecture notes.  For me these are far more accessible than Hartshorne and are relatively self-contained.
