Differential Geometry - A First Course
One property of introductory differential geometry is that it's a course that is frequently taken twice - dropped the first time, and passed the second time. Somewhere around tangent bundles is roughly the mean point at which this happens.
The main thrust of the course is roughly as follows - manifolds behave exactly the way you'd think that they would, only you have no way of really knowing what you mean by "the way you'd think that they would" until you've worked at it with diligence. The problem sets are all obviously trivial, in that the statement to be proved is "A is B", and the natural response is "Well, Jesus. What else could A be? I mean, really". You don't really have a firm grasp of what A is and you don't really have a firm grasp of what B is, but you'll be damned if they're not the same. And 5 hours of bashing your head against the definitions later, you have shown that A is indeed B, but you still don't really know what A and B are, but at least now you've shown that they're the same.
So in a sense, Differential Geometry is the opposite of Real Analysis/Introductory Measure Theory, in which the definitions are all highly intuitive and the statements to be proved make immediate sense, and the A and B sometimes don't even look the same at first (most of the time they do look the same, but straightforward exercises with Fubini's Theorem often look surprising at first), but it takes some discipline to figure out which steps are helpful at which points and to keep your epsilons in a row - that is, there are tools being used, and the definitions are all lists of useful properties rather than demons of well-definedness.
So in case you were wondering, I am not proving any deep truths about the universe. I'm guessing that there's eventually a payoff for persuing differential geometry far enough, but it's a little doubtful that I'll be taking that track that far.
I had somehow imagined that it would have been different than this. How long does it take before all these tools start to coalesce into a general sensibility for how to approach non-trivial problems with cleverness and sparkling economy, anyway? In other words, when do I get to feel smart?
The main thrust of the course is roughly as follows - manifolds behave exactly the way you'd think that they would, only you have no way of really knowing what you mean by "the way you'd think that they would" until you've worked at it with diligence. The problem sets are all obviously trivial, in that the statement to be proved is "A is B", and the natural response is "Well, Jesus. What else could A be? I mean, really". You don't really have a firm grasp of what A is and you don't really have a firm grasp of what B is, but you'll be damned if they're not the same. And 5 hours of bashing your head against the definitions later, you have shown that A is indeed B, but you still don't really know what A and B are, but at least now you've shown that they're the same.
So in a sense, Differential Geometry is the opposite of Real Analysis/Introductory Measure Theory, in which the definitions are all highly intuitive and the statements to be proved make immediate sense, and the A and B sometimes don't even look the same at first (most of the time they do look the same, but straightforward exercises with Fubini's Theorem often look surprising at first), but it takes some discipline to figure out which steps are helpful at which points and to keep your epsilons in a row - that is, there are tools being used, and the definitions are all lists of useful properties rather than demons of well-definedness.
So in case you were wondering, I am not proving any deep truths about the universe. I'm guessing that there's eventually a payoff for persuing differential geometry far enough, but it's a little doubtful that I'll be taking that track that far.
I had somehow imagined that it would have been different than this. How long does it take before all these tools start to coalesce into a general sensibility for how to approach non-trivial problems with cleverness and sparkling economy, anyway? In other words, when do I get to feel smart?