I’m really starting to notice the strain of taking too many classes this quarter. (I never learn, do I?) Classes that require a hundred pages of reading a week take a lot of time, especially for people who read as slowly as I do. I’m pretty lost in differential geometry, and I have been getting more and more lost since about halfway through the last quarter. There isn’t much excuse for that given that he is going very slowly. I’m not having too much trouble understanding the material in my other classes, fortunately.
But now the work is really starting to pile up. The main problem is that we have a homework assignment for quantum groups due on Monday, and the professor likes to give problems that are very difficult (at least for me). And I basically try to do them all by myself since there aren’t any other undergraduates in the class. So normally I’d spend all weekend on these problems and (hopefully) solve them. But this time I also have a history midterm due on Tuesday, a quiz in homological algebra on Monday to prepare for, and an abstract for a paper in music due on Thursday. And I should spend some time on algebraic number theory problems that have no deadline but really ought to get done at some point. And I should always be trying to catch up in differential geometry.
On the bright side, I found out that a problem we did in math club last quarter is really nice! The problem asks whether it is possible for a countably infinite set to contain uncountably many nested subsets. The answer, surprisingly, is yes. And if you have taken a course in real analysis, you already knew that. One of the problems on my homological algebra assignment I’m going to turn in in about an hour and a half is to show that if K is a field, the functor D from K-Mod to K-Mod given by D(V)=V* (the dual space) is not a duality. To do that, we show that a countably infinite dimensional vector space is not isomorphic to a dual of any vector space. So you find uncountably many linearly independent functionals on an infinite dimensional vector space, and we’re done! We were given the hint of finding uncountably many subsets of Q so that any two have finite intersection, and the obvious way of doing that is to take a Cauchy sequence of rationals converging to each real number. But I like mine better!