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Thursday, January 05, 2006
Comprehensively Fun
I am slowly beginning to become aware of the sheer crushing weight of material I will need to know for the comprehensive exams. It's roughly the equivalent of 6 3rd or 4th year math classes, spread over 3 exams taken in the same week. Now that I've started studying, I'm realizing how much this really is.
Of course (in theory) this shouldn't be so bad, as I (in theory) have learned most of this material earlier. Not so, as it appears. I started yesterday by going over very basic material - equivalent statements of the axiom of choice. All good until I got to the well-ordering principle, and to my great shock, learned that it was not what I had thought it was for the past several years (I thought well-ordered was just another wording of totally ordered; it's actually a slight strengthening of totally ordered). Not a major mistake, but somewhat worrying.
Nor does it help when the material I'm (re)learning from is wrong. Today I was going over the seperation axioms for a topological space - T1, T2, regular, normal, etc. They all seemed familiar, until I started trying to prove a statement in the book, namely that T3 implied T2. I worked at it for a bit, and was a bit confused. I could find no proof. And yet there it was, right after the definition of T3, ``it is obviously clear that T3 implies T2''. So I looked in another book, and it turns it out its not true at all. This ``obvious'' statement is incorrect. Yay for textbooks!
I like the idea of comprehensive exams - making show you are familiar with a broad selection of material. I even like the idea in practice, for me, because when working in category theory, it helps to know and understand lots of things in lots of different areas. What I'm not so sure about is this whole actual ``doing'' part of the comprehensive exams.
I am slowly beginning to become aware of the sheer crushing weight of material I will need to know for the comprehensive exams. It's roughly the equivalent of 6 3rd or 4th year math classes, spread over 3 exams taken in the same week. Now that I've started studying, I'm realizing how much this really is.
Of course (in theory) this shouldn't be so bad, as I (in theory) have learned most of this material earlier. Not so, as it appears. I started yesterday by going over very basic material - equivalent statements of the axiom of choice. All good until I got to the well-ordering principle, and to my great shock, learned that it was not what I had thought it was for the past several years (I thought well-ordered was just another wording of totally ordered; it's actually a slight strengthening of totally ordered). Not a major mistake, but somewhat worrying.
Nor does it help when the material I'm (re)learning from is wrong. Today I was going over the seperation axioms for a topological space - T1, T2, regular, normal, etc. They all seemed familiar, until I started trying to prove a statement in the book, namely that T3 implied T2. I worked at it for a bit, and was a bit confused. I could find no proof. And yet there it was, right after the definition of T3, ``it is obviously clear that T3 implies T2''. So I looked in another book, and it turns it out its not true at all. This ``obvious'' statement is incorrect. Yay for textbooks!
I like the idea of comprehensive exams - making show you are familiar with a broad selection of material. I even like the idea in practice, for me, because when working in category theory, it helps to know and understand lots of things in lots of different areas. What I'm not so sure about is this whole actual ``doing'' part of the comprehensive exams.
