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Curves and Tangents

In algebraic curves class on Friday, Joe Harris mentioned that the problem of whether there exist curves where any tangent intersects the curve at another point is open. This seemed like an interesting problem so I did some thinking about it. I realized that the case where one of these "second intersections" is really the point of tangency is probably a more doable case since you don't need to worry whether the points on the tangent lines you are getting are really part of the same curve. Anyway, restricting to C and doing some analysis, I managed to show that there exists a counter-example if and only if two polynomials of the form:

d/dx [(x^n-1)/(x-1)]

have a common root. Computer experiment shows that these polynomials are all irreducible, but I don't know how to prove it. If I could though, it would give me a perhaps novel proof of the conjecture in the genus 0 case, and provide some leverage on the genus 1 case.


No country for old men

I saw No country for old men at LSC last night. I was a little unimpressed by it. While they managed to do some interesting things as the characters were hunting for each other, the whole thing ended up pretty anti-climatic.


The contangent bundle of T is self-adjoint.

Not that it's actually possible to know too much math, I just wanted a catchy title.



So yeah, I finally created an account.  There was a long time when I didn't really see the point of a diary much less one publicly accessible, but I've determined several reasons to change my mind.

1) To waste time
Not that I currently need less time for other things.  Just I really don't like being bored, and it's good to have more things to fill the extra time with.  Also sometimes when switching activities I feel the need to do random stuff online to clean my plate.
2) To let my parents know what's going on
Hopefully this will mean that I don't need to call them as much.
3) For my later use
It might be nice to be able to look back at what I did in another year.  Also writing down what I do gives me the impression that I'm being more productive.
4) For my friends
I don't know if any of you care what I'm up to, but just in case, this is a lot easier than trying to maintain a conversation on email.

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March 2008


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