Monday, December 18, 2006
Solved a conjecture
I proved Conjecture 5.9 on page 14 of Barry Jessup's paper in the case of two-step nilpotent Lie algebras. I had a proof for the case where the dimension of the center was equal to 2 a few weeks ago, and discovered a nice generalization this morning after my supervisor suggested some simplified notation to clarify the argument. The proof is nothing special; there are some tricky bookkeeping details and it runs to about 5 pages since it is a double induction, however it does not use any advanced tools from algebraic topology. I'm still proud I managed to prove it, because I've been working on this since April. Generalizing it to the case of an arbitrary nilpotent Lie algebra (or finding a counter-example) should be fun, and might require the use of spectral sequences. The current proof relies on what is essentially a spectral sequence collapsing at the first term. This is the second original result of my thesis. My previous result shows that the central representation is faithful for a certain class of two-step algebras. This latest result shows that it is non-trivial for all two-step algebras (however we know of two-steps where the central representation is not faithful).
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Congrats, nothing compares to the sense of accomplishment of finding a proof for a hard problem, that moment when things seem to suddenly fall into place, but proof of an unsolved problem has the extra boost of knowing you did it first. :)
I think if you avoid wasting so much time on JL, your chances of a Fields Medal will improve greatly. :) Either that, of winning a Putnam competition. :)
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