ChapterZero

I’ve started reading Milnor’s “Topology from a Differentiable Viewpoint”, and I’m really enjoying it. I’ve made it up to the part where he uses manifold theory to prove the fundamental theorem of algebra– it’s amazing not only that it’s possible to do this, but that someone took the time to figure it out. Incidentally, while choosing a topology book, I came across another unexpected use of topological methods: a proof of the infinitude of the primes.

I’ve set myself the following goal, relating to the problems I mentioned earlier about the probability of the invertibility of random matrices: let be a smooth m-dimensional manifold, with , then where is the Lebesgue measure. Hopefully I’ll have all the machinery I need to examine this question after chapter 2.