One of my favorite questions is: for which \(g, n, p\) is the moduli space of \(n\)-pointed genus \(g\) curves \(\mathscr{M}_{g,n, \mathbb{F}_p}\) unirational/uniruled? Will Sawin has just posted a beautiful paper on the ArXiv answering this question in most cases, for \(g=1\). Indeed, he shows that for \(n\geq p\geq 11, \mathscr{M}_{1, n, \mathbb{F}_p}\) is not uniruled... (more below the fold)
Read MoreA "minimal" proof of the fundamental theorem of algebra
When I was in graduate school, I came up with what I think is a nice proof of the fundamental theorem of algebra. At the time, I wrote it up here somewhat formally; I thought it might make a nice blog post, since the formal write-up obscures the very simple underlying ideas. The goal was to use the minimal amount of technology possible -- in the end I use just a little algebra and some elementary point-set topology, as well as the implicit function theorem...
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