I gave this talk earlier today. It went somewhat well, though I didn’t say too much about the Brouwer fixed-point theorem. The main application I got to was the theorem of Monsky that one cannot subdivide a square into an odd number of triangles of equal area. Bizarrely, the only known proof of this result uses properties of the 2-adic valuation, in particular its extendability to the real line.
I could run latex2wp and make this into a proper blog post, but I think I’ll just leave it as a PDF for anyone interested to look at.