now we're up to sixteen.

.. lectures, that is. Lectures 14, 15, and 16 from last week and today are up for download at the usual place. Be warned:

1. In case the resolution looks a little strange, the original paper notes were on blue paper (it was the only 3-hole punch paper I could find without paying for it) and I saved these notes as greyscale JPEG graphics and then PDF files.
   
   
2. I oscillated with what the 'k' means in the definition of a CAT(k) space .. that is, if it is a nonpositive or a nonnegative parameter. The meaning, I hope, should be understood.
   
   
3. The latest lecture may be error prone. Mario himself refers to it as "the hardest theorem in elementary mathematics," and I might agree with him. It's a tricky manipulation of symbols, and now I know how my Calculus II students from yesteryear must have felt! q: