Here’s an analogy that I try to complete from time to time.

Integers:Reals::Free Group on Two Generators:?

(Under addition, the integers are the free group on one generator.) I’m not precisely sure what properties of the construction of the reals I’m trying to generalize to the right-hand side of the analogy, beyond the fact that the answer should be a group that is a path-connected topological space.

Two candidates I’ve considered are: 1) certain sets of paths on the plane (which is naturally a groupoid, but you can bully it into being a group) or 2) the Lie group corresponding to the free Lie algebra on two generators (I don’t know in what sense, if any, such an object exists).

Finally, somebody (Rudbeckia Hirta) willing to tell the truth about the harmonic series:

I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.

Via Let’s Play Math.

When I was writing the year in review post, I did a quick websearch to refresh my memory on the Perelman-Yau story. (Reading about it I found the idea that the big story of 2006 was a public personality conflict between prominent mathematicians was too depressing to contemplate, so I ended up skipping the details.)

Wikipedia has two incredibly detailed articles about the subject. One provides a summary of Manifold Destiny, even going so far as to list every interviewee. The other describes the war of words between Yau and Gang Tian waged in Chinese newspapers and on the web. The story is not all that interesting, but references to it appear from time to time.

One twist in the story reported by Wikipedia that’s new to me is that Sujit Nair discovered a section in the Cao-Zhu proof of the Geometrization Conjecture that duplicated some results in Kleiner and Lott’s manuscript. Cao and Zhu issued an erratum acknowledging the duplication.

Investigating John Baez’ shocking (but true) claim that I misspelled Bourbaki’s first name, I discovered that the MacTutor History of Mathematics site has a detailed biography of Bourbaki, despite the obvious objections to a biography of that illustrious personage.

Surprisingly, this thread at Not Even Wrong (attached to a post about Harvard’s alumni magazine) has drifted into a discussion of the merits or demerits of Bourbaki.

I would argue that whatever the merits of Bourbaki’s purely mathematical contribution, the influence on expository style was negative. (Though it’s possible that Bourbaki merely typified the style, but did not cause it.) The austere theorem-proof style of mathematical writing was dominant for much of the last century, only beginning to fade in the 90s. (Compare Bourbaki’s Commutative Algebra, or Matsumura’s text of the same name, to Eisenbud’s Commutative algebra with a view towards algebraic geometry. The earlier books aim for an effect akin to Moses descending from Sinai. Eisenbud’s book is much more idiosyncratic, full of motivations, hand-wavy gestures towards geometric intuition, and asides.)

Some subjects are so compelling that they require no external motivation — they sell themselves. For me, group theory would be an example. For other subjects, you need some idea of how human beings ever arrived at a topic so outre. The first time I saw the definition of Lie algebra, my reaction was “Huh?” I needed to see the geometric motivation, plus a few unsophisticated derivatives of matrix equations, to see the point.

I take a few days vacation, and I see (via Scott Aaronson) and things promptly spin out of control. Now people are publicly dividing by zero. I shudder to think what would have happened if I’d been gone for a full week.

Andrew Ranicki has an excellent webpage devoted to the the Hauptvermutung; the conjecture, now known to be false, that for a triangulable space, all triangulations are equivalent. Even more surprising, it’s false even if you restrict yourself to only manifolds. The discovery spelled the end of the original combinatorial approach to algebraic topology (though I think the approach was largely superceded by the time the falsity of the conjecture was discovered.) Ranicki includes a link to a PDF of The Hauptvermutung Book, an introductory collection of papers on the subject that he edited.

I also came across these lecture notes that describe Milnor’s counterexample in detail.

I just saw the movie Stranger than Fiction, and I noticed a strange pattern in the last names of characters: Pascal, Hilbert, Escher, Cayley, Mittag-Leffler. Also, the main character’s favorite word is “integer”. The main character has a penchant for counting, but other than that the movie is as far removed from mathematics as can be imagined, so it’s all very mysterious.

The December Notices of the AMS are out. The two features articles are:

• The Search for Symmetric Simple Venn Diagrams considers Venn diagrams as mathematical objects in their own right, rather than just as a graphical representation of sets.
• Better Ways to Cut a Cake discusses the game theory of cake cutting.

This month’s What is a… Quasiconformal Mapping introduces a generalization of conformal mappings that makes sense in general metric spaces.