Transitivity

The blog today is a result of transitivity: A implies B, and B implies C, so A implies C.

(A) I am at the Banff International Research Station (BIRS) in Banff for a workshop on random geometric graphs.

(B) The director of BIRS is Nassif Ghoussoub, who I recently interviewed.

(C) Nassif Ghoussoub recently posted on Facebook about Michael Atiyah’s paper on arXiv, which claims to settle a 60-year-old problem in complex geometry.

So from BIRS (A), I am led to Atiyah’s preprint (C). What’s especially extraordinary about the paper is that Atiyah is 86.

Atiyah’s discovery

The paper is concerned with the non-existence of a certain geometric structure in higher dimensions. The topology of the unit 6-dimensional sphere is distinct from 6-dimensional projective space over the complex numbers. The problem is to determine if there is a complex structure on the 6-sphere. There is an “almost” complex structure called J(0) for the 6-sphere, coming from the octonions (an 8-dimensional algebra over the reals).

Using topological tools from K-theory and KR-theory, Atiyah derives a short proof of the non-existence of a complex structure on the 6-sphere.  The paper is short and is leaving some experts scratching their heads on the details. Once the paper has been properly refereed, we will know for sure.

Who is Michael Atiyah?

Atiyah has one of the most accomplished mathematical careers, winning the Fields medal and a long list of other awards and honors. His Wikipedia article lists all of these.

His work focuses on geometry in a very modern sense. Fields like algebraic geometry, K-theory, and gauge theory all play a role in his tool chest. Although I don’t work in his area, his book on commutative algebra was a must-read for me as a pure mathematics graduate student. His latest work focuses on deep work in mathematical physics. Siobhan Roberts (who wrote biographies of Conway and Coxeter) interviewed Atiyah for Quanta Magazine. There is a great video there of him talking about mathematical beauty.

Octogenaricity

I’m inspired by the notion that mathematicians can continue doing ground breaking work into their 80s. Although I am no historian of mathematics, it is uncommon to find people researching mathematics at that age. Paul Erdős was a counterexample example, and Ron Graham is now 81 (he recently retired but still publishes). My field tends to focus on young people and their accomplishments. We are always on the lookout for prodigies. After all, the Fields medal is awarded only to those forty and younger.

As medicine and healthcare improves, we are likely to have an increasing cohort of mathematicians continuing their research well into their 80s.  On this note, I am reminded of the Longboat 10 K race I ran this September. Ed Whitlock ran the race too. It was a great accomplishment for me in 2015 when I passed him in the same race. This year I was jet-lagged, having landed the day before from a trip to Oxford, UK. In any case, that’s my official excuse. Despite my best efforts to keep up, Ed ran past me around the 9 K mark.

And Ed is 85.

Anthony Bonato