Some 30 or so years ago JH Conway posed a question about the ubiquity of the Dynkin Diagram - not necessarily in public, but I heard him ask it. I think it was in the context of "what would be Hilbert's questions for today?"
The context was the way they cropped up in relation to Lie Groups/Algebras, Group Theory and Geometry (obviously linked subjects), but in apparently different (not entirely explained or understood) ways.
My recent reading has covered the Octonions (which seem to be more fashionable than hitherto), and I wondered if there is a consensus that current ideas adequately "explain" the links between these areas, or whether there is work still to be done?
Are there any conjectures in this area which are unresolved?
My main interest is in the special/exceptional cases which give rise to some very interesting and idiosyncratic mathematical structures.
There is a whole area, originally due to Conway and extended by Borcherds, in which positive integral lattices (quadratic forms) are embedded in indefinite or Lorentzian lattices. This is being continued, in particular, by Daniel Allcock at U. Texas, Austin. I can't say I understood everything, but help from Borcherds and Allcock led to a nice little paper; meanwhile, see http://www.ma.utexas.edu/users/allcock/
As to something that I did understand, an even (positive) integral lattice is one in which the inner product of any two vectors is an integer, while the norm of any vector (inner product with itself) is even. With the GCD of all norms exactly 2, we have normalized enough, and this object has a covering radius: place a closed ball of radius $R$ centered at every lattice point, and all of $\mathbb R^n$ is covered, and $R$ is the smallest radius that will do that. Open problem: find all such lattices with covering radius exactly equal to $\sqrt 2.$ That was the only thing Allcock asked me, as I had indicated that Gabriele Nebe had found all such lattices with $R < \sqrt 2.$ See http://www.math.rwth-aachen.de/~Gabriele.Nebe/pl.html the item titled Even lattices with covering radius strictly smaller than sqrt{2} as a pdf with corrections, the original appeared in Beiträge zur Algebra und Geometrie, Vol. 44, No. 1, 2003, 229-234