Callan asked me if I was going to do a DG this Thursday (the day after tomorrow) so I wanted to ask if anyone else was interested in one. I'll run one if 3-4 people or so are interested, I'm not sure what the topic will be yet so feel free to suggest something here if you're interested (otherwise I'll choose it later on but probably won't advertise it on the front page :P).

This probably won't happen but I wouldn't mind one on algebraic number theory.

This probably won't happen but I wouldn't mind one on the theory of numbers which are algebraic.
For example the -invariant of curves with complex multiplication.

This probably won't happen but I wouldn't mind one on zones.

Ok, I guess I'll do one on Monday on algebraic number theory and advertise it properly, so we may get a few more people to turn up.

P.S. I'd love something on zones as well but I think the leading expert on them is pretty busy nowadays.

I could probably throw something together about Zones and their use in graph theory. This is quite an underrated area: you can use graphs to 'designate' (technical term) different zones, and then apply group actions on the set defined by each zone. Then you get some cool results out about the O-zone, given by: , the orbit of the action. One particular result is that using these O-zones allows us to partially order the integers.

That would be amazing! What I find really nice with the -zones is the idea that the -invariant of an -zone is defined, and gives you deep insight into its structure. For example, a result I really like is that if your zone has a Niloç metric defined then the -invariant is an algebraic integer.

I also quite like the result you quote, especially in view of its application to the symmetrisation problem, helping you classify all symmetric spaces depending on the holonomy group (also depending on the monodromy obviously, which is where the orbit of the action of the connection comes in).

I'm still a bit unsure about the Berger classification though, I mean clearly it shows that every hyperkähler manifold is a Calabi-Yau manifold, every Calabi-Yau manifold is a Kähler manifold, and every Kähler manifold is orientable, but I'm unsure if every orientable manifold actually admits a Niloç metric? I think you can get one by averaging over all loops, getting a parametrisation depending on the curvature at that point (for the geodesics); I'd like to hear a more rigorous approach though.

The rigorous approach you're looking for can be quite easily derived by using inclusion-exclusion on the Trawian of the Kähler manifold. Indeed, Schoof's algorithm already yields the required zonal superclosures so it's an elementary fact in the theory of megacommutative sums that using inclusion exclusion on the family of superclosures does indeed give you a well-defined Niloç metric.

Interestingly enough, this Niloç metric can be uniquely extended to the zone of all zones and can thus in practice be applied to any mathematical object whatsoever. In particular, one can see that the Niloç metric of any -set is exactly 1 and thus, by the trivial observation that the Niloç metric is always equal to the dimension when the latter is defined, all the zeros of the Riemann, Dedekind or Hasse-Weil zeta functions and generally any Dirichlet L-function all lie on a line (note that, in the case of the Riemann zeta function, this can also be seen by differentiating enough times).

Also, for a more elemetary approach to this problem, you should check out the seminal paper by Grothendieck and Awe which proves any such function has only a finite number of zeros (a consequence of the fact that the Reimann sphere is bounded), reducing the problem to a simple matter of calculation.

Clearly the knowledge of other members of this forum with regards to Zones far outreaches my own interest.

For those less familiar, a Zone (also referred to as a Po-Group by some) is a more general object than a group.

This definition gives our first fundamental theorem.

The Fundamental Theorem of Zones. Every group, , is a zone.

Given a particular zone, normally denoted we can find a group that generates with exactly three generators: normally denoted . In addition to this: , and if we have a bizone.

The intersection of two zones is called the interzone and has a representation as a Niloc Metrix (see elsewhere on forum).

People interested in numerical analysis can then use this matrix (specifically the eigenvalues of it) to calculate the density of the zone. Density is an interesting (and currently under developed area), but one particularly startling use is that it allows us to prove the 'Birthday Problem' in just one line, showing that the probability that I share a birthday with somebody else is around half.

In particular what Grothendieck and Awe showed was that the EWQ conjecture was essentially equivalent to some of the more trivial results of combinatorial zone theory (i.e. All bounded (and therefore finite (see for example one of Awe's earliest papers) zones may be classified precisely by the so called D.U.C. polynomial which was elegantly described as: where is the canonical megacommuative sum, are the singular Niloç points (when considered over the natural zonal bundle inclusion mapping), is any monadically diverse curve lying in a strictly semi-stabular-K7 presentation of our zone, denotes the nth Awe number which is simply the size of our associated zonal group (see fundamental theorem above) and finally p denotes all primes except those satisfying the relation i.e. the Grothendieck-Awe containing numbers) and this of course allowed quite a few theorems of finite zonal theory to be solved as mentioned above and was one of the more beautiful results deduced in the paper with what surprised many in the field by the elementary methods used.

Edit: I forgot to mention that I really love density.