"Pound Cost Averaging"

Another math-y post. I worried about offending those who have no taste for math. My readership is sufficiently small that I cannot reasonably run the chance of turning anyone away. But this did irk me.

The reading today is again taken from the Financial Times. I read a piece by a trader who advocated what he called 'pound cost averaging'. This was his term for buying the same value of, say, shares at regular intervals. He pointed out that this enables one to buy more shares when the price falls, and fewer when the price rises. All very well, I hear you say. But wait, there's more!

He followed this up with an example that made my heart sink. It ran something like this: "Suppose you buy £100 worth of shares in January at £1 per share, then £100 worth of shares in February at £1.50 a share. You end up with 100 shares in Jan and 67 shares in Feb, giving you 167 shares. If, on the other hand, you had spent all £200 at the average price of £1.25, you would only have 160 shares."

I wanted to weep. I have grown accustomed to meeting otherwise very bright people who have forgotten how to do basic arithmetic. But this was in the FT - the newspaper for people interested in money and numbers! The example proved nothing, as examples always don't; the final numbers came out so close that one might suppose the opposite result could be achieved in other circs; and most importantly, it missed a better point.

The point is about the arithmetic-geometric mean inequality. The point is that simple maths helps in simple finance problems, and interesting math helps in interesting finance problems. If the FT doesn't take every chance to illustrate this simple point, who will? (Apart from me.)

Everything you've ever wanted to know about Etale Cohomlogy but were afraid to ask

This is going to be a short post to attempt to explain the 'point' of etale cohomology and the etale fundamental group. The latter is often called the algerbaic fundamental group.

In case you don't know what any of these things are, they are important in the study of arithmetic algebraic geometry. It is an area of mathematics somewhere between algebraic geometry and number theory. Lang called it Diophantine geometry. If you don't know what those words mean, what follows will not make sense to you.

To begin with, the most readable introduction to this area across which I have come is in Milne's online notes. They are incomplete and lack many proofs, but they give one the general idea. I don't aim here to teach anyone about etale cohomolgy or the fundamental group, but if you are learning about them and you happen to stumble across this site (a concurrence which is mind-bogglingly unlikely), then this may help you to see the big picture.

First of all, start by understanding the fundamental group. Consider two cases. If our base space is a field, the fundamental group is its absolute Galois group. If the base is a complex variety, the etale fundamental group is closely related to its regular fundamental group. For any other scheme, it is different, but you should understand that it generalises these very different notions, one purely geometric and the other purely algebraic.

Its importance in the study of etale cohomolgy lies in the following statement: Connected etale covers of a scheme correspond to transitive sets acted on by the etale fundamental group. The analogy with the geometric case is perfect, because connected covering spaces of a nice (if these comments are making sense to you so far, you'll know what nice means) space correspond to transitive sets acted on by the regular fundamental group.

Then the basic construction of the machinery of etale cohomology is very similar to that of regular sheaf cohomology, but extra algebraic input is often need for standard proofs.

However, the cohomology groups produced by etale cohomology are very different from those produced from Serre-style quasi coherent chomolgy theory. The real value of etale cohomology lies in these differences. They are sufficiently many and large to require little comment.

It is often said that etale cohomology (especially in characteristic zero) has two main ingredients: Galois cohomology and clasical topology. This is made most concrete by comparing the cohomlogy of a k-scheme to that of its base change to the algebraic closure of k. The relationship is broken down by Grothendieck's spectral sequence theorem into precisely the algebraic and geometric data for which one would hope.

If you make it this far, please leave a comment. I'd love to discuss this with someone.