SNU Topology of number fields, week of 15 July

I was looking over the notes and saw a few items omitted by Milne.

First, in the proof of Hilbert’s theorem 90, there is no proof of the linear independence of distinct characters on a group. You can try to figure it out yourself (quite easy) or look at the notes with that title Keith Conrad’s webpage. (For some strange reason, I can’t create a direct link to the paper.)

Secondly, Milne has a cryptic remark without definition or proof about a ‘Verlagerung’ (transfer map) in Proposition 3.2. This is reference to the fact that the restriction map for Tate cohomology in degree minus 2 corresponds to a classical formula for a map between abelianizations. This proof is not so straightforward, and can be found in Prop.11.12 of these nice notes by Holden Lee. It seems they’re from a reading course he took with Professor Shin Sug Woo.

If some of you would like to read a more leisurely exposition of group cohomology, I found online these notes by Kenneth Brown.

Finally, this old paper of Oort contains a discussion of extensions and cup products.

SNU Topology of Number Fields

Here are some documents to help you with the course.

1. General Motivation:

Lecture at the Cambridge workshop on non-abelian fundamental groups in arithmetic geometry (2009)

Lecture at Bordeaux conference in honor of Martin Taylor’s 60th birthday (2012)

Lecture at British Mathematical Colloquium (2011)

Lecture at AMC, Busan (2013)

2. James Milne’s lecture notes:

Class Field Theory

Algebraic Number Theory

Arithmetic Duality Theorems

3. Langlands programme:

Langlands’s lecture at Helsinki ICM

Report on work of Bao Chau Ngo (with Sugwoo Shin)

4. Topology of number fields:

Mazur’s paper on etale cohomology of number fields

Morishita’s paper on knots and primes

Furusho’s paper on Galois action on knots

Is air blue?

This blog is now mostly about my teaching at Merton and Oxford. However, I thought I would occasionally insert a post about my learning as well. For the most part, this means learning from my colleagues, whom I tend to pester endlessly over meals with silly and nerdy questions. One such appears in the title of this post; the victims were Alan Barr and Alex Schekochihin.

The answer of ‘yes’ can be justified as follows:

When we speak of the color of an object, it is the visual sensation* correponding to the mixture of frequencies in the light it scatters. This color is not constant, but there is a dominant one determined by an interpolation of usual experience.

From this point of view, it seems sensible to say that air in small quantities is transparent, but in large quantities**, it is blue.

So when a young child asks,

‘Why is the sky blue?,’

a reasonable response is

‘Because air is blue.’

It’s not that the usual answer in terms of Rayleigh scattering is wrong. But this is going into the deeper explanation of why air is blue. On the other hand, the shallower response above corresponds to something like:

Q: Why are leaves green?

A: Because they contain a lot of chlorophyll, which is green.

The Rayleigh scattering and so forth would then be analogous to an explanation of why chlorophyll is green. (I don’t know. Of course there must be a chemical explanation of sorts, but I seem to recall that there is also an interesting evolutionary explanation.) You can go into this later when the child is older.

By the way, ‘air’ here refers to the substance making up the Earth’s atmosphere. Atmospheric colours seen from other bodies in the solar system seem to be quite diverse.

Invoking the classification of different interactions, one might argue that there is a distinction between a process that involves a good deal of absorption-emission (which is the case for most solid objects we see) and one that only has elastic scattering. However, Alan and Alex assure me that these are really no different from a physicist’s view: It’s all scattering.

At the end of the Warden and Tutor’s meeting today, Simon Hooker contributed the interesting remark that liquid oxygen is a very pretty blue, although that is likely to be a different phenomenon from Rayleigh scattering.

Apparently, Philip Larkin was aware of this question and answer:

High Windows (1967)

When I see a couple of kids
And guess he’s fucking her and she’s
Taking pills or wearing a diaphragm,
I know this is paradise

Everyone old has dreamed of all their lives—
Bonds and gestures pushed to one side
Like an outdated combine harvester,
And everyone young going down the long slide

To happiness, endlessly. I wonder if
Anyone looked at me, forty years back,
And thought, That’ll be the life;
No God any more, or sweating in the dark

About hell and that, or having to hide
What you think of the priest. He
And his lot will all go down the long slide
Like free bloody birds. And immediately

Rather than words comes the thought of high windows:
The sun-comprehending glass,
And beyond it, the deep blue air, that shows
Nothing, and is nowhere, and is endless.

As you know, a poet chooses his words pretty carefully, especially in a short poem. So Larkin must have meant something nontrivial writing the last stanza: ‘blue sky’ would have been the phrase coming more readily to mind.

I think Alan summarized his annoyance quite succinctly:

‘They fuck you up, these mathematicians.’

Well, we don’t mean to…


* ‘Visual sensation’ here is referring to the fact that color seen by the eye is an equivalence class of light. You may know that the space of colours is three-dimensional, a mere projection of the space of physical light, which is infinite-dimensional. For an intriguing overview of this topic, I recommend the article ‘Geometry in Color Perception’ by A. Ashtekar, A. Corichi and M. Pierri, in: Black Holes, Gravitational Radiation and the University, (Kluwer Dodrecht, 1999), p. 535-549, CGPG pre-print 97/12-7.

** By the way, the fact that colour is an aggregate effect applies to usual objects as well. Gold is rather yellow, but I doubt it would be if we broke it down into molecules. Obviously, how much stuff needs to be present for us to start experiencing colour will depend on the substance.

Probability, sheet 4, number 8

I didn’t have time to fully discuss number 8 of sheet 4 . But Matej Balog has kindly agreed to let me post his solution on the blog.

Probability sheet 4, number 9

I would like to highly recommend working out number 9 on probability sheet 4, simply because it’s so amusing.

It reads like this:

Passengers arrive at a bus stop at rate 1 per minute. Find the distribution of the number
of passengers boarding a typical bus in two cases: (a) buses arrive regularly every 10
minutes; (b) buses arrive as a Poisson process with rate 1 per 10 minutes. Which one has
higher variance?

I arrive at the bus stop at 2pm. Find the distribution of the number of other passengers
boarding the same bus as me in the two cases above.

Most of the problem is straightforward, and I won’t go over it here. For example, the mean number of passengers boarding will be 10 in either case, but, naturally, the variance will be higher in case (b). But what’s amusing is the second part, where we assume you arrive at the bus at some given time, and calculate the mean number of *other* passengers boarding with you. For case (a), you find 10 as before, but for case (b), the mean turns out to be 20!

This situation is sometimes referred to as the ‘inspector’s paradox’. That is, if you’re an inspector trying to check up on the mean number of passengers boarding at a given stop by arriving at 2 PM for a number of days to take the bus, you will tend to find a larger mean than the true mean for the average bus, at least in the model (b). You should ask yourself why this happens.

Automorphism groups of groups

After last week’s GAGA tutorials, I wrote some notes related to sheet 5.

Quadratic forms

A quadratic form is essentially the same thing as a symmetric bilinear form, at least over a field of characteristic different from 2. A while ago, I wrote a brief note on quadratic forms dealing with diagonalization and geometry. I hope it’s useful for HT2013, sheet 4, problem 8, in particular.

Orthogonal bases

I was a bit confused during tutorials today about orthogonal bases, so I wrote up a short note on how to find them. This method is illustrated with problem 5 on sheet 4 for Linear Algebra II.

Probability sheet 4

Here is a sketch of problem 10 on sheet 4. The convergence question that came up in the tutorial is clarified in an elementary way.

Added: Towards the end of the document, I should have stated that S_N is a Riemann sum for the integral with mesh size 2^{-N+1}, Since this goes to zero and we are integrating a continuous function, we get convergence of the sum to the integral.

Homomorphism of formal groups

Wei Yue asks the following question: In the definition of a homomorphism f(T) between two formal groups F and G, what is the reason we required f\in TR[[T]]? The short reason is that we want f to preserve the origin, that is, f(0)=0. A more complicated reason is that in general, the expression G(f(X), f(Y)) may not make sense for a power series G(X,Y) if f has a non-zero constant term. This is a point we have been somewhat cavalier about: When can we substitute one power series in another and get a well-defined power series as the answer? You should ponder this for yourself a bit, and then try to formulate some conditions precisely using the definition

R[[T]]= \lim_{\leftarrow} R[T]/(T^n)

of the ring of power series with coefficients in R.

Here is an exercise worth trying out: Suppose F(X,Y) is a polynomial that defines an algebraic group law on the field \mathbb{C} with 0 as the origin. Then F(X,Y)=X+Y is the only choice.
Hang on, what about


Well, you have to formulate a bit carefully what is meant by an algebraic group law. Anyways, the conclusion is that deforming the usual group structure in any reasonable sense requires us to move out of the realm of polynomials.