## Archive for the ‘Uncategorized’ Category

### ICM2014 — Bhargava laudatio

August 15, 2014

I ended up writing more than I expected to about Avila. I’ll try not to fall into the same trap with Bhargava, not because there isn’t lots to write about him, but simply because if I keep writing at this length then by the time I get on to some of the talks I’ve been to subsequently I’ll have forgotten about them.

Dick Gross also gave an excellent talk. He began with some of the basic theory of binary quadratic forms over the integers, that is, expressions of the form $ax^2+bxy+cy^2$. One assumes that they are primitive (meaning that $a$, $b$ and $c$ don’t have some common factor). The discriminant of a binary quadratic form is the quantity $b^2-4ac$. The group SL$_2(\mathbb{Z})$ then acts on these by a change of basis. For example, if we take the matrix $\begin{pmatrix}2&1\\5&3\end{pmatrix}$, we’ll replace $(x,y)$ by $(2x+y, 5x+3y)$ and end up with the form $a(2x+y)^2+b(2x+y)(5x+3y)+c(5x+3y)^2$, which can be rearranged to
$(4a+10b+25c)x^2+(4a+11b+30c)xy+(a+3b+9c)y^2$
(modulo any mistakes I may have made). Because the matrix is invertible over the integers, the new form can be transformed back to the old one by another change of basis, and hence takes the same set of values. Two such forms are called equivalent.

### ICM2014 — introductory post

August 11, 2014

Four years ago I blogged from the ICM in Hyderabad. The posts are amongst the most popular I have written — my statistics show that some of them are still being read quite regularly even now. Right now I’m sitting in Charles de Gaulle airport waiting to board a plane to Seoul, where I will be attending this year’s ICM, or rather, as I did last time, attending the first half of it. I’m not sure I’ll have the time or energy to write quite as much about ICM2014 as I did about ICM2010, but I’ll do what I can. In particular, I’ll try to convey exactly what I manage to understand from some of the main talks — especially the talks about the work of the new Fields Medallists. Given all the rumours about the likelihood of one of them being female, I am particularly glad to be going to the opening ceremony to witness (I hope) an important moment in mathematical history.

Just as the last ICM was the first (and still only) time I had been to India, this one will be my first visit to Korea. I’m looking forward to that aspect too, though my hotel is right next to where the congress is taking place and the programme looks pretty packed, so I’m not sure I’ll see much of the country. Talking of the packedness of the programme, I can already see that there are going to be some agonising decisions. For example, Tom Sanders is giving an invited lecture at the same time as Ryan Williams, two speakers I very much want to listen to. I suppose I’ll just have to read the proceedings article of the one I don’t go to. Equally unfortunate is that Ben Green’s plenary lecture is not until next week, when I’ll have gone. But I hope that I’ll still be able to get some kind of feel for where mathematics is now, what people outside my area consider important, and so on, and that I’ll be able to convey some of that in the next few posts.

I’d better stop this now, since I’ll soon be getting on to an Airbus 380 — a monstrously large double-decker plane. One of my children is something of a transport enthusiast and told me in advance that this would be the case (he had looked it up on the internet). I had hoped to end up on the top floor, but that turns out to be for business class only. The flight is about 11 hours: it leaves at 9pm French time and arrives at around 2:30pm Korean time. The challenge will be not to be utterly exhausted by the time of the opening ceremony on Wednesday morning. My memory of Hyderabad is that by the end of the four days I was so tired that I was almost getting anxious about my health. I plan to look after myself a bit better this time, but it may be difficult.

### Holding a country to ransom

October 15, 2013

Here is a quick thought about the mathematics of the US shutdown, not to be taken too seriously (the thought I mean — the shutdown obviously is to be taken seriously). It’s for the benefit of anyone who is puzzled that the Tea Party can have such a large influence, and more generally how a political system can be stable when almost nobody likes it. I’m going to prove that in a country of $n$ people, it is possible to devise a democratic system in which $n^\alpha$ of those people control the decisions, where $\alpha=\log 2/\log 3$. For example, in a population of 100,000,000, all you need is a band of fanatics with about 112,000 people — or approximately 0.1% of the population. Although we do not have such a system and the distribution is unlikely, the systems and distributions we do have still allow a minority to have undue influence, and for similar reasons. What I’m about to describe is the extreme case.
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### Another test

April 2, 2013

This time I want to test whether I can have polls where the results are not visible until the poll closes. So if you have a few seconds to vote, that would be very helpful. If the facility works, then my next post will include some secret ballots.

### An experiment concerning mathematical writing

March 25, 2013

Update: comments on this post are now closed, since my latest post would compromise any further contributions to the experiment.

Most of this post consists of write-ups of proofs of five simple propositions about metric spaces. There are three write-ups per proof, and I would be very grateful for any comments that you might have. If you would like to participate in the experiment, then please state your level of mathematical experience (the main thing I need to know is whether you yourself have studied the basic theory of metric spaces) and then make any comments/observations you wish on the write-ups. The more you say, the more useful it will be (within reason). I am particularly interested in comparisons and preferences. For each proof, the order of the three write-ups has been chosen randomly and independently.

It would also be useful if you could rate each of the 15 write-ups for clarity and style. So that everyone rates in the same way, I suggest the following rating systems.

Clarity.

-2 very hard to understand
-1 hard to understand
0 neither particularly hard nor particularly easy
1 easy to understand
2 very easy to understand

Style.

-2 very badly written
0 neither badly written nor well written
1 well written
2 very well written

I stress that ratings should not be regarded as a substitute for comments and observations, or vice versa. What I really need is both comments and numerical ratings.

I do not want people to be influenced by the answers that other people give, so all comments on this post will go to my moderation queue. When I have enough data for the experiment, probably in a week or so, I will publish all the comments (unless for some reason you specifically request that your comment should not be published).

The more people who participate, the more reliable the results of the experiment will be. I realize that it may take a little time, so thank you very much in advance to everybody who agrees to help. (Update 26th March: I now have over 30 responses; they have been very helpful indeed, so I am extremely grateful for those. If they keep coming in at a similar rate over the next few days it will be wonderful.)
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### Test

March 24, 2013

I am testing the WordPress feature that allows me to moderate all comments before allowing them to appear. This has nothing to do with the discussion on the last post. Rather, I want to be sure that the feature works before my next post, where it will be important for people to comment without seeing what others have said. So if someone could make a quick comment on this post, that would be helpful. Once I’m sure the feature is working, I’ll put up the post for which it matters.

In an ideal world, I would use the feature just for that post. However, as far as I can tell, my only options are allowing all comments, moderating all comments, or disabling comments completely on individual posts. Sending comments to the moderation queue on a post-by-post basis doesn’t seem to be possible, but if anyone knows a way, then I’d be very pleased to hear about it. Assuming there isn’t a way, then for a short while, all comments on this blog will be moderated, but I will try to approve comments on other posts regularly, so I hope this won’t be too annoying.

Update. Good job I did this test. I changed the relevant setting but didn’t click “Save settings”. Hence the three comments below.

Further update. OK, now it seems to be working just fine. Many thanks to those who sent test comments. I’ll put up the new post later this evening (British time).

### A more formal statement about mathematical publishing

February 8, 2012

A group of mathematicians have been putting together a statement that explains some of the background to, and reasons for, the Elsevier boycott. This statement, which has been signed by 34 mathematicians (we are confident that many more would be happy to endorse it, but we had to stop somewhere), is now ready for release. If you are interested in reading it, then click here.
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### Abstract thoughts about online review systems

February 2, 2012

As many people have pointed out, to get to a new and better system for dealing with mathematical papers, a positive strategy of actually setting up a new system might work rather better than complaining about the current system. Or rather, since it seems unlikely that one can simply invent ex nihilo a system that’s satisfactory in all respects, one should set up systems (in the plural) and see which ones work and catch on.

I’ve already had a go at suggesting a system, back in this post and this post. Another system that has been advocated, which I also like the sound of, is free-floating “evaluation boards” that offer their stamps of approval to papers that are on the arXiv. (I associate this idea with Andrew Stacey, though I think that in this area there are several good ideas that have been had independently by several people.) But instead of discussing particular systems, which runs the risk that one ends up arguing about incidental details, I want to try to adopt a more “axiomatic” approach, and think about what it is that we want these new systems to do. Once we’re clear on that, we have a more straightforward problem to solve: how do we achieve most efficiently what we want to achieve?
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### AV vs FPTP — a supplementary post

April 29, 2011

In this post (the first past the previous one) I want to consider some further arguments, most of which have arisen from the interesting comments I have received. You’ll have to be pretty interested in voting theory to have waded through my earlier post and still want more, but NO2AVers who are looking for ammunition may find some here (as long as they are selective about which arguments they go for).

I have also written a shorter post on AV versus FPTP. I plan to post it at the weekend (when people are less distracted by the Royal Wedding, though perhaps the audience for that is rather different from the audience for this). (more…)

### Status update

November 30, 2010

I would not normally edit my own Wikipedia page, but there was an inaccuracy I felt strongly about. I hope that Quomodocumque will understand.