## Archive for 2012

### What maths A-level doesn’t necessarily give you

November 20, 2012

I had a mathematical conversation yesterday with a 17-year-old boy who is in his second year of doing maths A-level. Although a sample of size 1 should be treated with caution, I’m pretty sure that the boy in question, who is very intelligent and is expected to get at least an A grade, has been taught as well as the vast majority of A-level mathematicians. If this is right, then what I discovered from talking to him was quite worrying.

The purpose of the conversation was to help him catch up with some work that he had missed through illness. The particular topics he wanted me to cover were integrating $\log x$, or $\ln x$ as he called it, and integration by parts. (Actually, after I had explained integration by parts to him, he told me that that hadn’t been what he had meant, but I don’t think any harm was done.) But as we were starting, he asked me why the derivative of $e^x$ was $e^x$, and what was special about $e$.
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### What actually happened

November 9, 2012

The short version is that I’ve had the ablation (see previous post) and the surgeon who did it says that he has a good feeling about it. It’s taken till now to write this because, unlike most people who have ablations, I felt terrible for two days after it — with a headache (normal) and a fever (less normal but not unheard of). The fever was not very high, but high enough to be unpleasant, and meant that the only thing I could bear to do was go to bed, except that on the second night after the operation I had to spend part of the night sitting up on a sofa because my chest hurt too much when I was horizontal. (That was normal, and nothing to worry about.) So today is the first day that I am well enough to do anything as strenuous as writing a blog post.
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### Mathematics meets real life

November 5, 2012

I’ve been in two minds about whether to post this. On the one hand, I try to keep personal matters out of this blog — though there has been the occasional exception — but on the other hand I have a topic that fits quite nicely with some of what I’ve been writing about recently, since it concerns a fairly important medical decision that I have had to make based on what felt like inadequate information. Since that is quite an interesting situation from a mathematical point of view, and even a philosophical point of view, and since most people have to make similar decisions at some point in their lives, I have opted to write the post.

The background is that over the last fifteen years or so I have had occasional bouts of atrial fibrillation, a condition that causes the heart to beat irregularly and not as strongly as it should. It is quite a common condition: I’ve just read that 2.3% of people over the age of 40 have it, and 5.9% of people over 65. Some people have no symptoms. I myself have mild symptoms — I can feel a slightly strange, and instantly recognisable, feeling in my chest, and I experience a few seconds of dizziness almost every time I stand up from a relaxed seated position — otherwise known as orthostatic hypotension, which I often used to get anyway (as do many people).
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### EDP27 — the modular version of Roth’s AP-discrepancy theorem

September 19, 2012

Recall from earlier posts Gil’s modular conjecture for HAPs. It states that if $n$ is large enough and $f$ is a function from $\{1,2,\dots,n\}$ to $\mathbb{Z}_p$ that never takes the value 0, then for every $a$ there exists a HAP $P$ such that $\sum_{x\in P}f(x)\equiv a$ mod $p$. It is easy to see that this implies EDP, so it may well be very hard, or even false. However, one can hold out a little hope that, as with some strengthenings of statements, it is in fact easier, because it is in some way more symmetrical and fundamental. Given that, it makes good sense, as Gil has suggested, to try to prove modular versions of known discrepancy theorems, in the hope of developing general techniques that can then be tried out on the modular EDP conjecture.

A very obvious candidate for a discrepancy theorem that we could try to modularize is Roth’s theorem, which asserts that for any $\pm 1$-valued function $f$ on $\{1,2,\dots,n\}$ there exists an arithmetic progression $P$ such that $|\sum_{x\in P}f(x)|\geq cn^{1/4}$. That gives rise to the following problem.

Problem. Let $p$ be a prime. What is the smallest $n$ such that for every function $f:\{1,2,\dots,n\}\to\mathbb{Z}_p$ that never takes the value 0, every $a\in\mathbb{Z}_p$ can be expressed as $\sum_{x\in P}f(x)$ for some arithmetic progression $P$?

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### EDP26 — three generalizations

September 6, 2012

This short post is designed as a possible way in to EDP for anyone who might be interested in participating but daunted by the idea of reading large amounts of material. One of the natural strategies for proving EDP is to try to formulate and prove stronger statements. At first that sounds paradoxical: isn’t it even harder to prove a stronger statement? But the answer to that question is often no. To give a slightly silly example, suppose you were asked to prove that for every $c>0$ there exists $N$ such that for every $n\geq N$ if $n$ is odd and has at least $c\log n$ prime factors (counted with multiplicity), then $2^{\phi(n)}\equiv 1$ mod $n$, where $\phi$ is Euler’s totient function. You could make the problem easier by proving Euler’s theorem, that $a^{\phi(n)}\equiv 1$ mod $n$ for every $n$ and every $a$ that is coprime to $n$. You wouldn’t have as many hypotheses to use, but that’s good, since they can’t be used. Perhaps a better and more relevant example is when you generalize the class of numbers you are working with so as to allow a wider set of methods. For instance, suppose you want to prove that the largest possible product of three positive integers that add to 300 is at most $10^6$. If you replace positive integers by positive reals, then you suddenly have available methods that you didn’t have before — for example, you could use compactness plus a lemma that says that if any two numbers are not equal then you can increase the product by replacing both of them by their average. (I’m not saying that’s the easiest proof — just that it’s a proof that you can’t do without first generalizing the statement.)
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### EDP25 — third guest post by Gil Kalai

September 4, 2012

## The Polynomial Method

The polynomial method is another basic combinatorial technique that occasionally works. One way to describe the method is as a way to translate a combinatorial statement into the vanishing of a certain polynomial modulo $p$.

### A demonstration of the method

Theorem: Every graph (or hypergraph) G with n vertices and 2n+1 edges contains a nontrivial subgraph H with all vertex-degrees divisible by 3.

(This is a theorem of Noga Alon, Shmuel Friedland, and me from 1984.)

Before the proof: If we want to get a subgraph with all vertex degrees even then we need n edges (or n+1 edges for hypergraphs). This has a simple linear algebra proof which also gives an efficient algorithm.
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### EDP24 — an attempt to get back into the diagonal-decomposition approach

August 31, 2012

Gil has a not quite finished third post that will appear soon on this blog. Meanwhile, here are a few thoughts I’ve had recently as I tried to get my brain into EDP mode again.

The approach to the Erdős discrepancy problem that, rightly or wrongly, I found most promising when we were last working on it was to prove a certain statement about matrices that can be shown quite easily to imply a positive solution to the problem. In this post, I’m going to treat that matrix statement as the problem, and think about how one might go about trying to prove it. I’ll give the brief explanation of why it implies EDP, but not of what the possible advantages of the approach are (discussions of which can be found in some of the earlier material).
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August 27, 2012

## The Large Deviation Heuristic: an example – triangle-free graphs

Here is a very general probabilistic-based heuristic that seems to give good predictions for questions related to EDP. I will refer to this heuristic as “LDH”. (In my polymath5 comments I referred to it as PH – probabilistic heuristic)). I am thankful to Noga Alon and to Yuval Peres for some helpful help.
Here is an example: Suppose we want to study the following extremal problem.

What is the largest number of edges in a graph on n vertices with no triangle.

If we use the probabilistic method we can ask what is the probability that a random graph in $G(n,m)$ contains no triangle. As long as this probability is positive we know that a triangle-free graph with n vertices and m edges exists. (Being a little careful we can consider $G(n,p)$ instead of $G(n,m)$ where $m=p{{n}\choose {2}}$. Looking at random graphs gives us a perfectly correct proof of the assertion that there are triangle-free graphs with $n$ vertices and $Cn$ edges for every $C$.

LDH:

1) Estimate naively the probability that a random graph in G(n,m) contains no triangle.

2) Choose m so that this estimated probability behaves like 1 over the number of graphs with n vertices and m edges.
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August 22, 2012

### The general form of a discrepancy problem

Let $H$ be a hypergraph, i.e., a collection of subsets of a ground set $A$. The discrepancy of $H$, denoted by $disc(H)$ is the minimum over all functions $f:A \to \{-1,1\}$ of the maximum over all $S \in H$ of

$|\sum \{f(s):s\in S\}|$.

We will mention one additional definition, that of hereditary discrepancy. When $H$ is a hypergraph and $B\subset A$, the restriction of $H$ to $B$ is the hypergraph with vertex set $B$ whose edges are all sets of the form $S \cap B$ for edges $S$ of $H$. The hereditary discrepancy $herddisc(H)$ of $H$ is the maximum over all $B \subset A$ of the discrepancy of $H$ restricted to $B$.
Here is a link for a recent post discussing discrepancy and the famous Beck-Fiala theorem. The Beck-Fiala theorem assert that if every element in $A$ is included in at most $t$ sets in $H$ then $disc (H) \le 2t$. (Of course, the theorem applies also to the hereditary discrepancy.)
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### EDP: a possible revival

August 22, 2012

A few months ago, Gil Kalai got in touch to suggest that it was time to revisit Polymath5, the attempt to prove the Erdős discrepancy problem. I agreed, but we have both been rather busy in the interim, so it is only now that we are ready. One of the things that Gil did was write three posts, which I shall be putting up as guest posts, the first one today and the others over the next few days. I hope that people who contributed to the project the first time round will consider having another go, and that people who didn’t, but would be interested, will find that they can use the wiki that we created to record our progress to get up to speed with what is already known. Once Gil’s posts are up, I’ll probably write a post myself, and I hope that some serious work might start in early September. I always felt that when the original attempt petered out, it was just a kind of loss of energy that individual mathematicians often feel, rather than the result of hitting a brick wall. And just as having a break from a problem is often useful for individuals, I hope it will turn out to be for this polymath project. If it doesn’t, then maybe I’ll do something I meant to do much earlier, which is write up in paper form the progress that has already been made. (Of course, if I do that, then anybody is free to contribute to the writing.)

If you want to look at some of the earlier posts, they are collected together in the polymath5 category on this blog.