In this post we will look at random partitions. A partition $\pi$ of $\mathbb{N}$ is a set of disjoint subsets $\{\pi_i\}_{i=1}^\infty$ of $\mathbb{N}$ such that $\bigcup_i \pi_i=\mathbb{N}$. We arrange these sets $\pi_i$ in the order of their least element, so that $\inf \pi_1 < \inf \pi_2< \dots$, and if there are only finitely many subsets, we trail the sequence with emptysets, e.g. $(\{1\}, \mathbb{N}\backslash \{1\},\emptyset,\dots)$. Denote the set of partitions of $\mathbb{N}$ by $\mathcal{P}_\infty$.

Notice that each $\pi \in \mathcal{P}_\infty$ induces an equivalence relation $\buildrel \pi \over \sim$ on $\mathbb{N}$ by $i \buildrel \pi \over \sim j$ if and only if $i$ and $j$ belong to the same block of $\pi$. Now let $\sigma$ be a permutation of $\mathbb{N}$, and we define a partition $\sigma \pi$ by saying that $i$ and $j$ are in the same block of $\sigma \pi$ if and only if $\sigma(i) \buildrel \pi \over \sim \sigma(j)$.

Suppose now that $\pi$ is a random partition. We say that $\pi$ is exchangeable if for each permutation $\sigma$ which changes finitely many elements, we have that $\sigma\pi$ has the same distribution as $\pi$.

There is a wonderful theorem by Kingman which will follow shortly but for now let us look at some basic properties of random exchangeable partitions.

Now that you’ve decided on your outfit, you take a trip to the cinema with a friend. When you get there, you see that there are only two seats left and they are next to each other. In how many different ways can you and your friend occupy these two seats? Well, once seated, either you will be sitting on the left and your friend will be to the right of you or you are sitting on the right with your friend on your left. So there are two ways in total. This is very easy to see, but what if there were three of you and three seats left? In how many ways can you and your two friends occupy the three seats?

This post is in the ‘Mathematics for Everyone’ category. These posts will be short snippets and will, I hope, increase slowly in complexity.

Suppose that you are going out to the cinema tonight and that you do not know what to wear. You have narrowed down your choice considerably but you cannot decide which of three different tops and which pair of two different pairs of jeans you would like to wear. You would wear any top with any pair of jeans, but you need to choose which top and which pair of jeans. How many different complete outfits are you choosing from?

This post is just to link Tim Gowers’s recent blog post on Plünnecke-type sumset estimates in groups because he discusses the recent work of his soon-to-be-former student Giorgis Petridis.

I had the pleasure of being taught (outstandingly) by Giorgis on more than one occasion during my undergraduate years and it was great to be in the audience for his recent talk on his work at the Isaac Newton Institute. His new papers are on the ArXiv here. Congratulations!

### Affinity in Generality

Let V be a vector space. Consider the set of all affine transformations of V: An affine transformation of V is a map from V to itself which can be expressed as

$x \mapsto L(x) + b$,

for some invertible linear map (automorphism) L and some vector b in V. The set of all affine transformations of V forms a group under composition and is called the affine group of V. Note that the set of automorphisms of V is a subgroup of the affine group and also that the set of translations of V is also  subgroup of the affine group. Note that every element of the affine group can be expressed as a composition of an invertible linear map followed by a translation. Note also that the only affine transformation which is both an automorphism and a translation is the identity map. It is also quite easy to see that the translation group is a normal subgroup of the affine group. However, the automorphism group of V, which, since V is a vector space is known as the general linear group of V, is not.

Is the affine group the internal direct product of the translation group and the automorphism group? No, but it comes close. In the previous post, we saw that if G is the internal direct product of H and K, then not only can every element of G can be expressed (uniquely) in the form g = hk, but also the elements of H commute with the elements of K. This is the sense in which the two groups H and K do not interact with each other inside G. In the case of the affine group, there is interaction: The translation group of a vector space does not commute with its general linear group. This breakdown is evidenced by the fact that one of the two groups fails to be normal.

Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how much I used to like algebra. The material here is by no means advanced, but relies on some basic definitions from group theory.

Suppose I have two groups H and K. Can we combine them together? I learnt some years ago now that there is a straightforward way of forming a ‘sum’ of these two objects: One takes the cartesian product H$\times$K, and bestows it with a group operation in the most basic and obvious way possible: (h,k)$\ast$(h’,k’) = (hh’,kk’). The group that results is known as the direct product of H and K.

The same can be done with vector spaces: Given two vector spaces V and W, I can form the direct sum of them, which is thought of as the vector space consisting of ordered pairs of the form (v,w) with v in V and w in W. It is easy to guess what the rules for addition and scalar multiplication must be. However, soon after learning these two definitions, I began to realise that at least a small amount of wool was obscuring my eyes: Suppose I have a vector space X and two subspaces Y and Z such that every vector x in X can be written uniquely in the form x = y + z, for y in Y and z in Z. I would then be invited to observe how this means that X was the direct sum of Y and Z. This didn’t sit well with me. It clearly wasn’t quite the same thing: There was no cartesian product; there was no way that x = y + z was the same as x = (y,z).

There are plans to build a Museum of Mathematics in Manhattan and over \$20 million has already been raised for the ’cause’. The Math Midway, which already exists as a traveling exhibition, has something of a preview of coming attractions. There is also a discussion about what mathematicians would like to see at the museum taking place on MathOverflow here.

What do you think of this idea? Personally, I am skeptical, though I may just be being snobbish – it is likely that my academic leaning precludes a fair judgement of such a thing as a museum of mathematics because I will always be tempted to say “If you want to learn about maths, then maybe you should study some maths: Read some books, pay attention at school, discuss maths with teachers, discuss it online, join a math club/circle etc.” I’d be worried about the negative effect of people going in and saying “Oh so this is maths? It’s boring”. Which seems to me even less desirable than “I know nothing about maths”, because my response to the latter can be “If you studied it like I do, then you too would love it!”

So, if it can be accepted that not everything is necessarily amenable to ‘museumization’, then I would definitely argue that mathematics falls into this category and would make a poor subject for a traditional-style museum. (I’ve never been a huge fan of Science Museums either, so perhaps I’m just the wrong person to ask – what do ‘proper’ academic scientists think of the Science Museum in London, for example?)

It sounds a bit romanticised by it’s also arguable that one really does need to invest some effort into appreciating mathematics. Every mathematician knows the feeling when, while at the pub, having just defended the beauty and awesomeness of mathematics, a friend says defiantly “Go on, tell me some maths”…  Show me the beauty and awesomeness. But this is impossible. The passivity of this stance immediately places the friend outside of those who are capable of appreciating mathematics on the scale that the mathematician does. You can’t just sit there and be shown. You have to do. You have to show yourself! The mathematician knows that her friend isn’t actually prepared to spend the next hour struggling to appreciate some minute idea which the mathematician seems to assert is worth understanding.

This comment was sort-of inspired by seeing the MathOverflow discussion, but there is a character limit on comments in the forum!

### Higher Dimensions

I want to extend the treatment to we just went through to higher dimensions. In my view, this provides a very nice way of thinking about why the generalisation of the derivative to higher dimensions is what it is.  It has certainly taken me a while to appreciate the definition of the derivative of a a function of many variables and I am attempting to share some of my thoughts.

In the last post we thought about graphs of functions and tangent-lines. Now, the graph of a function is intially something 0ne thinks of a picture of that function. In my opinion, there is a sense in which one ought to continue to do so. It’s just that now, we’ll be thinking about $\mathbb{R}^m$-valued functions f defined on  an open subset U of $\mathbb{R}^n$. Recall that the graph of f is the set of points $(x, y)$ in $U \times \mathbb{R}^m$ such that f(x) = y.

What might we mean by the derivative of f ?

In these posts I will not be explaining all of the basic mathematical concepts which are required to appreciate the main discussion, so they are not aimed at the novice, as it were. I would guess maybe that a second-year university student in mathematics will be able to fully appreciate these posts.

Recently, I have been thinking about differentiation and in this post I would like to discuss some ways of approaching the concept, starting right from the basics. In this first post, we’ll discuss some approaches to the idea of differentiation. I want to settle on a certain geometric perspective and then generalise it to higher dimensions. The aim is to shed light on why the definition of the derivative of a function generalises as it does.

### But Did Homer Catch the Bus?

Before we get on to the continuum hypothesis, let’s talk briefly about what we did in the last post because it’s a nice introduction to an important idea. We saw that it was possible to add up infintely many things. Let’s discuss this in the context of one of the paradoxes we introduced last time.

Let’s think back to Homer running for the bus (introduced at the start of the previous post). Suppose the distance to the bus is 100 metres. The way the paradox is phrased makes it seem as though, by splitting up 100 into infinitely many bits, it is impossible for Homer to traverse the entire distance because he cannot do infinitely many journeys in a finite amount of time. The mathematical definition we discussed last time (that of a series summing up to a finite quantity – one of the basic ideas of mathematical analysis) ignores this potential problem.