There are a lot of facts in mathematics. Knowing more facts in mathematics certainly improves the ability of a mathematician to a certain extent. As such it’s natural to show that you do know a lot of mathematical facts in your personal statement, which you can do by writing that you’ve joined activity X, read book Y or got award Z.

However, the personal statement needs to be more personal than the facts that you know. How can one make facts “personal”?

Order of learning

Picking up a textbook / set of lecture notes for linear algebra, you would realise there are a lot of ways of introducing linear algebra. You could start with systems of linear equations going to matrices, you could start with vectors in the coordinate plane or you could start with the abstract definition of a vector space.

Similarly, detailing how you have gone from learning X to Y to Z may be quite interesting and unique to you! Explain the impacts of learning something before others. Maybe the things you’ve learnt before inspired you or gave you more examples or (unfortunately) limited your understanding.

Alternate definitions/proofs

A lot of definitions and theorems in mathematics have alternate ways of formulating them. For example, a real sequence is convergent if and only if it is Cauchy. However, while both concepts are equivalent, their definitions—Cauchy sequences and convergent sequences—differ significantly in terms of their formulations.

This is where personal preference comes in. You may find one definition more intuitive than others. Explaining why this is the case to you allows room for personal opinion. You could write how you’ve encountered one definition of a certain object and found it confusing until you met another definition. Again the emphasis should be on explaining why you felt that way instead of stating everything factually.

Choice of examples

If someone asked you an example of a vector space, what would you give? Would it be \(\mathbb{R}^2\)? \(\mathbb{R}^3\)? Solutions to linear differential equations?

In other words, What’s the most natural example of a mathematical object to you and why? It would show what area of mathematics you are most familiar with and interested in.