Whether you are applying for a Master's, PhD or postdoc, the personal statement is your chance to present a coherent picture of your traits and abilities, and convey a bit about your personality. Here is an example of what not to do:

“I am fantastically wonderful brilliant student. I got 110% in all my exams and I will surely win eleventeen Field's medals in the next five years. Your university would be insanely lucky to have someone as awesome as me...”

#### ⚠️ Disclaimer ⚠️

Even more so than elsewhere in this notes, what follows are mere guidelines and suggestions for best practices.

You should conduct thorough research of your own into the requirements of the universities you are applying to.

##### Not a mathematician?

These notes may still be helpful even if you are not a mathematician.

If you are applying to do a different science degree, simply substitute in your subject every time it reads “maths”.

The less your subject is related to pure mathematics, the less it is likely that my advice will be helpful.

Though some aspects of the advice remain universal: e.g. don’t lie and write good English!

### The Purpose of the Statement

For the remainder of today's post we say Statement, with a capital S, to mean some or all of the various personal prose communications that you may be asked to attach to an application. Depending on the country or university, this Statement comes under the heading of personal statement, cover letter, statement of purpose, statement of objectives, and so on.

Suppose for a moment that you were not required to submit a Statement, just your academic record and your references. In that case, a member of the application committee could still learn a lot about you:

• your grades reflect your ability to learn for exams and perform under that kind of pressure;
• your references describe the impression you give off as a student and comment on your talent.
1. What would be missing is your own voice, arguing your case.
##### The two arguments

The Statement is the only place in your application where you are allowed to make two essential arguments that are not explicit elsewhere:

1. Why you think you would make a good Master's or PhD student or postdoc.
2. Why you think you should be a taken on as a Master's or PhD student or postdoc at a particular institution or by a particular researcher.

The two arguments are not always disjoint, but everything else that is not serving these arguments directly—the formatting, the phrasing, the random additional information you add—is there to pad out the text and prove you can string together a few decent sentences.

### Relevant information

Many Statements include the wrong kind of information, such as:

• too much personal history, which is either outdated (stories from childhood) or irrelevant to the subject (hobbies);
• too few appropriate subject-specific detail: a lot is said about maths in general, and not much about what makes the applicant a worthy mathematician;
• too few precise references to the university of choice (e.g. it is in a good geographic location and it is prestigious), which is a bare, almost embarrassing minimum that distinguishes you in no way from dozens of other applicants.

Let’s see how to avoid these pitfalls.

##### Personal history in the Introduction

Especially for American Statements, it is often recommended that you try to engage the reader in the introduction by recounting a memorable personal anecdote.

1. You do not have to include an anecdote if there are none.

However, if you decide to include an anecdote, you do yourself no favour by starting with a trite memory with little to no substance.

1. BAD: I fell in love with maths in fifth grade because I noticed I was good at geometry.
2. BAD: I became passionate about pursuing a degree in engineering after I played with origami as a child.

Actually you can make such statements relevant.

To begin with, try to include a maths fact or some maths terminology.

1. OK: I fell in love with maths in fifth grade when the teacher stated you could not square a circle with compass and straightedge in a finite number of steps.

Secondly, try to avoid general words like love; instead, relate a concrete incident.

1. GOOD: When my fifth grade teacher stated you could not square a circle with compass and straightedge in a finite number of steps, I spent the whole evening trying to do just that, convinced I could disprove her.

This is already much better. Missing now is the relevance to your current aspirations (Galois Theory, for example).

1. BETTER: When my fifth grade teacher stated you could not square a circle with compass and straightedge in a finite number of steps, I spent the whole evening trying to do just that, convinced I could disprove her. When I failed, I wanted to know why. Eventually, I went on to learn about $\pi$, transcendental and algebraic numbers, and polynomials, and finally Galois Theory at university, but by then my interest in the subject was firmly established and had evolved far beyond the ancient geometry problem that started me on this path.

This would be a solid first draft of an introductory anecdote. Depending on the remainder of your Introduction and the rest of your Statement, you would remove some of the elements.

Regardless, you would need to edit these sentences to about half the current length, because your Statement should be chiefly about your most recent work, not your enthusiasm as a child.

Only ever include relevant incidents, and be specific about how they are relevant. Use maths terminology whenever possible; avoid opinion words.
##### Personal history elsewhere

Personal history details that occur in the Statement fall into two groups:

• facts mentioned in the records,
• facts not mentioned in the records.

The records encompass any other formal, factual documents whose contents you are familiar with; most commonly, these are your academic record and your CV (unless you have also read your recommendations).

##### Facts mentioned in the records

Your CV will contain dates, names, and GPAs. When drafting your Statement, you can assume the reader is familiar with all of that information or has it at hand. Therefore, you do not need to repeat yourself.

On the other hand, you can and should freely reference facts.

Here are some examples:

1. BAD: I finished my Bachelor's degree in the Summer of 2017.

The whole sentence is redundant; it should be obvious from the CV. It becomes relevant only if there is a some greater point you are about to make. For example, that you took a gap year to do charity work, and it was important that this gap year was 2017–2018 because that is when a particular disaster happened—a socio-historical fact not mentioned in the CV.

This is extreme, but you do need an extraordinary reason for explicit repetition.

1. BAD: During my internship (May–June 2016 at Zrootech), I learned how to…

The parenthetical information belongs in the CV.

1. GOOD: The conferences I attended last summer gave me an opportunity to meet various researchers in my field, and talk to them about…

The CV should include information about the conferences, locations, dates, topics, so this is an appropriate reference that can be followed up via the CV.

Whilst you should not repeat yourself, you should also not tend to the other extreme of avoiding all references to facts from your CV.

Specifically, any details that you mention in the Statement that are also mentioned elsewhere are viewed as one of the following:

• A “highlight”: if you have won an award, had a particularly successful thesis, or have achieved anything that is outstanding in recent years (not in kindergarten!), then you are entitled to talk about this event in the Statement and say how it has influenced you.

What is more, you are expected to do so, and it would be strange if you chose not to.
• An “explanation”: if there are any unexplained gaps in your CV, or you took a non-standard route towards accomplishing your degree, or you have a medical reason why you do badly in exams, then you are expected to help the reader understand the circumstances and how they have influenced you.
As a rule, do not complain or seek sympathy, instead always talk about the lessons you took away from difficult experiences.

Events that make you stand out will fit in either of those categories, highlight or explanation, or even might be considered both.

For example, a six-month internship with a railroad company looks like it needs explaining if you are applying to do a PhD in statistics, until you say that you were learning on the job about route and time-table optimisation, in which case it could become a highlight.

##### Facts not mentioned in the records.

Hard facts (dates, names, numbers) really should be mentioned in the records.
The rest: your anecdotes may or may not be verifiable, but be careful of exaggeration in general, and cultural sensibilities in particular.

Some assertions might come across as appropriately bold in an American application, but might seem like boasting on a British application.

1. I am especially keen on your program because it would allow me to interact with and learn from top specialists in my field, specifically Professor Juniper, the Fields medallist who has recently taken up a post at your university.

Forceful, forthright, confident. To British ears it might sound presumptuous that you think you would get to interact with such an esteemed member of staff.

1. I would especially appreciate the opportunity to attend the seminar series given by your Fields medallist, as I am interested in one the research areas she developed. Namely, I would…

Subdued, realistic, humble, and in line with British sensibilities. Might sound unambitious to American ears.

Do you notice the difference in tone? Neither is better or worse, and you should not try to affect a tone foreign to you, but be aware that small word choices, viewed cumulatively, will influence the way you come across.

### Subject-specific detail

The first argument (and more important one in a lot of ways) says why you think you would make a good Master's or PhD student. Traditionally this argument is made by directly answering the following questions in your Statement.

• General motivation: Why do you want to do an advanced degree?

This is a natural, open-ended question, but you are not expected to give a creative answer. It will suffice to say that you have long-term interest in a particular subject, or wish to gain insight into the subject before going into industry or continuing a PhD.

• Skill-specific question: What research skills or experience do you have that make you think you would be a good Master's or PhD student?

For this you mention your most relevant skills: specialised courses, reading courses, projects, seminars, programming experience from an internship, other practical experiences. This is where you are aiming to “highlight” facts from the CV.

• Personal motivation: Why have you chosen your subject area and not some other? What interests you within it?

This is your opportunity to deliver an insight into your mathematical thinking that appears nowhere else, and that shows your interest is not merely superficial.

1. EXCELLENT: I particularly enjoy investigating which structures an invariant can “see” and applying its properties to distinguish two otherwise similar spaces, ranging from elementary examples: $\mathbb{C}P^2$ and $S^2 \vee S^4$ are not homeomorphic as their cohomology ring structure is different, to more complex ones: there exist diffeomorphic Calabi-Yau 3-folds with different quantum cohomology rings.

You should plan to spend between one and three sentences answering these questions somewhere within the structure of your Statement.

### Application-specific detail

You have to convince whoever is reading your application that you are applying to them or their department, and not that you copy-pasted the text for some other university.

There is a good reason for this: no advanced program wants to accept people who do not actually want to attend.

In other words, as much as you are worried whether a position will suit you, the university offering that position is worried about the same thing. Because, once accepted, if you are unhappy, you are likely to be unsuccessful and that will also likely make the university look bad.

Thus, in your statement you are expected to answer these basic questions (when applicable):

• When applying to a person: What drew you to this particular academic? Which of their research do you find most interesting and why?

Here it is good to list a specific paper or book they have written and you have at least glanced at, or if you took a course or seminar with them, even better, comment on what you enjoyed and why. Obviously, if you have met the person, even briefly, then remind them of this if the encounter was at all positive.

• When applying to a department (but also to a person): What drew you to this particular department or university? What research conducted in the department do you find interesting?

If you can, avoid more than the briefest mention of general ranking or prestige, or of location. The second question suggests the kind of answer you are expected to give: mention research groups at the department, any facilities that they provide and that you need for your work, specific courses that you need for your degree that may not be offered elsewhere, etc.

• When applying to a taught Master's program: If the program envisions a particular curriculum, what makes you particularly suited to that study that curriculum? If there is flexibility, which areas do you intend to focus on?

This is where you can “highlight” facts from your CV and say how this background has prepared you well for the curriculum, or what areas you will build on.

• When applying for any research degree: How do you see yourself contributing to the research or to the research life at the department?

This is an open-ended question where you are not expected to give a particularly creative answer. Most well-meaning answers will do: contributing to a particular research group, hoping to develop a skill set that will complement those of others, working closely with other researchers in a similar area.

This is a bit of a chicken-and-egg cycle. to answer these questions you have to know where you are applying, but in order to know where you are applying, you need to have answered these questions at least partially (in your head).

Before writing each individual application, remind yourself of why you are applying to a particular program.

1. TERRIBLE: I am applying to the University of Oxbridge because it is almost as brilliant as I am.

Go and find out why that professor recommended this program, and write down those reasons. In particular (when applicable):

• Find out more about the person you are applying to. Look at their CV, at their official department page, at a list of their research, at the abstract of those research papers that seem most relevant to you.
• Find out more about the department you are applying to. If it is a taught Master's program, look at the details of the coursework.
• Find out which research groups the department has, what their specialities are, whether you would fit in and how. It is good to see some famous names, but it is more important to see the speciality areas that you might be looking to join.
• Sure, learn more about the prestige of the institution and its location. But even though these factors may play a huge role in your own decision process, do not emphasise them in the statement because the institution is well-aware of its own prestige and the attractiveness of its location (as are all the other candidates), so emphasising this will not gain you anything.

### Writing

Here are some general tips on the actual writing of your Statement.

Get the name right!

If you are applying to a professor in person, then make sure to get their name right as well as any titles.

If you are applying to a department and no name is given, or an impersonal online application form then it may acceptable to use the standard impersonal form of address: To Whom It May Concern. Though, whenever you can, try to find out whether there is someone you can address it to (perhaps the Head of Department).

Do not use bombastic language or cliches

Throughout this course we have urged you to remove redundant words, as well as, when appropriate, to remove opinion words. In your Statement some opinion words are welcome, though whenever you can back them up with concrete reasons.

1. GOOD: I find knot theory interesting, especially the idea of refining Floer-theoretic methods to investigate the properties of knots

The second part of the sentence does not actually say why you find knot theory interesting, but it does indicate that you have been thinking about the subject (itself an indication of interest).

You should scrutinise your use of adverbs: always, best, never, well, and all those ending with ly.

You should have no need for very, especially in phrases such as: Very interested, very excited, very enthusiastic, very good, very helpful.

The same is true of really.

In the similar vein, avoid creative words that may not use superlatives, but are meant to sound like superlatives. Also avoid cliches and colloquialisms. Here are some further samples of what not to write under any circumstances:

1. BAD: Mathematics is beautiful.

Yes, it is, but you can assume whoever is reading your statement is already aware of this fact (seeing as they have devoted their career to mathematics) and does not need reminding of it.

1. BAD: I am bursting with enthusiasm.

This metaphorical meaning of bursting does not belong in formal writing.

1. BAD: I am highly motivated.

The adverb should be removed, but then so should motivated unless you are following it up with a good reason.

1. BAD: I love my subject.

This is understood and expected.

1. BAD: The realisation that I wanted to do a PhD blew me away.

The phrase blew me away is informal and does not belong in your Statement.

1. BAD: Mathematics is not a spectator sport, and hence I always spend many hours solving difficult problems.

This is (a) cliched (b) patronising (since a professional educator is likely reading it) and (c) boasting.

1. BAD: My childhood dream is... My utmost desire...

You are too old to refer to dreams and desires in this way.

1. BAD: Since the age of five I have been fascinated with science.

Whilst this may be true, the statement is too general to be helpful, and must be either deleted or modified as illustrated above.

1. BAD: My greatest ambition... I am ambitious... My ambition has paid off so far...
2. AWFUL: I want to Make Mathematics Great Again.
##### Be truthful!

This may sound obvious, but it is surprising how easily one can get carried away with embellishments and small alterations that seemingly slant the Statement in your favour.

Typically this involves one of the following scenarios:

• You decide to inflate a (programming or language) skill. For example, you might write:
1. During my internship, I learned to use MATLAB.

But actually you opened MATLAB twice during that time. Commonly, this happens when you think no one can actually check whether your claim is true or not until after they have decided to hire you.

• You decide to unduly highlight a subject area. For example:
1. In my third year I attended an advanced seminar on Kirby Calculus.

This may sound impressive, until it becomes apparent that you attended it by sitting in the back, staring blankly at the board, or playing with your phone.

This kind of truth slippage occurs most commonly when you need a list of two items to look longer, so you add a few more things you attended, or a few more things that would look good if they were true (a.k.a. “alternative facts”).

• You decide to get help with writing advanced maths in your Statement. You have correctly identified that this would be a good idea, so you ask an older friend to supply you with a maths insight. Alternatively, you open a textbook or paper, select an important-looking result, and regurgitate it in your statement.

Depending on your sophistication and on how far away from the truth you have strayed, each of these scenarios will be more or less detectable. Your Ideal Reader has probably seen hundreds of such applications, and can detect a lie (let us call it what it is) without even thinking about it.

It is best if you self-police, and detect such scenarios in your own Statement before sending it off. The question you should always be asking yourself is as follows:

If I am asked to demonstrate a skill, talk about an experience, or discuss technical knowledge would a charitable interviewer buy my story?

##### Humour

Everyone likes a joke or two, but the Statement is hardly the place to try to be funny.

Avoid humour, puns, and anecdotes that are supposed to be amusing.

If you have met the person you are applying to or have seen them speak and are aware of their sense of humour, and you are confident you can say something that would fit said sense of humour, then you could maybe include an appropriate reference or comment.

Though, we cannot emphasise enough how cautious you should be when doing so.

##### Tone

Be courteous and professional.

Yes, the boring bit comes last, but it must be done.

• Ask a friend to take a look at the Statement and tell you what they think (does it leave a good impression, does that impression match who you are, are there any glaring errors, etc).
• Google phrases that you are unsure of.
• Check the spelling and grammar.