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So, the title speaks for itself, but essentially over the course of the last few weeks I've been doing a course in complex analysis, and some of the identities that pop up are immaculate, and I think including one of these will enable me to convey that not only am I passionate about my subject (mathematics), but I am also well informed. Is this an acceptable thing to do?

In my opinion, it feels "too unconventional to be acceptable," but I'd like to hear your input rather than my own.

Any responses are appreciated.

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    What are you applying for (PhD, postdoc, faculty)?
    – astronat
    Mar 5 '18 at 15:15
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As a general principle, I would say that there is rarely going to be a situation where mathematical statements are going to add value in a personal statement. I am in a mathematical field, and I have not seen an exampe that contradicts this. Some reasons why this is so:

  • You are either going to be talking about a math problem that is well known or one that is unknown to the reader. If it is well-known you should be able to describe it by name, without using any maths. If it is a problem that the reader has not encountered, with any real substantive depth, it is unlikely that the reader is going to be able to digest the mathematics in a short read.

  • For applicants for graduate programs, it is usually the case that their undergraduate training has given them the capacity to manipulate equations, but they are not yet adept at backing up calculations with intuitive explanation --- they can't yet "see the matrix". For this reason, it is much more impressive to see a good textual explanation of the intuition of a problem than to see mathematical equations describing or solving a problem.

  • If you are an applicant to graduate school, your mathematics is far below the standards of the professors who will be reading your statement. (Yes, I am even talking to you Mr summa cum laude.) What you think is a demonstration of your mathematical competence is probably just a demonstration of how clunky you are at setting out your work. I am not talking about the correctness of your equations (which should be taken for granted), but other aspects of the presentation that your professors will notice (e.g., notation choices, brevity, completeness, rigour, etc.). Becoming good at writing mathematical work in a clear and parsimonious way is a skill that takes longer than your undergraduate degree --- it might be best not to remind the professors of how far you have to go!

  • You are talking about presenting mathematical results that you only just learned in a complex analysis course you are doing. Okay, so you can write down the formulas. But have you had time to fully mull over their implications and intuition? Have you seen fifty other results in other mathematical fields that are connected to these results in a deep way? Do you really understand them? Are they in your bone marrow? If not, it might be best to avoid.

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    Yesterday I was thinking of giving an example of a good way and a bad way to do this, but decided not to take the time. However, many of the things you said are what I was thinking of (although not in as well an expressed way that you gave). For what it's worth, my example was going to be the identity theorem for analytic functions, with the bad way having symbols and jargon and quantifiers and other such clutter, and the good way being how this result tells us (for example) that all the trig identities one learns in high school continue to hold when complex numbers are used. Mar 8 '18 at 14:00
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Here's the thing about personal statements in math: a bad one can hurt you, but a good one doesn't help you all that much. Your grades and letters are what will get the committee's attention and convince them that you're smart. You don't have to use the personal statement to keep convincing them that you're smart. The goal is more to demonstrate that you know what grad school in math is generally all about. It's also good if your enthusiasm shines through, but don't try too hard. For example, talking about how much you loved math as a kid would just be trite.

To answer the question: yes, it's probably fine to include a formula that you really like, and talk a bit about how it captured your imagination. But it's better if you can relate this to your future plans, because that is what the statement should be focused on.

And again, don't formula-drop to look smart, especially not if the formula is something everyone learns in a first course in complex analysis (because knowing it doesn't set you apart from other applicants).

The above advice assumes you're applying for a graduate program. If you're applying for a summer REU or something similar, the same advice mostly applies, but there is less of an expectation that you know what your future plans are. But you can still tie your enthusiasm for Cool Identity X to your desire to learn more advanced math and get a taste of research through whatever program you're applying to.

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Disclaimer: biomedical field here so not totally in tune with the math department culture.

Based on the very limited input, I'd advise against it, for the following reasons:

  1. In some more diverse department, the hiring/admission committee may contain members from different fields, using mathematical expression may risk losing their understanding.

  2. For some more serious people, this may appear to be too informal. I think it'd be cool to do that in the job talk, but not so in a more one-way communication like a personal statement.

  3. The biggest risk is that this may switch on some unintended areas in the committee member's brain, which can be high risk high return, or a total flop. If you decided to go that route, you really need to make sure the math is right so that it wouldn't divert their attention and start grading it.

Having said that, it's all about being tastefully done. To tell the truth I too would be attracted by an application that looks like a patient's chart, a data analysis plan, a graphic novel, or even a statistical software output. The competition is indeed intense and I appreciate your innovation to stand out. If you're so inclined to try it, I'd suggest showing it to some professors (or people resembling whoever will judge your application) for some honest input.

Also, consider attaching the work as a writing sample if the submission system allows it. Mention briefly about this attachment in the personal statement so reviewers who are more inclined can decide to see it.

Best of luck to your endeavor!

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My opinion.

In a personal statement you're trying to attract their attention to you and is a way to increase your chances for meeting them which will hopefully lead to a job or other position. It's also a way to link your CV to a specific role too.

So if you think an actual equation will increase your chances or make you seem better for the job, go for it.

In reality though, it might be better to mention these equation rather than stating them. This might also benefit you as you can talk about the equations once you get to meet them, you might be able to write them down and explain their appropriateness to them in person, rather than making them try and understand them on their own.

Good luck.

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