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I am writing my PhD application statements now. I have sentences such as

The way in which compactness is employed to reason about ℂn using fields of finite characteristic exemplifies much of what has captivated my interest in this area.

and

I am beginning a senior thesis this spring with Dr. Weissman on using the asymptotic reals to better understand representations of SL(n, ℝ) algebraically.

A grad student friend of mine suggested I replace the mathematical notation here with "complex affine space" and "the special linear group over the reals" respectively. Is this a standard convention or substantially preferred? I like the way the math looks more, and its very minimal.

I am not too worried about clarity (in this case), the three instances of formulas appearing in my statement should be readily understood by anyone on the admissions committee. I was told that it is bad form to include the symbols, and I wanted to know if this is true as it seems unbelievable to me. Thanks!

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Using mathematical notation is appropriate. This is certainly good advice for common terms such as those employed in the two sentences, which every mathematician will understand. In general, however, these letters are addressed to a committee of people from across mathematical disciplines. As a consequence, you should not expect them to understand technical terms or specific symbols of your subdiscipline. So no sheaves, cohomologies, pseudo-differential operators, or polynomial approximation estimates please without further elaboration at a level understandable to every professional mathematician.

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Use both.

Looking at the specific examples given:

  1. The notation $\mathbb{C}^n$ does not necessarily mean a "complex affine space". Fundamentally, it is just the Cartesian product of n copies of the complex numbers—there are lots of structures which might be placed on top of that set. My recommendation would be something like

    The way in which compactness is employed to reason about the complex affine space $\mathbb{C}^n$ using fields of finite characteristic exemplifies much of what has captivated my interest in this area.

    (NB: I find this sentence somewhat hard to parse—I would recommend rewriting it.)

  2. So far as I know, the notation $\mathrm{SL}(n,\mathbb{R})$ is only used to denote the special linear group. This notation is, to my knowledge, completely unambiguous, and I wouldn't be worried about using it without further comment. However, it doesn't hurt to write something like

    I am beginning a senior thesis this spring with Dr. Weissman on using the asymptotic reals to better understand representations of the special linear group $\mathrm{SL}(n,\mathbb{R})$ algebraically.

More generally, in an application to a PhD program, you can assume that anyone reading your personal statement is going to have enough background to get the general gist of what you are talking about (pretty much anyone with a PhD in mathematics is going to have some familiarity with pretty much anything that an undergraduate can be expected to produce), but you cannot assume expertise. Hence I would recommend using both words and notation. This helps orient the reader your field, and demonstrates that you know standard terms and notation.

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    I think using both as you've described is good, and for what it's worth, doing this is analogous to how acronyms are often dealt with when writing for a "general audience" -- for its first appearance, follow the acronym with the full name in parentheses. Here, however, parentheses should be omitted. Incidentally, for the first example, the OP might consider beginning with "A special interest of mine is how compactness ..." or "One of my interests is how compactness ...", and maybe replace "employed to reason about" with something less wordy (or something slightly more explicit).
    – Dave L Renfro
    Commented Nov 19 at 18:55
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    +1. This approach also implicitly shows that you have an important technical writing skill -- defining your notation! (even if it is standard)
    – Andrew
    Commented Nov 21 at 15:39
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Definitely use (standard) mathematical notation: it's easier (for actual mathematicians) to understand your point! :)

Less ambiguous, too.

... and shows that you know about standard notation...

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In the few cases you describe, you can use both, write "complex affine space" and "the special linear group over the reals" followed by the mathematical symbol. This combines the advantages of words and symbols.

In longer mathematical texts with many symbols, it would inflate the text. This is no problem in this case.

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    I don't see the advantage of doing this -- in fact also writing the words "the special linear group over the reals" would make it look more weird and awkward to me. (Not as much with complex affine space.) Can you elaborate?
    – Kimball
    Commented Nov 19 at 14:06
  • I don't mind the exact words. In my experience, it helps to give some name to help those not familiar with the symbols and it prevents from misinterpreting symbols as different branches use same symbols for other things.
    – usr1234567
    Commented Nov 20 at 7:13
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I recommend using English. Using notation may create an impression that one doesn't know how the things are called... or worse: that the one knows to write formulas but do not really understands them. This might be not a big handicap in math, but in fields like physics it would be seen as a disqualifying factor.

If you feel that the existing terminology is not sufficient or ambiguous, a possible option is to use words and write mathematical notation in parentheses or the other way around.

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    By this logic, just using words may create the impression that one knows what things are called but doesn't know how to write formulas about them!
    – ajd
    Commented Nov 19 at 23:20
  • @ajd writing a fancy mathematical symbol doesn't prove that one can do math any more than writing words. The problem is that science students spend most of their studies learning advanced math, and often get an impression that math is an objective in itself. In reality, math is basic literacy for a scientist. Knowing to read and write is not a big deal, using words to communicate ideas is another matter.
    – Roger V.
    Commented Nov 20 at 6:08
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    @ajd I would argue math is more than basic literacy to an aspiring mathematician. it is the ends and the means. thank you
    – 4u9ust
    Commented Nov 20 at 21:31

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