Pure Mathematics Statement of Purpose: How much detail should be included?

[Löb'sTheorem]

I'll be applying to graduate schools this fall, and I've been doing quite a bit of research into my various options (of course I've been looking into grad schools for the duration of my undergraduate degree, but now I need to actually fill in the minutia and avoid making any gaffs that hurt my chances, if possible); in any event, I'm a bit unsure of

(a) How much weight (if much at all) is given to the statement of purpose

(b How much detail should be included in the way of interests, interest in faculty research specialties etc.

My primary concern is with length, and due to the breadth of my interests I can only include so much. To give you an idea, my primary areas of interests are

Algebraic Areas: algebraic geometry (I'm presently part way through Eisenbud, and Atiyah McDonald, and I've covered a significant amount of material in the book Ideals Varieties and Algorithms and have had an independent study in the basics of the area), algebraic cryptanalysis (I've done some research in this), arithmetic geometry, category theory (specifically categorical logic and the application of category theory to cognitive science, though there are only a few researchers looking into the latter), Geometric complexity theory is super fascinating to me

Logic Areas: Game semantics in proof theory, Girard's linear logic and "Ludics" programme and Japaridze's work in computability logic, constructive mathematics and Martin-Löf type theory (a large part of this is due to the work by Voevodsky and Awodey in homotopy type theory, which seems very interesting to me), applications of model theory to algebraic geometry and tropical geometry, reverse mathematics (I've been reading Simpson's book on subsystems of second order arithmetic and it is fascinating to me)

Applied Areas: Machine learning, specifically reinforcement learning and Bayesian methodologies in pattern recognition (I've been reading Barto and Sutton as well as Bishop), Computational cognitive science and computational neuroscience, specifically representational learning

Some of these areas are more recent interests and some are surely irrelevant to certain departments, so I would probably not include my interest in linear logic or machine learning in the case of a school that has no researchers doing anything related to either area.

So now that I've blasted you with a wall of boring self indulgence, can anyone tell me whether I should include as many interersts as possible, if the school has people working in a lot of them? Or should I totally avoid doing that? Basically, I need to know how to write a statement of purpose.

Edited August 20, 2011 by Löb'sTheorem

[OH YEAH]

I'm a theoretical computer scientist as opposed to a pure mathematician, but since some of your interests overlap, I'll go ahead and give my two cents anyway.

1) I'd say your SoP is the second most important part of your package, well below your letters of recommendation and research experience though (these are more or less the same thing, which is why I say "second most").

2) Don't give a laundry list of your interests. Instead, spend 90% of your SoP talking about your research experience. Your research experience should be (hopefully) related to your interests; show your passion by highlighting past accomplishments. You can use the (smaller) part of your SoP to talk about your future research interests, *firmly grounded in what the dept does*. Say what you like, and then relate it to a specific faculty member or research group.

can anyone tell me whether I should include as many interersts as possible, if the school has people working in a lot of them?

You shouldn't have space for this... I'd say pick 3 or 4 people you want to work with and talk about them.and their interests.

If you want me to read your SoP, feel free to send me a PM! Good luck.

Edited August 20, 2011 by OH YEAH

[Löb'sTheorem]

Ok, that sounds like solid advice; I could definitely write quite a bit on my research, and it is fairly closely related to my interests.

So 90% on ideas from that research (which will be invariant program to program) and then talk about other interests and faculty that interest me, to be tailored for each program.

I'm glad to hear that a good SoP can make a difference, my GPA (~3.5) is fairly solid but not quite up to snuff, and my institution is very small (though I've taken graduate courses at a neighboring institution starting my third year; however, they unfortunately limit the number of courses I can take as if they were from my home institution, hence if I wanted to take as many as I could have handled it would have been quite costly).

Unfortunately, I won't have any publications as of the time I apply (though I'm continuing my research this year, so I may acquire one on the way out, but that will be no help on my applications).

Edited August 21, 2011 by Löb'sTheorem