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My question is far more specific than the one in the title, and so, I urge you to not reply to the question in the title alone, and to try and reply to the questions I shall pose in this note.

I am a Master's student in an above-average European university, interested in geometry and topology, and will be applying to graduate schools in the U.S. this December. While I have done the standard graduate-level introductions to algebraic topology, differential topology, representation theory and differential geometry, I have no background in commutative algebra and algebraic geometry. I would like to work in low-dimensional topology, and my thesis will be in this area.

Will the fact that I know little to no algebraic geometry hinder my application? Do graduate committees specifically look for background in certain related areas, when candidates apply for a particular area? I ask because while people often say it doesn't matter much, it (possibly concidentally?) is also quite clear that most entry-level graduate students in top-tier graduate schools know substantial knowledge (at least a first course of all, and a second course of maybe two or three) of algebraic topology, differential geometry, representation theory and algebraic geometry.

P.S. In your answers, please consider that the graduate schools I am applying to in the United States will be part of the top-ten lists by USNews for topology schools or geometry schools.

closed as off-topic by Jon Custer, Brian Borchers, Solar Mike, David Ketcheson, corey979 May 29 at 17:48

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    The committees look for what they want to look for, and they don't tell us. So, you have to apply... – Solar Mike May 29 at 17:06
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    For what it's worth, based on being a grad student and postdoc at two top-20 US schools, I would say that practically none of the entering PhD students had any knowledge of algebraic geometry or similarly advanced fields. If you get in, you'll certainly have the opportunity to take courses in any of those areas that you wish. – Nate Eldredge May 29 at 18:38
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    will be applying to graduate schools ... I have done the standard graduate-level introductions to --- I think your concerns are misplaced. Having already taken several graduate-level classes, you're already ahead of many beginning graduate students. Much more important than having taken the classes is how well you did in them and what your letters of recommendation say about your potential (and how well supported the evidence is for said potential). Especially in mathematics, it's more about what you CAN do (and the evidence for this) rather than WHAT you know. – Dave L Renfro May 30 at 0:09
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Mathematics as a whole is very broad. It has been about a hundred years since any individual could say s/he knew all of it. Mathematicians are specialists - often extremely so. So, if you are working in a different area, having little knowledge of an unrelated one is not considered much of a disadvantage.

On the other hand, it is possible in the US to gain admission to doctoral programs, even very good ones, with only an undergraduate education. In the US the undergraduate education is very broad with only a little specialization. A math major also studies history and philosophy for example. So, the undergraduate math education, while, itself broad, can't cover everything.

However, if you are interviewed for a doctoral position the interviewers will decide on the questions. You need to make sensible answers, of course, but most interviewers will be looking for general math ability, not expertise in everything. That would be different, of course, if you are interviewed by a potential advisor who will want to know about you background in a particular specialty.

But from your description, your lack of specific knowledge in a few areas isn't likely to be a handicap. However, the competition for any slot will be intense at the level for which you aim. Nothing is guaranteed.

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