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if there is one thing that going to math competitions has taught me it is that I suck at high school olympiad level geometry. However I often find solace in the fact that not a lot of mathematicians are working on compass and ruler constructions anymore.

However: A lot of people seem to be working or at least use techinques from an area of mathematics known as algebraic geometry (in fact: today the abel prize was given to an algebraic geometer).

My question is the following:are the same kind of mental abilities required to understand and solve problems in algebraic geometry or are they more similar to other areas such algebra, topology and combinatorics?

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Link to the prize thing? Thanks. –  Git Gud Mar 20 '13 at 21:37
abelprize.no –  Lepidopterist Mar 20 '13 at 21:38
Definitely not. Algebraic geometry is a much odder beast. (FWIW, I was pretty good at math competitions, but I always ended up translating geometry problems into Cartesian coordinate problems and solving those "algebraic" problems. But that is only very loosely related to what mathematicians call "algebraic geometry.") –  Thomas Andrews Mar 20 '13 at 21:40
I have a tendency to think of olympiad-style geometry as being combinatorics, actually. But modern algebraic geometry doesn't really involve any of that. (Evidence: I understand some modern algebraic geometry, but combinatorics is still beyond me!) –  Zhen Lin Mar 20 '13 at 21:41
There really is no connection. @ZhenLin, I wouldn't say it involves combinatorics, but regardless it's something completely different. Just as olympiad algebra or olympiad number theory have little connection to modern algebra or algebraic number theory, respectively. –  Adeel Khan Mar 20 '13 at 22:29

2 Answers 2

up vote 3 down vote accepted

Your question on mental abilities is of course too vague to admit a definitive answer, but I'll try to give some reflections on the subject.

1) Essentially, I strongly believe that the differences in abilities necessary to tackle the different branches of mathematics are vastly exaggerated.
In my experience good mathematicians are good at any subject.
The difference between their choices results from mathematics having become so vast that it is very difficult or impossible to have expertise in several subjects, unless you are Serre or Tao.
But my conviction, formed by introspection and anecdotal evidence, is that the subject mathematicians end up with very much depends on chance: books found in a library when 16 years old, teachers had in high-school or university, admired friends,...
To be quite honest, some subjects like combinatorics seem to require special gifts and be a little isolated, but even that is changing: I'm thinking of combinatorists like Stanley who use quite sophisticated "mainstream" mathematics, commutative algebra for example.

2) As for algebraic geometry, it certainly requires no special gifts.
Its origin is Descartes's (and Fermat's) fantastic invention of coordinate geometry, which allows one to solve difficult geometric problems by algebra, in an essentially purely mechanical way (which by the way Jean-Jacques Rousseau didn't like: read the extract from his Confessions in the epigraph to Fulton's Algebraic Curves, page iii)
In the 19-th century and in the 20-th century up to about the 1960's hard algebraic geometry was taught (under the name "analytic geometry") in high schools and a Swiss friend of mine showed me problems he solved when 17 years old, which would baffle most Ph.D holders in algebraic geometry nowadays.
The problem is that, in order in particular to solve quite classical problems and also for arithmetic reasons, algebraic geometers like van der Waerden, Zariski, Weil, Serre, Grothendieck,... had to introduce quite sophisticated machinery, culminating in the notion of scheme.
The unfortunate consequence of those developments is that too many introductory courses spend a semester (say) setting up this wonderful modern machinery and have no time left for showing how to apply it to concrete problems.

The good news is that an antidote to this state of affairs exists: it is called math.stackexchange !
I am amazed at the quality, concreteness and pertinence of many questions and answers relating to algebraic geometry here, and I can only advise you to become a frequent user of our site: just use the tag [(algebraic-geometry)] and start reading, pen in hand!

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While the following book: Elementary Algebraic Geometry by Klaus Hulek (American Mathematical Society, Volume 20, Student Mathematical Library), 2003 is not aimed at high school students (rather I would say about the level of mathematics major juniors in college) looking at it might give you some idea of the kind of concerns and issues, connections if you will, that this subject tries to make between algebra and geometry.

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This doesn't answer the question. –  Adeel Khan Mar 21 '13 at 1:40

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