Re: Motivation to do Algebraic Geometry

On 3 Feb., 15:28, Timothy Murphy <gayle...@xxxxxxxxxx> wrote:
mynameisrab...@xxxxxxxxxxx wrote:
Since a few month I try to understand things in algebraic geometry and
I'm searching for a similar motivation to do algebraic topology as the
above ones for algebraic topology. All the things I know so far which
demonstrate the use of algebraic geometry are, well, a little
complicated: I.e. the theorem of Fermat which has been proved by Weil
using algebraic geometry will be out of reach for me for many many
years (I hope one day I will understand something like that). So I am
searching for an illuminating easy example to demonstrate the use of
algebraic geometry methods, which can be a motivation for me to study
such things and which I will be able to understand in, say, one year
or so. Perhaps there are some examples in number theory?

Does anybody know examples like that?

The Congruent Number Problem -
can you find a right-angle triangle with integer sides and integer area?

Is this really the problem you mean?
Pythagorean triangles always have integer area...

(Or rational sides and rational area.)

This reduces to a problem on elliptic curves.

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland

.

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