I don’t have a mathematical background and am looking for an “intuitively satisfying” explanation (so, for example, the Wikipedia article is way too technical). Perhaps this is not possible in which case, fair enough.
I understand (I think) what a polynomial is and I believe algebraic geometry is about understanding the solutions to polynomial equations using abstract algebraic techniques and geometry. I rapidly get lost when the discussion shifts to rings, fields, schemes and so on. However, I’m not looking to understand all these different concepts but rather get a high level overview.
One day, I’d like to understand how Grothendieck revolutionized the discipline but that may be far too ambitious 🙂
In: Mathematics
I’ll try the eli5.
Geometry study the “shape” of things.
Algebra study the “structure” of mathematical notions.
So let’s take an exemple : a rotation is a geometric transformation that does not change the shape of an object. This is a geometrical notion.
Now we know that if we do two rotations from the same center , the outcome is the same as one rotation. (Like if you rotate 90° and then 30°, it is like rotating 120°. Or if you rotate 40° , then -40°, it is like rotating 0°).
So assuming we keep the center, a rotation combined with a rotation is a rotation. And you have a rotation that does nothing. And you can always cancel a rotation by rotating in the other direction.
This gives a lot of information about the *structure* of this set of rotation. This structure is a well known algebraic structure (in my example it is a group). But we also know A LOT about group and what’s going on in this kind of steucture, so we can use our knowledge about group structures to get knowledge about rotation. And by doing that, we did algebraic geometry.
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