What's algebraic geometry good for?

[AdrianZ]

I'm currently studying abstract algebra from Herstein's interesting book "Topics in algebra", I've learned different definitions so far and I've solved most of the problems covered in the book. I've so far studied groups, subgroups of them, normal subgroups, quotient groups, isomorphism theorems, products of groups, conjugacy classes and the conjugacy class equation and I've understood theorems like Cauchy theorem, but so far I've excluded Sylow theorem from my group theory knowledge because I find a bit hard to for self studying. In rings, I've got acquainted with basic definitions and I've gone further and realized that every domain can be extended to a field and I've understood results like R/M is a field if and only if M is a maximal ideal provided that R is a commutative ring with a multiplication identity element. I've also studied the ring of polynomials with coefficients in F from Herstein's "topics in algebra" and Hoffman-Kunze linear algebra book.
Today I was studying Euclidean rings and I found the subject very beautiful and subtle and I guess by the end of today I'll try to solve some problems on Euclidean rings.

Having said all of these things, my favorite area of mathematics is geometry, but I also love abstract algebra, analysis and topology. The name algebraic geometry suggests that it must be an interesting field that links geometry to algebra. Is that true?

What is algebraic geometry about? What are the main theorems in algebraic geometry? What are the applications of algebraic geometry in pure mathematics and applied mathematics or in physics? Is there any book that explains algebraic geometry for an undergraduate student? What are the prerequisites to study algebraic geometry? Is it a good idea that I study algebraic geometry now?

Thanks in advance

 

[morphism]

At the core of it, algebraic geometry is about studying polynomial equations f(x_1,...,x_n)=0 (or systems of polynomial equations) from a geometric viewpoint. Geometry enters the picture through the zero locus Z={(x_1,...,x_n) | f(x_1,..,x_n)=0} of the polynomial. E.g. the zero locus of [itex]f(x,y)=x^2+y^2-1 \in R[x,y][/itex] is a "circle" in R^2 (here R is a ring). The hope is that you can use intuition gained from studying the geometry of Z to help you with your algebraic problem of studying the equation f(x_1,...,x_n)=0.

Conversely, algebraic geometry allows you to transform certain problems in geometry into problems in algebra. For example, certain geometric phenomena that you see on curves, like nodes and cusps, etc., correspond to simple facts about the polynomial defining the curve. Here the hope is that the wealth of available algebraic techniques can guide your geometric intuition. Implicit here is a remarkable idea, namely that to every ring there should correspond some "geometric" object. For certain rings, like R[x,y]/<x^2+y^2-1>, the corresponding geometric object is obvious. But for others, like [itex]\mathbb Z[/itex], not so much.

That's the simple-minded explanation of algebraic geometry: it's a two-way dictionary (in some precise sense) between algebra and geometry.

I have to run now - I'll say a bit more about applications later.

 

[mathwonk]

it makes old men young, ugly guys handsome, faint women courageous. In short it heals all ills, salves all wounds, pacifies all strife. Get yours today.

 

[chiro]

it makes old men young, ugly guys handsome, faint women courageous. In short it heals all ills, salves all wounds, pacifies all strife. Get yours today.

I don't even think Jesus Christ could compete with that!

 

[Alesak]

it makes old men young, ugly guys handsome, faint women courageous. In short it heals all ills, salves all wounds, pacifies all strife. Get yours today.

:D

I guess holy-grail is inscripted with polynomials, then..