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I want to learn more about classical varieties. How do I proceed?

I was looking for more information on Classical Varieties and what exactly it means. I worked through the wikipedia article - http://en.wikipedia.org/wiki/Algebraic_variety, but I do not understand what they are used for.

I have taken calculus, multivariable calculus, linear algebra, ODEs, PDEs, (engineering math background). This sort of math has always been a mystery to me. What are the financial implications of such math? I have little experience in geometry (Is this geometry?) and would like to unravel a bit of the mystery, mostly for curiosity's sake. My goal in this endeavor is to understand a bit more of how collaboration, problem solving, and "engineering" is done in mathematics, as well as learn some interesting things (which I am assuming are not practically useful??)

A little nudging of "start with this" or "This should be at your level" or "Whoah you totally have the wrong idea, look at this" would be great

Thanks

edit:

From the amazon review of Undergraduate Algebraic Geometry - "

The style is friendly, straightforward and unpretentious. Everything is well motivated, and one occasionally gets to hear the author's personal perspective or view about a certain topic. I will quote two examples. When discussing the Zariski topology, the author writes "The Zariski topology may cause trouble to some students; since it is only being used as a language, and has almost no content, the difficulty is likely to be psychological rather than technical". This was very calming for me to read, as I have been previously struggling with the "deep meaning" of the Zariski topology, and no book has had the honesty to tell me that I shouldn't worry that much about it. As a second example of the author's style, after a Q.E.D. in page 53 the author explains that "The proof of (b) is a typical algebraist's proof: it's logically very neat, but almost completely hides the content: the real point is that ..."

"

I think I'll get this book thanks

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4  
Well, you first need a background in abstract algebra. –  M Turgeon Jun 13 '12 at 20:18
    
Could you explain what you mean by financial? –  Dylan Moreland Jun 13 '12 at 20:41
    
Get a copy of Miles Reid's Undergraduate Algebraic Geometry from your library and read it through. That's as good a plan as I can think of! –  Mariano Suárez-Alvarez Jun 13 '12 at 20:56
    
By Financial, I mean something that the knowledge can be used to create a sellable product, other than educating others about the math. Ok I'll check that out and page through it. If you were to explain Algebraic Varieties and its implications to an executive manager in one or a couple paragraphs, what would that contain? Here is my version, but I made up the exec details as I don't know them -- Algebraic Varieties is the study of the emergent patterns from simple shapes and rules. If a faster converging solution can be found to the GTi projection, then this would mean tighter manufacturing –  SwimBikeRun Jun 13 '12 at 21:14
8  
You are more in the "totally wrong idea" part of the spectrum... –  Mariano Suárez-Alvarez Jun 13 '12 at 21:21
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