# Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ?

• My understanding (probably naïve): Algebra deals with structures and equations, geometry deals with spaces and figures and algebraic geometry combines structures with spaces and equations with figures. – k.stm Apr 19 '14 at 22:12
• What does abstract algebra got to do with this? – usukidoll Apr 19 '14 at 22:20
• Algebra and Geometry and simple, Algebraic Geometry is not. – user142299 Apr 19 '14 at 22:20
• @NotNotLogical what a strange comment :s – rschwieb Apr 19 '14 at 22:22

Here are some oversimplified blurbs about what each one does.

(Abstract) algebra deals with operations on sets, especially binary operations.

Geometry deals with sets which have groups acting on them. Part of this involves shape.

Algebraic geometry applies commutative algebra to sets described by algebraic equations. It gives information about the shape of such sets. As its name implies, it uses both algebra and geometry. It might be better to say it uses algebraic techniques to answer geometric questions.

• Then what are the groups that acts on the objects in algebraic geometry? – Fredrik Meyer Apr 20 '14 at 17:53
• Dear @FredrikMeyer : There isn't really any way I could offer a response to this. The main problem is that it seems to challenge a claim I didn't make. Another thing is that I don't know much about algebraic geometry, but I understand it's evolved and abstracted to great heights of its own, so that the connections with geometry may not be as obvious as they should be. Anyhow, if you'd like to pull apart a specific statement above that I made to help me improve it, that would be great. Thanks! – rschwieb Apr 20 '14 at 20:46
• Dear @rschwieb: I'm sorry if I sounded unclear or picky. What I ment to adress was that it isn't clear what you ment by the statement in the third line. Of course you're reffering to Felix Klein's Erlangen program, in which he proposes to base geometry upon group theory (sort of). But this only applies to certain spaces, as far I can see, namely the homogeneous ones. – Fredrik Meyer Apr 20 '14 at 23:51
• @FredrikMeyer thanks, that's much clearer. I'm not familiar with geometries that don't specify transformations. Perhaps this is a point of departure from the main body of classical geometry that newer branches of geometry take. Do you think this is the case? – rschwieb Apr 21 '14 at 1:11
• I don't know much about anything other than algebraic geometry. However, there's still groups hanging around. Two affine varieties are isomorphic if and only if there's a ring homomorphism between their corresponding rings. So isomorphic varieties are classified by $\mathrm{Aut}(R)$, just as, say, isomorphic circles in the Euclidean geometry is classified by $\mathrm{Aff}(\mathbb R^2)$. – Fredrik Meyer Apr 21 '14 at 2:01