What is geometry, algebra, or topology? I have trouble grasping the notion of geometry, algebra, and topology. An example is when someone might say, "I study the geometry of jet spaces" or "I study the lie algebras". What does it mean for a problem to be geometric, or algebraic, or topological? What separates an algebraic problem from a geometric or topological one? I feel like if I better understood these words I would understand the big picture more. There are probably more to add besides geometry, algebra, and topology but I couldn't think of any more.
This is a bit simplified, but it's my view on these areas:


Geometric studies consider planes and curves, and how they bend and twist.
Algebraic studies consider structures of arithmetic and of symmetry.
Topologic studies consider continuity of functions.
Analysis studies differentiation and integration, series and limits.


These can also be mixed. For instance:


In algebraic topology you construct a structure of continious functions that you can combine in some specified way, and you study the arithmetical structure that arises (there are loads of ways to do it, and they all have different names like fundamental group, homology groups and cup product).
In differential geometry you use analysis (differentiation) to analyse how curves curve, or to study the possible vector fields in a space (an example of a vector field on a space is the globe of the Earth, with details on how the wind is blowing on each point).
Try these simpler words; Algebra is the use of simple equations and variables to solve a problem. Geometry uses these equations and variables and applies then with various theorems to shapes. Algebra II is more complex equations and can deal with several variables in one problem whilst general topology is based solely on set theory and concerns itself with structures of sets. Hope these basic mental sets help.