Algebraic geometry
Algebraic geometry takes a central place in modern mathematics.  The concepts it uses connects it to diverse fields such as complex analysis, topology and number theory.  At the start, algebraic geometry was about studying systems of polynomial equations in several variables. Algebraic geometry starts at the point where equation solving leaves off: In many cases, finding the properties of all the solutions in a given set of equations have, are more important than finding a particular solution: this leads into some of the deepest place in all of mathematics, both conceptually and in terms of technique.