The foundation of math. Math is elegant because everything is on a solid foundation. Set theory explores this.

Ah! One of my favorites. Also known as Abstract Algebra. It is a math that studies algebraic structures. Those are numbers on which operations are defined, like addition and multiplication.(2) Modern Algebra

One can say a specific case of Abstract Algebra. But it studies Vector Spaces.(3) Linear Algebra

No idea. Perhaps you mean the standard basic High School Algebra?(4) Classical Algebra

Already asked me this.(5) Partial Differential Equations