The Cutting Edge of Mathematics

What would you say are the areas of mathematics that are currently on the cutting edge of research and are still being developed. What new fields of math have emerged in the last few decades?

I find it's interesting that most of math education progresses historically; first we learned the algebra and geometry of the Greeks in elementary and middle school, then the graphs and functions of Descartes time in high school, the Calculus of Newton's time in college, and so on. After Calculus was made math really branched out and that shows with the wide variety of undergraduate and graduate courses.

I wonder what math education will be like a hundred years from now!

Re: The Cutting Edge of Mathematics

Hey MathKnot.

A lot of new pure mathematics has been motivated by the study of theoretical physics. Stuff that ranges from operator algebras and functional analysis from Quantum Mechanics all the way to the stuff inspired from String Theory, Quantum Gravity theories and other kinds of theoretical physics research.

Looking at the directional of theoretical physics is one way to gauge this because it does create a lot of direction for research.

Re: The Cutting Edge of Mathematics

New Mathematics that is NOT motivated by physics are things like qualitative properties of non-linear differential equations- stability, absorbing states, etc.