My area of research, if you can say that I still have an area of research, is geometric topology. Yes, despite everything, I’ve managed to stay moderately active.
One big development in the past decade and a half is the solution to the Poincare Conjecture and the use of Ricci Flow to solve it (Perelman did the proof).
As far as what the Poincare Conjecture is about:
(If you’ve had some algebraic topology: the Poincare Conjecture says that an object that has the same algebraic information as the 3 dimensional sphere IS the three dimensional sphere, topologically speaking).
Now the proof uses Ricci Flow. Yes, to understand what Ricci flow is about, one has to understand differential geometry. BUT it you’ve had some brush with vector calculus (say, the amount that one teaches in a typical “Calculus III” course), one can get some intuition for this concept here.
Watch the video: it is fun. 🙂
Now when you get to the end, here is what is going on: instead of viewing a space (such as, say, the 2-d sphere) as being embedded in a larger space, one can talk about the space as being intrinsic; that is, not “sitting in” some ambient space. Then every point can be assigned some intrinsic curvature, and Ricci flow works in that setting.
Of course, one CAN always find a space to isometrically embed your space in (Nash embedding theorem) and still pretend that the space is embedded somewhere else; some “first course in differential topology” texts do exactly that.