Traditional areas of maths like algebra, calculus or trig don't seem a good way to think about subdividing the subject in the modern world.
You might ask, why subdivide at all?
In a sense, you shouldn't. The expert mathematician utilises whichever maths areas helps them solve the problem at hand. Breadth and ingenuity of application is often key.
But maths represents a massive body of knowledge and expertise, subdividing helps us to think about different areas, for curricula to focus their energies enough that there's sufficient depth of experience gained by students at a given time to get a foothold.
However I believe the subdivisions should be grouped by modern uses of maths, not ancient divisions of tools.
So here goes with our 5 major areas:
- Data Science (everything data, incorporating but expanding statistics and probability).
- Geometry (an ancient subject, but highly relevant today)
- Information Theory (everything signals--whether images or sound. Right name for area?).
- Modelling (techniques for good application of maths for real world problems)
- Architecture of Maths (understanding the coherence of maths that builds its power, closely related to coding).