Talk:Mathematics/@comment-82.28.228.24-20200830040554

Replying to the previous comment's reply, seemingly a different person this time:

I'd stick with the usual program material, an undegraduate section that would involve abstract algebra, some limited number theory, linear algebra and matrix topics, combinatorics and graph theory, a calculus sequence along with an advanced calculus sequence which is also real analysis by some people, basic complex analysis and fourier topics, some basic topology, curves and surfaces as differential geometry and perhaps some basic manifold theory. The "applied" aspect of the a mathematics program involves probability, differential equations, numerical analysis, basic statistics and some applied mathematics course involving modelling with fields, fluids etc from physics. Further than roughly these and you're beyond the core of mathematics which all mathematicians ought to know(hopefully im not forgetting any major part). Optional topics such as nondifferential geometry, more number theory, calculus of variations, are extras one can learn depending on what they want out of their mathematics education.

For the graduate part one can go many ways depending on the degree, but once the above material is mastered one can use this exact page to go any direction he wants. Usually a pure mathematics msc involves measure and integration theory, functional analysis, manifolds, riemannian geometry, group theory and further algebra with modules categories etc. An applied mathematics msc involves lots of PDEs and numerical analysis, lots of variation beyond that. Ive even seen masters in mathematics with courses primarily on logic and combinatorics, without any of the usual material beyond maybe complex analysis.

But beyond that anyone should recognize that "applied mathematics" is a convention to signify particular topics in mathematics and doesnt represent anything necessarily more useful than so called "pure mathematics", in fact the pure mathematics curriculum is decided based on how useful the concepts are over the whole of mathematics. Which is why ward cheney wrote something along the lines "my recipe for a good applied mathematician is first educating someone in pure mathematics, and then throwing him into practical problems".