Difference between pure and applied analysis?

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Difference between pure and applied analysis?

Pure mathematicians try to generalise and make more abstract the pre-existing concepts , they delve deeper into seemingly simplistic mathematics.

There are two things that can be done with a concept or an idea, you can go uphill or you can go downhill ( look deeper into the concepts).Pure mathematics goes downhill.

Lets say that we have the Cartesian co-ordinate system. A pure mathematician defines a field of numbers, develops the concept of vectors , define vector spaces,find some of the properties of vector spaces ,generalise to functional spaces, define hilbert spaces and so on.

On the other hand an applied mathematician would find how to use this concept of Cartesian co-ordinate system to solve some problems, in other words how can this mathematical concept be applied.

In the light of the above example, they may realise that quantum mechanics is having some sort of link with linear algebra, incorporate the concept of hilbert spaces , get more result out of the equations (as those concepts have already been developed in pure mathematics )

Applied mathematics like pure mathematics plays a crucial role in science. In physics many concepts of pure mathematics are now applied ( so in a sense physics is applied mathematics) .

Take the example of General Theory of Relativity. Differential geometry ,( in layman terms - calculus in manifolds / in higher abstract spaces) is used extensively .And now even topology is used.

Another example - Group theory ( abstract algebra) . A pure math stuff , but used extensively in physics and chemistry ( in economics too ) .

So the work of an abstract mathematician is to apply these abstract seemingly unrelated concepts to problems faced in science or in real life.

One other thing applied mathematicians usually do is , create mathematical models.

A mathematical model is a description of a system using mathematical concepts and language.

It may seem from my answer that pure mathematics and applied mathematics are somewhat different things, the reality is not quite so . it is often hard to distinguish between the two,because there is a huge overlap.

This is roughly what a pure mathematics student is required to study before research.

Analysis , Abstract Algebra ( groups , rings , fields , ... ) , Topology and differential geometry , Theory of numbers .

This is roughly what an applied mathematics student is required to study before research.

Analysis , differential equations , linear programming , Abstract algebra ( not in that much detail ) , numerical analysis , dynamics and some other things.

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