What portion of mathematics has been, “applied?”

Factual Questions
1

I was wasting some time on Wikipedia last night and stumbled across this interesting article about mathematics and eventually came across this PDF:

http://www.american.edu/academic.depts/cas/mathstat/Events/wood_visit/MWoodInterview.pdf

The PDF is an interesting read about a math prodigy that ended up doing her undergrad at Duke because of their commitment to undergraduate research in mathematics. She mentioned a lot of different types of mathematics or principles that have been discovered in mathematics that I had never heard of before. I finished math in high-school through differential equations and multivariable calculus but I haven’t taken a, “pure,” math class since. I’ve taken physical chemistry and a little bit of engineering that touched on topics like a Fourier transform, but it seemed like most examples of math that had been applied to the real world tapped out around diff-eq’s or multivariable calculus.

So, I admit that I have no idea about what portion of all the math out there that I’ve been exposed to, let alone all of the necessary applications of the math that I have been exposed to, but out of all of the mathematical fields that are relatively well developed, what portion of the work in those fields has even been applied to the real world in some manner?

Even if it doesn’t have some, “practical,” application but only helps to describe some physical phenomena, that counts as having been applied. Also, if a given math subject is necessary to develop some further or more esoteric principle in math, that also counts.

So, what portion of the math out there was been applied to the, “real-world,” in the physical sciences, economics, information technology, or similar fields?

I’m not necessarily looking for a rigorously quantitative answer to this question, just trying to get a qualitative idea of the applicability of the problems that contemporary mathematicians are working on.

2

It at least goes far beyond basic calculus. For instance, game theory can uses lots of that “pure” math and can be ridiculously difficult, and it has broad applications in political science, economics, and other social sciences.

ETA: Of course, physics relies heavily on advances in math, especially theoretical physics.

3

I think this will be incredibly hard to answer in any meaningful way, and will vary widely depending upon the particular area of mathematics that you choose to examine. Those who use mathematics in an applied sense often are unaware of the extent of the theoretical results, and those who develop the theory are often unaware/uninterested in how it is used.

As an example, from my own area of research, which is in automated inductive theorem proving and (nominal) methods for handling name binding (i.e. computer science), an awful lot of mathematical logic is used regularly.

4

Pretty much all of it, except maybe the very latest developments that haven’t been around long enough to make their way into applications. At the very least, there’s no large branch of mathematics that has no use outside of keeping mathematicians employed.

5

I’m not sure about that; at least, I think there have been times when that was not true. For the thousands of years that it was studied before its uses in cryptography were discovered, and even most of the last couple centuries as it assumed its modern form, what application did most of number theory have?

6

Number theory has uses in algebra, which fits the criteria laid out in the OP:

It has been the exception historically, but even it can be applied in other fields now.

7

threemae, I’d recommend you read up on the field of operations research. This field uses many different areas of mathematics to solve real-world problems. Despite its many contributions to our everyday lives, it’s still a relatively obscure field.