Firstable i explain my situation: On my University most of the careers are doing videos to explain what we do and try to atract more people to our careers.

Im in a really bad position, because the people who are in charge of the video want me to explain what a pure mathematician do and how it help society ( with practical examples and maybe naming some companies that work with pure mathematician and what they do in that companies ) in only 5 or 10 mintues, so i think that the best that i can do is make a example.

Other reason that have me in a bad position: In my University we have the career "Mathematical engineering" and they do most applications and some research in numerical analysis, optimization. (*)

I know that pure mathematics are increasing its importance in society every year. Many people in my country think that mathematics has stagnated over time and now only engineers develop science. I think that the most practical thing i can do is give some examples of what we are doing with mathematics today ( since 2000  ).

If some of you can help me i need the following:

1. A topic in mathematic that not appear in (*). Preferably dinamical systems, logic, algebraic geometry, functional analysis, p-adic analysis or partial differential equations.

2. A research in that topic.

3. Practical applications of that research and the institution that make the application.

Extra 1. If you know a institution ( Not a University  ) that contract pure mathematicians and you know what they do there you can tell me too.

Extra 2. If you have a very good short phrase explaining "what a mathematician do" or "how mathematics help society" i will regard it too.

Thanks in advance.

First, I will explain my situation. In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.

I am in a really bad position, because the people who are in charge of the video want me to explain what a pure mathematician does and how it helps society. They want practical examples, and maybe naming some companies that work with pure mathematicians, and what they do in those companies. All this in only 5 or 10 minutes, so I think that the best I can do is give an example.

Another reason that I am in a bad position: In my University we have the career "Mathematical engineering" and they do mostly applications and some research in numerical analysis and optimization. (*)

I know that pure mathematics is increasing its importance in society every year. Many people in my country think that mathematics has stagnated over time and now only engineers develop science. I think that the most practical thing I can do is give some examples of what we are doing with mathematics today (since 2000).

If some of you can help me, I need the following:

1. A subject in mathematics that does not appear in (*). Preferably dynamical systems, logic, algebraic geometry, functional analysis, p-adic analysis or partial differential equations.

2. A research topic in that subject.

3. Practical applications of that research and the institution that made the application.

Extra 1. If you know an institution (not a University) that contracts with pure mathematicians and you know what they do there, please tell me also.

Extra 2. If you have a very good short phrase explaining "what a mathematician does" or "how mathematics helps society" I will appreciate it too.

Thanks in advance.