I have just posted the following message in my first-year pure maths module’s announcements forum.
With its emphasis on abstraction and rigorous logical thinking, you may wonder whether or not the Pure Mathematics you are learning in G11FPM actually has applications in the “real world”.
In fact I did mention some applications in the first lecture and workshop: for example, secure transactions on the internet depend on encryption algorithms, many of which which have their basis in Number Theory, especially the theory of prime numbers. See, for example, http://en.wikipedia.org/wiki/RSA_(cryptosystem)
In fact, whichever area of mathematics you work in, rigorous logical thinking is rather important (although this may not always be apparent at undergraduate level).
Have a look at the page
for a rather good discussion of just how important Pure Mathematics is.
However, I will admit that the main reason that I do research in Pure Mathematics is because I think it is beautiful and fascinating, rather than because I expect my work to have applications in the “real world”.
The area of mathematics you choose to specialize in is largely a matter of taste. I know that not all of you will find Pure Mathematics to your taste. Still, I will do my best to introduce you to the flavour of the subject. After that it is up to you!
However, I plan to post a few more messages in this thread giving some more applications of “pure” mathematics.