I sent the following email to my colleagues recently. I have had numerous responses, and will say something general about these below.
I will shortly be teaching G12MAN Mathematical Analysis to our second year students again.
One issue that has sometimes arisen (OK, always!) is the issue of what applications the theory has.
In particular, G12MAN is compulsory for students on the Mathematical Physics course, but is not a prerequisite for any “Applied Maths” modules before 4th year.
I would like to find a few examples where a “non-Pure” mathematician has (recently?) used a theorem from mathematical analysis (at any level) in order to justify their work, or something similar. If this has direct “real-world” applications, that might be good. And if it has been done by a lecturer the second-years will be familiar with, that might be even better!
For example, has anyone had to deal with interchanging the order of limit and integral for a sequence of functions? Did you need to use uniform convergence rather than pointwise convergence? Or perhaps the Dominated Convergence Theorem? Or perhaps the interchange was NOT valid?
This is just one area where it might be relatively easy to discuss a specific example (although students find the abstract theory hard).
If possible, I would like to show the students a slide, and say “Look! This is what Professor/Dr X needed to quote last week in order to justify …”
This system for motivation has been used successfully in Cork in order to motivate Statistics to Biology Students, although there it was as easy as “Look! Here is a data set that Dr X, who teaches you Biology, had to investigate last week!”
I have had a number of responses from colleagues in so-called “Applied” Mathematics, and in Statistics, listing applications of mathematical analysis in their work.
However there was also a note of caution from “Pure”, reminding me that “Pure” Mathematics is, in any case, important for its own sake. We need existing pools of expertise in all areas of mathematics. Apart from anything else, it would be dangerously short-term thinking if we were to abandon those areas where we could not see immediate applications, because important applications often only become apparent years after the crucial work is done.
Of course, in many countries the distinction between “Pure” and “Applied” is regarded as artificial. It is all mathematics anyway, and so we can also talk about applications of mathematics within mathematics.