I’m teaching graph theory this year. It was one of my favourite areas of mathematics when I was a student. It contains many gems, ranging from with Euler’s solution to the problem of the seven bridges of Konigsberg to the power of Ramsey’s Theorem. The arguments seem to me to be unusually varied, and often sufficiently elementary that great depth of study is not required.
I have had very little contact with graph theory in the time since I graduated. As an undergraduate I used Robin Wilson’s Introduction to Graph Theory, and I am now using it as the basis of my course. I remember enjoying the book in my youth, and finding it approachable, but I don’t remember finding the material as straightforward as it now seems. (My students aren’t finding it entirely straightforward, either, but that may be my fault.)
Why is this? I don’t think I’m a better mathematician than I was 35 years ago. In terms of solving exam questions, I would not perform as I did when I was twenty. Even with practice, I am sure I could not get back to that level, and not only because I no longer value that kind of cleverness enough to put the effort in. I now have a much better general understanding of mathematics and how it all fits together, but I no longer have the ability to master detail that I once did.