I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different approach.
(I do hope this is not inappropriate for MO).
Let me start with some books I would like to read (again with self-explanatory titles)
1) The Weil conjectures for dummies
2) 2-categories for the working mathematician
3) Representations of groups: Linear and permutation representations made side by side
4) The Burnside ring
5) A functor of points approach to algebraic geometry
6) Profinite groups: An approach through examples
Any other suggestions ?
Best Answer
I don't know for certain that this doesn't exist, so I'm in a no-lose situation: if this is a rubbish answer then it means that a book that I want to exist does exist. Many mathematicians of a pure bent have taken it upon themselves to get a good understanding of theoretical physics. And many have actually managed this. But it seems to me that they usually go native in the process, with the result that I cease to be able to understand what they are saying. It could be that this is just an irreducibly necessary feature of physics, but I doubt it. Out there in book space I believe there exists a book that explains theoretical physics in a way that physicists would dislike intensely but mathematicians would find much easier to read. It may well be that if you want to do serious work in mathematical physics then you have to understand the subject as physicists do. However, this book would be aimed at pure mathematicians who were not necessarily intending to do serious work in mathematical physics but just wanted to understand what was going on from a distance.
I used to have a similar view about explanations of forcing, but I think Timothy Chow's wonderful Forcing for Dummies has filled that gap now.