One thing that I have definitely benefited from in graduate school is my undergraduate background/degree in mathematics. It’s not that I use the contents of the math classes I have taken — Dynkin diagrams and cohomology don’t often appear in information theory or signal processing problems — but I am pretty comfortable with mathematical formalism. Grad students who didn’t do as much math usually get caught up on measure theory and so on by taking the graduate probability sequence or analysis sequence. What we don’t get is a survey of all the mathematical tools which we could use to attack different problems so that we know what kind of tools may relate to a given problem and where to look for it.
I think a first-year topics lecture series that introduces some new mathematical concepts to first-year graduate students could be great. Subjects like algebraic graph theory, auctions and mechanism design, random matrices, and so on may form a small unit in a regular class or may not be covered in classes at all. Not everyone wants to sit through a reading group on percolation theory, but they might want to get a basic idea of what percolation results are like and how they are useful (e.g. network connectivity).
On the other hand, maybe if such a lecture series were offered nobody would come and it would be useless. Any thoughts?