Smith invariants and differential posets

We will start by introducing differential posets (defined by Stanley in 1988), and seeing some examples. In these posets we have "up" operators, "down" operators, and compositions thereof. Using a little linear algebra, it turns out that the up operator composed with the down operator, DU, has eigenvalues that are easy to write down. I will speak on a conjecture relating the Smith invariants of DU+tI over Z[t] to the eigenvalues of DU+tI. I will discuss the current progress in Young's lattice, as well as in Young-Fibonacci posets. This is joint work with Vic Reiner.