Title: Crystal structure of certain PBW bases Speaker: Ben Salisbury (Central Michigan University)
Time: Wednesday, March 29, 2017 at 11:15 am Place: MONT 214Abstract: Lusztig's theory of PBW bases gives a way to realize the crystal $B(\infty)$ for any complex-simple Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations: one for each reduced expression of the longest element of the Weyl group. There is an explicit algorithm to calculate the actions of the crystal operators, but it can be quite complicated. In this talk, we will explain how, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. Conditions describing these reduced expressions will be given in every type except $E_8$, $F_4$ and $G_2$ and several examples will be provided. This is joint work with Peter Tingley and Adam Schultze.

Title: Symmetric powers of the Legendre and Kloosterman family Speaker: Douglas Haessig (University of Rochester)
Time: Wednesday, March 29, 2017 at 2:25 pm Place: MONT 214Abstract: This talk will discuss two families, the Legendre family of elliptic curves and the Kloosterman family of exponential sums. While the two families are quite different they share many of the same arithmetic properties. For instance, the L-function of symmetric powers of both families are polynomials of known degree and satisfy functional equations once trivial factors are accounted for. This suggests that results on one family may shed light on analogous results for the other. The aim of this talk is to explore this line of reasoning.