it has been observed since the early work of richard stanley that several well-known permutation statistics are ``compatible'' with the operation of shuffling permutations. in joint work with ira gessel, we formalize this notion of a shuffle-compatible permutation statistic and develop a unifying framework for studying shuffle-compatibility, which has close connections to the theory of p-partitions, quasisymmetric functions, and noncommutative symmetric functions. in this talk, i will survey the main results of our work as well as several new directions of research concerning shuffle-compatibility.

by a gluing construction, we produce steady kahler-ricci solitons on equivariant crepant resolutions of quotient singularities $c^n/g$, with the same asymptotics as cao's soliton on $c^n$. this is joint work with olivier biquard.

let $g$ denote a $p-adic$ reductive group, and $i_1$ a $pro-p-iwahori$ subgroup. a classical result of borel and bernstein shows that the category of complex $g$-representations generated by their $i_1$-invariant vectors is equivalent to the category of modules over the (pro-p-)iwahori-hecke algebra $h$. this makes the algebra h an extremely useful tool in the study of complex representations of $g$, and thus in the local langlands program. when the field of complex numbers is replaced by a field of characteristic $p$, the equivalence above no longer holds. however, schneider has shown that one can recover an equivalence if one passes to derived categories, and upgrades $h$ to a certain differential graded hecke algebra. we will attempt to understand this equivalence by examining the $h$-module structure of certain higher $i_1$-cohomology spaces, with coefficients in mod-$p$ representations of $g$. if time permits, we'll discuss how these results are compatible with serre weight conjectures of herzig and gee--herzig--savitt.