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.