Using a combinatorial description of the Bernstein operator and its action on Schur functions we are able to describe formal power series solutions to various families of partial differential equations. Some recent examples of combinatorial interest include the (hyper) map series, the Hurwitz series and the Monotone Hurwitz series. In this talk we will introduce and describe the 2-Toda hierarchy of partial differential equations along with a family of solutions which are of combinatorial interest. We will then discuss some of the enumerative results that have been obtained using the 2-Toda hierarchy.
Does there exist a root-system uniform and manifestly nonnegative combinatorial rule for Schubert calculus? The answer to the analogous question for representation theory of complex semisimple Lie groups is "yes", thanks to the Littelmann path model. Famously, the two stories share a common ancestor in the Littlewood-Richardson rule. However, while that rule covers the entire $G=SL_n$ case of the representation theory problem, it only handles a small part of the $G=SL_n$ case of the intersection theory problem on generalized flag varieties $G/P$. I'll examine the question through the lens of "root-theoretic Young diagrams". The main new case of analysis is the family of (co)adjoint varieties. While these varieties extend the (co)minuscule varieties for which there is a uniform rule, our rules for them have both uniform and non-uniform features. This is joint work with Dominic Searles (University of Illinois at Urbana-Champaign).