Harvard-MIT Algebraic Geometry Seminar: Webs and Schubert calculus for Springer fibers

Classical Schubert calculus analyzes the geometry of the flag variety, namely the space of nested subspaces $V_1 \subseteq V_2 \subseteq \cdots \subseteq \mathbb{C}^n$, asking enumerative questions about intersections of linear spaces that turn out to be equivalent to deep problems in combinatorics and representation theory. In this talk, we'll describe some recent results in the Schubert calculus of Springer fibers. Given a nilpotent linear operator $X$, the Springer fiber of $X$ is the subvariety of flags that are fixed by $X$ in the sense that $XV_i \subseteq V_i$ for all $i$. The top-dimensional cohomology of Springer fibers admits a representation of the symmetric group first discovered by Tonny Springer as the seminal example of a geometric representation. Where classical Schubert calculus describes geometry governed by permutations, that of Springer fibers incorporates the combinatorics both of permutations and of partitions. We'll describe new results about this geometry in more detail, including evidence that from a geometric and topological perspective, the best combinatorial model for Springer fibers comes from representation-theoretic objects called webs.