In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan–Lusztig-type conjecture for the characters of the irreducible and tilting modules in the Bernstein–Gelfand–Gelfand category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan’s conjecture and its variants associated to all Borel subalgebras in full generality.

We prove an unobstructedness result for deformations of subvarieties constrained by intersections with another fixed subvariety. We deduce smoothness and expected-dimension results for multiple-point loci of generic projections, mainly from a point or a line, or for fibers of embedding dimension $2$ or less.

We propose a version of the quantum ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of “most” eigenfunctions. We consider expander graphs with few short cycles (for instance random large regular graphs). Our method mimics the proof of quantum ergodicity on manifolds: it uses microlocal analysis on regular trees, as introduced by the second author in an earlier paper.

We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov–Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov–Eliashberg theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach and Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a $C^0$-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in $C^0$-Hamiltonian dynamics: uniqueness of generators for continuous analogues of Hamiltonian flows.