Abstract: A toric vector bundle is a vector bundle over a toric variety which is equipped with a lift of the action action of the associated torus. As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization. Toric vector bundles were first classified by…

Abstract: Multiplicity free Hamiltonian manifolds are the nonabelian
generalization of symplectic toric manifolds. I will explain how F. Knop
classified them using the combinatorics of smooth affine spherical
varieties and illustrate, by means of joint work with G. Pezzini, K.
Paulus, O. Goertsches and N. Wardenski, how one can concretize Knop's
classification.

I will review a sequence of results starting with Yicheng Liu’s construction of stability conditions on products with curves, and my joint work with Lie Fu and Chunyi Li on characterizing geometric stability conditions in this case, to more recent joint work with Li, Macrì, Perry, Stellari on invariant stability on self-product of curves. This paves the way…