Together with Jihyeon Jessie Yang, we are resurrecting an old idea of Raoul Bott for using large torus actions to construct canonical bases for unitary representations of compact Lie groups. Our methods are complex analytic; we apply them to families of Bott-Samelson manifolds parametrized by $\mathbb C^n$. Our construction requires the vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles over the total spaces of such families; this vanishing is conjectural, hence the question mark in the title.

Neural network in machine learning has interesting similarity with quiver representation theory. In this talk, I will build an algebro-geometric formulation of a `computing machine', which is well-defined over the moduli space of representations. The main algebraic ingredient is to extend noncommutative geometry of Connes, Cuntz-Quillen, Ginzburg to near-rings, which capture the non-linear activation functions in neural network. I will also explain a uniformization between spherical, Euclidean and hyperbolic moduli of framed quiver representations.