Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck K-theory of simplicial toric stack bundles and study the Chern character homomorphism.

We discuss two topics in non-commutative Iwasawa theory. One is on the ranks of the dual of the Selmer groups over Iwasawa algebras. Another is a new proof for a result of Ochi-Venjakob.

We generalize Yau’s estimates for the complex Monge-Ampère equation on compact manifolds in the case when the background metric is no longer Kähler. We prove $C^∞$ a priori estimates for a solution of the complex Monge-Ampère equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.

In this paper, we first get a subgradient estimate of the $CR$ heat equation on a closed pseudohermitian $(2n + 1)$-manifold. Secondly, by deriving the $CR$ version of sub-Laplacian comparison theorem on an $(2n + 1)$-dimensional Heisenberg group $H^n$, we are able to establish a subgradient estimate and then the Liouville-type theorem for the $CR$ heat equation on $H^n$.

Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural global action-angle coordinates. This paper is concerned with the quantization of these manifolds and of their action coordinates. Applying the geometric quantization procedure, one is lead to consider invariant subspaces of a tensor product of irreducible representations of $SU(2)$. These quantum spaces admit natural sets of commuting observables. We prove that these operators form a semi- classical integrable system, in the sense that they are Toeplitz operators with principal symbol the square of the action coordinates. As a consequence, the quantum spaces admit bases whose vectors concentrate on the Lagrangian submanifolds of constant action. The coefficients of the change of basis matrices can be estimated in terms of geometric quantities. We recover this way the already known asymptotics of the classical $6j$-symbols.