Abstract: I will discuss the property of polynomial convexity of compacts in C^n, and the outline the proof of local convexity of two Lagrangian submanifolds in C^n at a point of transversal intersection.

Abstract: Higher Gauge Theory (HGT) studies categorical analogs of constructions in the geometry of connections on bundles. In this talk, I will describe an HGT analog of a phenomenon that occurs in gauge theory: a groupoid of such connections arises both as a category of functors and also from a group action. I will describe a 2-categorical analog of this phenomenon, showing a relation between global and local symmetry in categorical geometry.