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.

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.

I shall survey the main steps in the heat equation proof of the Atiyah-Singer index theorem for Dirac operators on spin manifolds.

The Orlik-Terao algebra of a hyperplane arrangement is the algebra of the rational functions with poles on an arrangement of hyperplanes. This is a commutative analogue of the cohomology algebra of the complement and defines the so-called reciprocal plane which is a an irreducible projective variety. In a recent joint work with G. Denham and S. Tohaneanu, we give a decomposition result about the OT algebra whenever the underlying combinatorics has a certain genericity structure. This result implies that the OT algebra is Koszul if the arrangement is nicely filtered. As consequences, we give several other results such as characterizations for the OT algebra to be a quadratic complete intersection.