Let $T$ be a torus. On a manifold with a $T$-action which has finitely many fixed points, the integration of an equivariantly closed form can be evaluated as a finite sum of its moment map on the fixed point set. We will give a proof of this localization formula for circle action (assuming certain facts about finite-dimensional representations of circle). We will then work out an explicit example of integrating volume form of the 2-sphere using this formula.

We will discuss what it means for a holomorphic foliation (possibly with singularities) of a complex surface into complex curves to be “nef”. Then we will look at some examples. These nef foliations are important for a classification program proposed by Brunella and others in the spirit of the Enriques classification of complex surfaces.

We give a brief overview of the best lower bounds on the packing density that can be obtained from lattice sphere packings in n-dimensional Euclidean space, focusing on large n. We then show how, using division rings, various existing constructions can be extended and new effective lower bounds can be uncovered in many dimensions. This is joint work with Nihar Gargava.