A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. We answer this question in the case of indecomposable abelian groups by classifying the indecomposable abelian groups that are realizable as the group of units of a ring of any given characteristic. This is joint work (arXiv:1505.03508) with Keir Lockridge

Higher category theory is an emerging area of modern mathematics with applications to areas as diverse as algebraic geometry and theoretical computer science. However, there are multiple proposed foundations for this subject. Jacob Lurie believes that higher category theory should be founded on the theory of quasi-categories, which are studied using the Joyal model structure on simplicial sets. On the other hand, Carlos Simpson believes that higher category theory should be founded on simplicial presheaves. This talk will be the beginnings of trying to reconcile these two views, by developing a model structure for simplicial presheaves on a topological space where the weak equivalences are maps which are Joyal weak equivalences on stalks.