Abstract: Let $p$ be an odd prime. A field $F$ is said to be $p$-rigid if certain conditions on the cyclic algebras constructed over $F$ are satisfied. $p$-rigid fields have been studied through the last decades. In this talk I will present the properties of $p$-rigid fields together with new characterizations of such fields and their $p$-Galois groups (proved in joint work with S. Chebolu and J. Minac). In particular, given a field $F$, it is possible to detect whether $F$ is $p$-rigid simply by small quotients of $G_F(p)$, or by the cohomological dimension of $G_F(p)$, or by the $\mathbb{F}_P$-cohomology ring of $G_F(p)$, where $G_F(p)$ is the maximal pro-$p$ Galois group of $F$. In this case it is also possible to describe completely and explicitly every $p$-extension of $F$ (in a rather nice way) and every $p$-Galois group of $F$. Our results extend (and simplify) some previous results obtained by R. Ware, A. Engler and J. Koenigsmann; and are related to some important work of I. Efrat, A. Topaz, and others; and last but not least, they provide a new point of view upon such topics.

Abstract: Because our construction is done via Riemannian orbifolds, we will continue from last week by discussing some important theory and examples related to the Riemannian geometry of orbifolds. In fact, we will see some very special examples that show not all orbifolds are constructed via a group action on a manifold. We will then look at some planar isospectral domains that were constructed in 1994 by Buser and Conway as a segue to proving our main theorem about isospectrality and nonisometry of the two plane domains constructed by Gordon, Webb and Wolpert.