In this talk, we discuss the critical threshold phenomena in a large class of one-dimensional pressureless Euler-Poisson (EP) equations with non-vanishing background states. First, we establish local-in-time well-posedness in appropriate regularity spaces, specifically involving negative Sobolev spaces, which are adapted to ensure the neutrality condition holds. We show that this negative homogeneous Sobolev regularity is necessary by proving an ill-posedness result in classical Sobolev spaces when this condition is absent. Next, we examine the critical threshold phenomena in pressureless EP systems that satisfy the neutrality condition. We show that, in the case of attractive forcing, the neutrality condition further restricts the sub-critical region, reducing it to a single line in the phase plane. Finally, we provide an analysis of the critical thresholds for repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system in the context of cold plasma ion dynamics, where the electron density is governed by the Maxwell-Boltzmann relation. This talk is based on joint work with Dong-ha Kim, Dowan Koo, and Eitan Tadmor.

We develop a new construction of complete non-compact 8-manifolds with holonomy equal to $\Spin(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($AT^2C$) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry. We use the construction to produce infinite diffeomorphism types of $AT^2C$ $\Spin(7)$-manifolds and to produce the first known example of complete toric $\Spin(7)$-manifold.