Title: Geometric inequalities for axisymmetric black holes Abstract: In general relativity, isolated, dynamical black holes arise from asymptotically flat initial data for the Einstein equations. Such data are characterized by invariants such as the ADM mass and angular momentum. When angular momentum is conserved in the evolution, the standard picture of gravitational collapse suggests an inequality relating the ADM mass and angular momentum. Such geometric inequalities have been rigorously proved for axisymmetric, asymptotically flat maximal initial data for the vacuum Einstein…

Title: Factorization of positive definite functions through convolution and the Turàn problem Abstract: If $G$ is a finite abelian group, we call a subset $S\subset G$ symmetric if $0\in G$ and $-x\in S$ whenever $x\in S$. We also let $S^*=(G\setminus S)\cup\{0\}$. We consider the problem of expressing an arbitrary positive definite function $F$ on $G$ as the convolution product of two positive definite functions, one supported on $S$ and the other one supported on $S^*$. We show that, in the…

Title: Some tame or wild Cantor dynamical systems Abstract: A topological dynamical system is a pair $(X,T)$ where $X$ is a compact metric spaces and $T$ is a group or semigroup acting continuously on $X$. One algebraic invariant of a such a dynamical system is the Ellis semigroup.  The Ellis semigroup $E(X,T)$ of a topological dynamical system is defined to be the compactification of the action $T$ in the topology of pointwise convergence on the space of all function $X^X$. …

Title: Nodal deficiency via equipartition energy functionals and the Dirichlet-to-Neumann map Abstract: A classic result in differential equations is that the nth eigenfunction of a Sturm-Liouville boundary value problem has precisely n-1 zeros. Courant’s nodal domain theorem provides a natural generalization of this result to higher dimensions, but it is generally not sharp. The lack of sharpness is measured by the “nodal deficiency” of an eigenfunction. Despite over a century of intensive study, this quantity is still not very well…

Title: Linear stability of slowly rotating Kerr spacetimes Abstract: I will describe joint work with Dietrich Häfner and András Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. The proof uses…

The Dalhousie-AARMS Analysis-Applied Math-Physics Seminar takes place on Fridays from 4 - 5 pm Atlantic Time over Zoom.  If you would like to attend, please email the organizers for connection details.

The Dalhousie-AARMS Analysis-Applied Math-Physics Seminar takes place on Fridays from 4 - 5 pm Atlantic Time over Zoom.  If you would like to attend, please email the organizers for connection details.