Thomas Bäckdahl is a Post-Doctoral Research Assistant in the School of Mathematics at the University of Edinburgh.

Conserved quantities, for example energy and momentum, play a fundamental role in the analysis of dynamics of particles and fields. For field equations, one manifestation of conserved quantities in a broad sense is the existence of symmetry operators, i.e. linear differential operators which take solutions to solutions. A well-known example of a symmetry operator for the scalar wave equation is provided by the Lie derivative along a Killing vector field.

It is important to note that other kinds of objects Continue reading →

Caleb Meier is a postdoctoral researcher in mathematics at the University of California, San Diego.

In the n+1 formalism of general relativity, the (n+1)-dimensional space-time is decomposed into n-dimensional space-like slices that are parametrized by a time function.  This is the basis for formulating Einstein’s equation as an initial value problem.  In an effort to understand which space-times are constructible, an important question is, “What is the admissible class of initial data for this problem?”  This question is addressed by analyzing the so-called Einstein constraint equations, which are an undetermined system of equations to be solved for a metric and an extrinsic curvature tensor.
Continue reading →