EVENT: Pure Mathematics Seminar

Michael Pauley: Decoupling the equations for null Lie quadratics

Riemannian cubics are solutions to a variational problem used to interpolate points in (semi-)Riemannian manifolds. They generalise cubic polynomials in Euclidean space. If the manifold is a Lie group, the equation for a Riemannian cubic can be converted to an equation on the corresponding Lie algebra, solutions of which are called Lie quadratics.

A particular subset of Lie quadratics are called null, and their corresponding cubics are called null cubics. In Euclidean space the null cubics are precisely the parabolas, and in general Lie groups they have different dynamics from the other cubics. Null Lie quadratics in SO(3,R) and SL(2,R) have received a lot of attention, but in larger Lie algebras not much is known. In this talk I will explain how, in any finite dimensional Lie algebra, the differential equations for null Lie quadratics can be exactly linearised. This also applies to a variant, null Lie quadratics in tension.

Lyle Noakes: A Nonlinear 3rd Order ODE

The talk will discuss a nonlinear differential equation for a single real-valued function, showing how solutions of the ODE correspond to Riemannian cubics in SO(3). If time permits, we will review the main steps in an investigation of asymptotics of non-null Riemannian cubics using the ODE.