In dynamical systems with continuous symmetry much can be gained in understanding of phase space topology by ``moding out the symmetry''. I will discuss how this can be achieved in the example of a system of ODE's describing a single mode laser with SO(2) symmetry. I'll then discuss the problems one faces when trying to mode out symmetry in an O(2)-equivariant PDE, the Kuramoto-Sivashinsky equation.

In this talk, I will discuss symmetry and how it can be used to understand dynamical systems such as plane Couette flow and the Lorenz system. In particular, we will talk about what the group structure of plane Couette flow looks like and how we can use this to our advantage in trying to understand it. The first half of the talk will be pedagogical and in the second half I will show some new work.

Global warming and fishing have made a negative impact on coral reefs around the world. The diversity and biomass of coral reef fish is directly connected to the magnitude of coral reef cover in their local environment. It has been recently observed in pristine coral reefs that the biomass of fish is dominated by carnivorous fishes. This inversion of the fish biomass pyramid is in contrast to coral reefs where commercial fishing is practiced. We provide a predator-prey model which shows the connection between the size of coral reefs and biomass pyramid of coral reef fishes.