L1

Coherent structures, or defects, are interfaces between wave trains with

possibly different wavenumbers: they are time-periodic in an appropriate

coordinate frame and connect two, possibly different, spatially-periodic

travelling waves. We propose a classification of defects into four

different classes which have all been observed experimentally. The

characteristic distinguishing these classes is the sign of the group

velocities of the wave trains to either side of the defect, measured

relative to the speed of the defect. Using a spatial-dynamics description

in which defects correspond to homoclinic and heteroclinic orbits, we then

relate robustness properties of defects to their spectral stability

properties. If time permits, we will also discuss how defects interact with

each other.

Marstrand's Theorem is a one of the classic results of Geometric Measure Theory, amongst other things it says that fractal measures do not have density. All methods of proof have used symmetry properties of Euclidean space in an essential way. We will present an elementary history of the subject and state a version of Marstrand's theorem which holds for spaces whose unit ball is a polytope.