SEMINAR: Mathematics & Statistics Colloquium: Encoding geometric information into combinatorial structure

Mathematics & Statistics Colloquium

Time and date: 4pm, Thursday 22nd October

Venue: Blakers Lecture Theatre

Speaker: Paul Baird (Universite de Bretagne Occidentale)

Title: Encoding geometric information into combinatorial structure

Abstract: There may be several reasons why we might wish to encode geometric information in this way: to transmit 3D-images; to provide algorithms by which an autonomous robot may navigate in the world; to optimise complex networks. My approach will be to consider how we ourselves reconstruct 3D-images, or frameworks, from suitably drawn lines on a piece of paper, the so-called Gestalt effect. One crucial aspect that I wish to capture is invariance with respect to similarity transformations, so that the geometry of the framework doesn't depend upon the way in which it is embedded in space, or in other words, the perspective from which it is viewed. To do this I will consider a set of equations that one can associate to a combinatorial graph which define its "geometric spectrum". An element of the spectrum allows one at least locally to realise the graph as an invariant framework in Euclidean space. Comparison with smooth counterparts provides insight into the interpretation of the equations.

Very little background is required to follow this talk: a basic knowledge of complex numbers, linear algebra and the geometry of curves and surfaces would be useful.