Inhomogeneous media (i.e. materials whose properties vary spatially) are
ubiquitous in the world around us. Indeed, one could say that everything
possesses inhomogeneity on some characteristic lengthscale. Various
theories propose certain simplifications which "smooth out" such
inhomogeneities, a classical one being the theory of continuum mechanics
where one can define material properties such as fluid viscosity, elastic
shear modulus, etc. Of course this comes with the warning that one should
always carefully consider the results of a calculation carefully to check
that they do not violate the original assumptions of the theory.
In this talk we shall discuss various aspects of the mathematics and physics of inhomogeneous media including applications to bone, composite materials and soft tissue. The talk will allow us to make the transition from a simple undergraduate calculus problem involving separation of variables to the results which led to the award of a Nobel Prize to P.W. Anderson in 1977.
The road colouring problem is a conjecture in graph theory with connections to symbolic dynamics and the theory of finite-state automata. It was first proposed almost forty years ago and was recently resolved by Avraham Trahtman, a Russian emigre who now works in Israel.
I first heard about the problem in the Spring of 2008, when one of my Discrete Maths students who had read a news story about Trahtman's success emailed to ask what I could tell him about road colouring: "Not much", I was obliged to reply. But an academic hates to say that he doesn't know, so last Autumn I talked a 3rd year project student, Harriet Woodward, into doing a reading project about it. In this talk I'll tell you some of what we have learned.