MSI Weekly Bulletin - Week starting Monday 16 October, 2006

A global optimization approach to solve a non-linear integral equation, used in modelling of planar waveguides. Planar waveguide-based optical components are used widely in industry. One important part of the post-manufacturing process of waveguides is to check that the refractive index profile closely matches the desired specification. The index profile can be reconstructed by solving a non-linear integral equation. Because the problem is ill-posed, its numerical solution relies on regularization and optimization techniques.

Many astrophysical phenomena can be modeled using hydrodynamical fluid flow solvers. However, the increasing demand for problem size requires a compromise between computational cost and accuracy. In this talk, I will review methods of adaptive mesh refinement and how they help reduce computational cost, while still maintaining the required accuracy. If time permits, I will show how adaptive mesh refinement is being used in my project to simulate relativistic jets.

The notion of spectral triple in A.Connes' non-commutative geometry is so far one of the most developed candidates for a theory of "non-commutative'' (Riemannian spin) manifolds. On the other side, a suitable notion of "morphism between non-commutative manifolds'' is still in the making. This means, in particular, that we do not have yet a "duality'' theory between a category of commutative spectral triples and a category of their "spectra'' that somehow "refines'' the well-known Gel'fand's duality between commutative C*-algebras and compact Hausdorff topological spaces. It is our intention, in this talk, to present a "panoramic overview'' of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry with particular emphasis on: - definitions of morphisms/categories of spectral triples, - "horizontal categorification'' of Gel'fand's duality. If time will allow, we will conclude with a summary of the expected applications of "categorical non-commutative geometry'' to structural questions in relativistic quantum physics: - (categorical) covariance, - quantum "spectral space-time'', - (algebraic) quantum gravity.

I will give several equally unsatisfactory answers to this question.