Research lectures, seminars and events
The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but wish to enquire about attending one of the events please contact the organiser or host department.
|April 2012||June 2012|
Events for 18 May 2012
Pierre-Phillipe Dechant : From E_8 to viruses - affine extensions of (non-crystallographic) Coxeter groups
Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H_2, H_3 and H_4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H_3 generate (twist) translations along 2-, 3- and 5-fold axes of icosahedral symmetry, and we classify these translations in terms of the Fibonacci recursion relation applied to different start values. We thus provide an explanation of previous results concerning affine extensions of icosahedral symmetry in a Coxeter group context, and extend this analysis to the case of the non-crystallographic Coxeter groups H_2 and H_4. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes). We furthermore show how such affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of the more familiar affine extensions of the root systems E_8, D_6 and A_4. This broader and more conventional context of crystallographic lattices (such as E_8) suggests potential for applications in high energy physics, integrable systems and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.