Week 08.11.2009 – 14.11.2009

The classic Faraday wave experiment consists of a horizontal layer of fluid that spontaneously develops a pattern of standing waves on its surface as it is driven by vertical oscillation with amplitude exceeding a critical value. Faraday wave experiments have consistently produced patterns with remarkably high degrees of symmetry. Quasipatterns, which are quasiperiodic in any spatial direction, are particularly interesting since there is, as yet, no satisfactory theoretical understanding of their formation. We use multi-frequency parametric forcing to investigate the formation of patterns and approximate quasipatterns in a model partial differential equation, which plays the same role for the Faraday wave experiment that the Swift--Hohenberg equation plays for convection. We exploit three-wave resonant interactions to design forcing functions that ought to produce complex patterns, and make quantitative comparisons between weakly nonlinear predictions and the solutions of the PDE. This comparison reveals the limitations of the theory, and we explore ways in which these limitations can be addressed. Based on: Design of parametrically forced patterns and quasipatterns, by A.M. Rucklidge and M. Silber. SIAM J. Applied Dynamical Systems 8 (2009) 298-347.

We study various matrix models as toy models of the gauge dual of the AdS spacetime and black hole. I will summarize what these matrix models teach us about the information paradox and emergence of the light speed limit.

Quantum phase transitions imply certain scaling relations among space and time. A subclass of quantum critical systems is in fact invariant under relativstic conformal symmetries. Such theories have recently been studied as a new class of duals in AdS/CFT. I will describe how one can exactly embed so-called holographic superconductors in M-theory and how this embedding has already lead to new insights about the low-temperature behaviour of holgraphic superconductors with an underlying quantum critical point.

Local string models are those where Standard Model degrees of freedom arise on a small region inside a large bulk volume. I study threshold effects on gauge coupling running for such models. The Kaplunovsky-Louis formula for locally supersymmetric gauge theories predicts the unification scale should be the bulk winding mode scale, parametrically large than the string scale where divergences are naively cut off. Analysis of explicit string models on orbifold/orientifold geometries confirms this. The winding mode scale arises from the presence of tadpoles uncancelled in the local model. I briefly discuss phenomenological applications to supersymmetry breaking and gauge coupling unification.