Abstracts and Talk Materials:

Geometrical Singularities and Singular Geometries

July 14-25, 2008

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Paul Clavin (UMR CNRS-Universites d'Aix-Marseille I&II) http://www.academie-sciences.fr/membres/C/Clavin_Paul.htm

Ablative Rayleigh-Taylor instability
Mon Jul 14 10:05:00 - 10:50:00

Ablative Rayleigh-Taylor (R-T) instability is a special feature of the acceleration phase in inertial confinement fusion (ICF). Ablation stabilizes the disturbances with small wavelength, introducing a marginal wavelength. Due to a large temperature ratio, the conduction length-scale varies strongly across the wave, and the attention is limited to the intermediate acceleration regimes for which the length-scale of the marginal wavelength is in-between the smallest and the largest conduction length-scale. The analysis is performed for a strong temperature dependence of thermal conductivity. At the leading order, the ablation front appears as a vortex sheet separating two potential flows ^{1, 2}, and the free boundary problem takes the form of an extension of the pure R-T instability with unity Atwood number and zero surface tension. It shows also some analogies with the Kelvin-Helmholtz instability described by the Birkhoff-Rott equation. However, the hot flow of ablated matter introduces a damping at small wavelength which has a form different from the usual damping (as the surface tension for example). The nonlinear patterns are obtained by the same boundary integral method as used for revisiting the R-T instability ³. Unfortunately, a curvature singularity develops within a finite time, even though the short wavelengths are stabilised. Scaling laws are derived from numerical fitting and a self-similarity solution of the problem is exhibited close to the critical time ⁴. The occurrence of a curvature singularity indicates that the modifications to the inner structure of the vortex sheet can no longer be neglected. A non-local curvature effect is obtained by pushing the asymptotic analysis to the next order ⁵. The corresponding small pressure correction is showed to prevent the occurrence of the curvature singularity within a finite time.

Randall D. Kamien (University of Pennsylvania) http://www.physics.upenn.edu/~kamien/

The geometry of topological defects
Mon Jul 21 14:00:00 - 14:45:00

The theory of smectic liquid crystals is notoriously difficult to study. Thermal fluctuations render them disordered through the Landau-Peierls instability, lead to anomalous momentum dependent elasticity, and make the nematic to smectic-A transition enigmatic, at best. I will discuss recent progress in studying large deformations of smectics which necessitate the use of nonlinear elasticity in order to preserve the underlying rotational symmetry. By recasting the problem of smectic configurations geometrically it is often possible to exploit toplogical information or, equivalently, boundary conditions, to confront these highly nonlinear problems. Specifically, I will discuss edge dislocations, disclination networks in three-dimensionally modulated smectics, and large angle twist grain boundary phases. Fortuitously, it is possible to make intimate comparison with experimental systems!

Robert B. Kusner (University of Massachusetts) http://www.gang.umass.edu/~kusner

Lengths and crossing numbers of tightly knotted ropes and bands
Wed Jul 16 09:00:00 - 09:45:00

About a decade ago, biophysicists observed an approximately linear relationship between the combinatorial complexity of knotted DNA and the distance traveled in gel electrophoresis experiments [1]. Modeling the DNA as tightly knotted rope of uniform thickness, it was suggested that lengths of such tight knots (rescaled to have unit thickness) would grow linearly with crossing numbers, a simple measure of knot complexity. It turned out that this relationship is more subtle: some families of knots have lengths growing as the the 3/4 power of crossing numbers, others grow linearly, all powers between 3/4 and 1 can be realized as growth rates, and it could be proven that that the power cannot exceed 2 [2-5]. It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/2 [6]. We will survey these and more recent developments in the geometry of tightly packed or knotted ropes, as well as some other physical models of knots as flattened ropes or bands which exhibit similar length versus complexity power laws, some of which we can now prove are sharp [7].