We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in p + 1 dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in p + 2 dimensions. The dual geometry has an intrinsically flat timelike boundary segment Σ c whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a "near-horizon" limit in which Σ c becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For p = 2, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.

The superradiant scattering of a scalar field with frequency and angular momentum (ω, m) by a near-extreme Kerr black hole with mass and spin (M, J) was derived in the seventies by Starobinsky, Churilov, Press and Teukolsky. In this paper we show that for frequencies scaled to the superradiant bound the full functional dependence on (ω, m, M, J) of the scattering amplitudes is precisely reproduced by a dual twodimensional conformal field theory in which the black hole corresponds to a specific thermal state and the scalar field to a specific operator. This striking agreement corroborates a conjectured Kerr/CFT correspondence.1 for Near Horizon Extreme Kerr.S BH = 2πJ . The analysis was subsequently generalized to a large variety of extreme black holes with perfect agreement in every case [3].An important next step is to extend this duality to near -extreme Kerr. In this case, the light cones do not quite coalesce at the horizon, and right-moving excitations are allowed.One therefore expects a non-chiral 2D CFT.A natural approach to this problem, continuing in the spirit of [2], is to try to generalize the near-horizon boundary conditions employed in [1] to allow non-chiral excitations. 2 Although puzzles remain, this approach has recently met with partial success. In [7] consistent boundary conditions were found -an adaptation of those analyzed in [8] -which allow rightmoving conformal transformations but exclude left-moving ones. Up to a potential scaling ambiguity the deviation of the near-extreme entropy from its extreme value is accounted for by right-moving excitations. Similar results were found in [9]. Consistent boundary conditions which allow both left and right movers have not been found.

The problem of gravitational fluctuations confined inside a finite cutoff at radius r = r c outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the re- The ratio is expected to run when quantum gravitational corrections are included.

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The authors employ three numerical methods to explore the motion of low Reynolds number swimmers, modeling the hydrodynamic interactions by means of the Oseen tensor approximation, lattice Boltzmann simulations, and multiparticle collision dynamics. By applying the methods to a three bead linear swimmer, for which exact results are known, the authors are able to compare and assess the effectiveness of the different approaches. They then propose a new class of low Reynolds number swimmers, generalized three bead swimmers that can change both the length of their arms and the angle between them. Hence they suggest a design for a microstructure capable of moving in three dimensions. They discuss multiple bead, linear microstructures and show that they are highly efficient swimmers. They then turn to consider the swimming motion of elastic filaments. Using multiparticle collision dynamics the authors show that a driven filament behaves in a qualitatively similar way to the micron-scale swimming device recently demonstrated by Dreyfus et al. [Nature (London) 437, 862 (2005)].