Parabolic bundles and representations of the fundamental group

Let X be as smooth complex projective variety with Neron-Severi group isomorphic to Z, and D an irreducible divisor with normal crossing singularities. Assume r is equal to 2 or 3. We prove that if the fundamental group of X doesn't have irreducible PU(r) representations, then the fundamental group of X-D doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth and X is a complex surface.Comment: 13 pages, 1 figure, LaTeX2

Symplectic Geometry of character varieties and SU(2)SU(2) Lattice Gauge Theory I

Given a finite connected graph $\Lambda$, the space of $SU(2)$ lattice gauge-fields on $\Lambda$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of Chern-Simons theory) of an associated surface. We present evidence that, in the limit of large $\Lambda$, integration over the character variety with respect to the Liouville measure approximates lattice-theoretic integrals. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over the character variety. A continuous version of the Verlinde algebra facilitates computations. In two dimensions we recover standard expressions. For the theory on a 3-dimensional periodic lattice $\Lambda$ with Migdal action we get a very pleasant expression for the symplectic partition function, and with the Wilson action a more elaborate one. Each is a sum of a series with positive terms. One can also write down expressions for plaquette-plaquette correlations and 't Hooft loops.Comment: 43 pages, typos correcte

Oded Schramm and the Schramm-Loewner evolution: in memoriam

This article does not have an abstract

Factorisation of generalised theta functions. I

We prove a version of "factorisation", relating the space of sections of theta bundles on the moduli spaces of (parabolic, rank 2) vector bundles on curves of genus g and g-1

Yang-Mills theory for bundle gerbes

Given a bundle gerbe with connection on an oriented Riemannian manifold of dimension at least equal to 3, we formulate and study the associated Yang-Mills equations. When the Riemannian manifold is compact and oriented, we prove the existence of instanton solutions to the equations and also determine the moduli space of instantons, thus giving a complete analysis in this case. We also discuss duality in this context.Comment: Latex2e, 7 pages, some typos corrected, to appear in J. Phys. A: Math. and Ge

Ultrasound sensing using the acousto-optic effect in polymer dispersed liquid crystals

Acousto-optic effects are demonstrated in polymer dispersed liquid crystal (PDLC) films, showing promise for applications in ultrasound sensing. The PDLC films are used to image two displacement profiles of an air-coupled flexural transducers resonant modes at 295 kHz and 730 kHz. Results are confirmed using laser vibrometry. The regions on the transducers with the largest displacements are clearly imaged by the PDLC films, with the resolution agreeing well with laser vibrometry scanning. Imaging takes significantly less time than a scanning system (switching time of a few seconds, as compared to 8 hours for laser vibrometry). Heating effects are carefully monitored using thermal imaging, and are found not to be the main cause of PDLC clearing

Aero-Heating of Shallow Cavities in Hypersonic Freestream Flow

The purpose of these experiments and analysis was to augment the heating database and tools used for assessment of impact-induced shallow-cavity damage to the thermal protection system of the Space Shuttle Orbiter. The effect of length and depth on the local heating disturbance of rectangular cavities tested at hypersonic freestream conditions has been globally assessed using the two-color phosphor thermography method. These rapid-response experiments were conducted in the Langley 31-Inch Mach 10 Tunnel and were initiated immediately prior to the launch of STS-114, the initial flight in the Space Shuttle Return-To-Flight Program, and continued during the first week of the mission. Previously-designed and numerically-characterized blunted-nose baseline flat plates were used as the test surfaces. Three-dimensional computational predictions of the entire model geometry were used as a check on the design process and the two-dimensional flow assumptions used for the data analysis. The experimental boundary layer state conditions were inferred using the measured heating distributions on a no-cavity test article. Two test plates were developed, each containing 4 equally-spaced spanwise-distributed cavities. The first test plate contained cavities with a constant length-to-depth ratio of 8 with design point depth-to-boundary-layer-thickness ratios of 0.1, 0.2, 0.35, and 0.5. The second test plate contained cavities with a constant design point depth-to-boundary-layer-thickness ratio of 0.35 with length-to-depth ratios of 8, 12, 16, and 20. Cavity design parameters and the test condition matrix were established using the computational predictions. Preliminary results indicate that the floor-averaged Bump Factor (local heating rate nondimensionalized by upstream reference) at the tested conditions is approximately 0.3 with a standard deviation of 0.04 for laminar-in/laminar-out conditions when the cavity length-to-boundary-layer thickness is between 2.5 and 10 and for cavities in the depth-to-boundary-layer-thickness range of 0.3 to 0.8. Over this same range of conditions and parameters, preliminary results also indicate that the maximum Bump Factor on the cavity centerline falls between 2.0 and 2.75, as long as the cavity-exit conditions remain laminar. Cavities with length-to-boundary-layer-thickness ratio less than 2.5 can not be easily classified with this approach and require further analysis