Rigidity of conformal functionals on spheres

In this paper we investigate the nature of stationary points of functionals on the space of Riemannian metrics on a smooth compact manifold. Special cases are spectral invariants associated with Laplace or Dirac operators such as functional determinants, and the total Q-curvature. When the functional is invariant under conformal changes of the metric, and the manifold is the standard n-sphere, we apply methods from representation theory to give a universal form of the Hessian of the functional at a stationary point. This reveals a very strong rigidity in the local structure of any such functional. As a corollary this gives a new proof of the results of K. Okikiolu (Ann. Math., 2001) on local maxima and minima for the determinant of the conformal Laplacian, and we obtain results of the same type in general examples.Comment: 29 page

The asymptotic expansion of a CR invariant and Grauert tubes

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46246/1/208_2005_Article_BF01446285.pd

Deformations of characters, metaplectic Whittaker functions, and the Yang-Baxter equation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 121-123).Recently, unexpected connections have been discovered between characters of representations and lattice models in statistical mechanics. The bridge was first formed from Kuperberg's solution to the alternating sign matrix (ASM) conjecture. Kuperberg's proof of this conjecture, which enumerates ASMs, utilized a Yang-Baxter equation for a square ice model from statistical mechanics. In earlier work, Tokuyama and okada gave representation theoretic quantities as generating functions on certain symmetry classes of ASMs or generalizations of them. Brubaker, Bump, and Priedberg used a Yang-Baxter equation to reprove Tokuyama's result and this work seeks to do the same for a generalization of Okada's results in type B. We begin by defining the particular lattice model we study. We then imbue the lattice model with Boltzmann weights suggested by a bijection with a set of symmetric ASMs. These weights define a partition function, whose properties are studied by combinatorial and symmetric function methods over the next few chapters. This course of study culminates in the use of the Yang-Baxter equation for our ice model to prove that the partition function factors into a deformation of the Weyl denominator and a generalized character of a highest weight representation, both in type B. We conjecture that the resulting function is connected to metaplectic spherical Whittaker functions. In the last two chapters, we deal with two rather different approaches to computing Whittaker coefficients of metaplectic forms - one using a factorization of the unipotent radical to perform an integration and the other via Hecke operators on the metaplectic group.by Sawyer James Tabony.Ph.D

Spherical wave reflection and transmission

This study is concerned with the reflection and transmission of spherical waves at a plane interface between two different media. The phenomenon of the reflection and transmission of spherical waves has been studied by means of analytical methods, numerical computation, and experimental tests. A new integral representation for a spherical wave is obtained by transforming Lamb/Sommerfeld's integral representation. The new integral has no singularity so it allows more accurate numerical integration. A new proof of Lamb/Sommerfeld's integral representation for a spherical wave is presented based on the new integral. By using the new form of solutions for reflected waves and existing solutions for transmitted waves, numerical studies have been carried out to examine. the phenomenon of reflection and transmission. of spherical waves at plane surfaces of discontinuity in material properties. It is shown that the effective critical angle for the total reflection of a spherical wave is greater than that of a plane wave at a hard boundary, and that when the source height increases the effective critical angle for the total reflection of a spherical wave tends to that of a plane wave. It is shown that recent predictions of spherical wave reflection and transmission coefficients greater than 1 at normal incidence under certain condition are probably due to numerical integration error. It also has been found that for spherical wave reflection and transmission, the time average energy flux, normal to a plane parallel to the plane of discontinuity, may locally be in the direction opposite to that of the direction of energy transmission over the plane as a whole. This so-called "backward wave" occurs in an interference between the direct and reflected waves, as well as in a transmitted wave. An indirect test on the theory has been performed to check the pressure field, above a rigid boundary, predicted by the spherical wave theory. Theoretical and experimental results were in good agreement

Approches tomographiques structurelles pour l'analyse du milieu urbain par tomographie SAR THR : TomoSAR

SAR tomography consists in exploiting multiple images from the same area acquired from a slightly different angle to retrieve the 3-D distribution of the complex reflectivity on the ground. As the transmitted waves are coherent, the desired spatial information (along with the vertical axis) is coded in the phase of the pixels. Many methods have been proposed to retrieve this information in the past years. However, the natural redundancies of the scene are generally not exploited to improve the tomographic estimation step. This Ph.D. presents new approaches to regularize the estimated reflectivity density obtained through SAR tomography by exploiting the urban geometrical structures.La tomographie SAR exploite plusieurs acquisitions d'une même zone acquises d'un point de vue légerement différent pour reconstruire la densité complexe de réflectivité au sol. Cette technique d'imagerie s'appuyant sur l'émission et la réception d'ondes électromagnétiques cohérentes, les données analysées sont complexes et l'information spatiale manquante (selon la verticale) est codée dans la phase. De nombreuse méthodes ont pu être proposées pour retrouver cette information. L'utilisation des redondances naturelles à certains milieux n'est toutefois généralement pas exploitée pour améliorer l'estimation tomographique. Cette thèse propose d'utiliser l'information structurelle propre aux structures urbaines pour régulariser les densités de réflecteurs obtenues par cette technique

Causal spaces and the application of critical point theory to general relativity

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