Some Progress in Conformal Geometry

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $\sigma_2$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Loss of Alx4, a stromally-restricted homeodomain protein, impairs mammary epithelial morphogenesis

AbstractPostnatal development of the mammary gland is determined by reciprocal interactions between the ductal epithelia and adjacent stroma. Alx4 is a mesenchymally restricted homeodomain transcription factor expressed in a number of developing tissues, including skin appendages such as hair follicles, whiskers and teeth. We show here that Alx4 is expressed in a subset of ERα-expressing mammary stromal cells adjacent to terminal end buds and alveoli during puberty and pregnancy, respectively. Alx4 expression is induced in mammary stromal cells at the onset of puberty and can be induced in prepubescent mice by administration of 17β-estradiol. In order to determine the role of Alx4 during mammary gland development, we characterized mammary gland development of mice homozygous for the null allele of Alx4, lstD. Mammary glands from animals lacking Alx4 activity exhibit profound alterations in ductal morphogenesis. Overall development is delayed, ducts being grossly distorted in size and structure. Terminal end buds are also disoriented, displaying aberrant architecture during bifurcation. Despite the developmental delay, the ductal network typically reaches the limits of the fat pad. However, during puberty and in the adult virgin mice, the frequency and density of branch points is significantly reduced. We show further that the defective ductal morphogenesis is due to defects in stromal cells. Specifically, when injected into the cleared fat pad of wild-type recipients, mixed populations of wild-type epithelial cells and Alx4-deficient stromal cells give rise to retarded ductal morphogenesis. Wild-type stromal cells mixed with Alx4-deficient epithelial cells result in normal progression of ductal development. Defective branching morphogenesis in Alx4-deficient females is not due to a loss in expression of HGF, since the level of HGF message in mammary stromal cells is similar in mutant and wild-type littermates. MMP3 is similarly expressed while a 40% increase in MMP2 and a 50% decrease in MMP9 message levels in Alx4-deficient mice relative to their wild-type littermates is observed. Thus, the activity of the stromally restricted homeodomain factor, Alx4, is required for normal branching morphogenesis of the ductal epithelia during pubescent mammary gland development

Pickoff and spin-conversion quenchings of ortho-positronium in oxygen

The quenching processes of the thermalized ortho-positronium(o-Ps) on an oxygen molecule have been studied by the positron annihilation age-momentum correlation techinique(AMOC). The Doppler broadening spectrum of the 511 keV gamma-rays from the 2gamma annihilation of o-Ps in O_2 has been measured as a function of the o-Ps age. The rate of the quenching, consisting of the pickoff and the spin-conversion, is estimated from the positron lifetime spectrum. The ratio of the pickoff quenching rate to the spin-conversion rate is deduced from the Doppler broadening of the 511 keV gamma-rays from the annihilation of the o-Ps. The pickoff parameter ^1Z_eff, the effective number of the electrons per molecule which contribute to the pickoff quenching, for O_2 is determined to be 0.6 +- 0.4. The cross-section for the elastic spin-conversion quenching is determined to be (1.16 +- 0.01) * 10^{-19} cm^2.Comment: 4 pages with 5 eps figures, LaTeX2e(revtex4

A conformally invariant sphere theorem in four dimensions

In this paper we provide a sharp characterization of the smooth four-dimensional sphere. The assumptions of the theorem are conformally invariant, and can be reduced to an L^2 inequality of the Weyl tensor and positivity of the Yamabe invariant.Comment: 39 pages, 0 figure

Overcoming data scarcity of Twitter: using tweets as bootstrap with application to autism-related topic content analysis

Notwithstanding recent work which has demonstrated the potential of using Twitter messages for content-specific data mining and analysis, the depth of such analysis is inherently limited by the scarcity of data imposed by the 140 character tweet limit. In this paper we describe a novel approach for targeted knowledge exploration which uses tweet content analysis as a preliminary step. This step is used to bootstrap more sophisticated data collection from directly related but much richer content sources. In particular we demonstrate that valuable information can be collected by following URLs included in tweets. We automatically extract content from the corresponding web pages and treating each web page as a document linked to the original tweet show how a temporal topic model based on a hierarchical Dirichlet process can be used to track the evolution of a complex topic structure of a Twitter community. Using autism-related tweets we demonstrate that our method is capable of capturing a much more meaningful picture of information exchange than user-chosen hashtags.Comment: IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, 201