Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space

The standard Poisson structure on the rectangular matrix variety Mm,n(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂ GLm+n(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of Mm,n(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GLm+n(C). Three different presentations of the T-orbits of symplectic leaves in Mm,n(C) are obtained – (a) as pullbacks of Bruhat cells in GLm+n(C) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GLm(C) and GLn(C). In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group Sm+n, such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed rowechelon form. Finally, decompositions of generalized double Bruhat cells in Mm,n(C) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.Comment: 20 page

Impact on construction project planning council of aqueduct Fruticas Chipaque

En los últimos años Colombia ha tenido un papel destacado en el escenario internacional por ser un país emergente económicamente, fortaleciendo significativamente los aspectos más relevantes para el desarrollo de un país. Dentro de estos factores relevantes se encuentra la construcción de infraestructurapública, que aporta grandes beneficios y brinda bienestar a la comunidad en general. Sin embargo muchas de estas obras se han visto afectadas por la mala planeación con la que se programan, incurriendo en sobrecostos que han llevado en el mejor de los casos a tiempos más largos de ejecución y en los casos más críticos al abandono de los proyectos, causando un notorio detrimento en el patrimonio de la nación y de los colombianos. Una vez revisada la información concerniente a la construcción del acueducto Fruticas del municipio de Chipaque se pudo determinar que son varios los factores que intervinieron para que su ejecución no se realizara dentro de los tiempos establecidos, siendo los más relevantes una pésima planeación de la obra dentro de su parte inicial y una gestión del riesgo inadecuada e inoportuna. El propósito fundamental de este artículo es hacer un análisis de un caso específico de un mal proyecto de construcción de infraestructura pública en Colombia, con el fin emitir algunas conclusiones, que de ser tenidas en cuenta, contribuyan a evitar que nuevos casos similares a este se presenten, afectando directamente a la economía del país y al mejoramiento de la calidad de vida de todos los colombianos.In recent years Colombia has had a prominent role on the international stage for being an economically emerging country, significantly strengthening the most important for a country's development issues. These relevant factors are the construction of public infrastructure, which provides great benefits and provides welfare to the community. However many of these works have been affected by the poor planning that are programmed with, incurring cost overruns that led to the best at longer runtimes and the most critical project abandonment cases causing a noticeable detriment to the heritage of the nation and Colombians. After reviewing the information concerning the construction of Fruticas aqueduct Township Chipaque it was determined that several factors intervened to your execution is not carried out within the established time, the most important being a lousy planning of the work within part of their initial management and inadequate and inappropriate risk. The main purpose of this paper is to analyze a specific case of a bad project to build public infrastructure in Colombia, to issue some conclusions, which if taken into account, help prevent new cases similar to this one submit, directly affecting the economy and improving the quality of life for all Colombians

The orbit structure of Dynkin curves

Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \cB be the variety of Borel subgroups of G and let e in Lie G be nilpotent. There is a natural action of the centralizer C_G(e) of e in G on the Springer fibre \cB_e = {B' in \cB | e in Lie B'} associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case \cB_e is a Dynkin curve. We give a complete description of the C_G(e)-orbits in \cB_e. In particular, we classify the irreducible components of \cB_e on which C_G(e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G.Comment: 12 pages, to appear in Math

Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.Comment: 68 pages; minor change

Character D-modules via Drinfeld center of Harish-Chandra bimodules

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg (Represent. Theory 3, 1–31, 1999). Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category of Lusztig (Adv. Math. 129, 85–98, 1997) can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category (Bezrukavnikov et al. in Isr. J. Math. 170, 207–234, 2009) this allows us to derive (under a mild technical assumption) a classification of irreducible character sheaves over ℂ obtained by Lusztig by a different method. We also give a simple description for the top cohomology of convolution of character sheaves over ℂ in a given cell modulo smaller cells and relate the so-called Harish-Chandra functor to Verdier specialization in the De Concini–Procesi compactification.United States. Defense Advanced Research Projects Agency (grant HR0011-04-1-0031)National Science Foundation (U.S.) (grant DMS-0625234)National Science Foundation (U.S.) (grant DMS-0854764)AG Laboratory HSE (RF government grant, ag. 11.G34.31.0023)Russian Foundation for Basic Research (grant 09-01-00242)Ministry of Education and Science of the Russian Federation (grant No. 2010-1.3.1-111-017-029)Science Foundation of the NRU-HSE (award 11-09-0033)National Science Foundation (U.S.) (grant DMS-0602263

Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we consider the universal Artin motive mapping to M and denote it ω0X(M). We use this to define a motive

The molecular and cellular origin of human prostate cancer

Prostate cancer is the most commonly diagnosed male malignancy. Despite compelling epidemiology, there are no definitive aetiological clues linking development to frequency. Pre-malignancies such as proliferative inflammatory atrophy (PIA) and prostatic intraepithelial neoplasia (PIN) yield insights into the initiating events of prostate cancer, as they supply a background "field" for further transformation. An inflammatory aetiology, linked to recurrent prostatitis, and heterologous signalling from reactive stroma and infiltrating immune cells may result in cytokine addiction of cancer cells, including a tumour-initiating population also known as cancer stem cells (CSCs). In prostate tumours, the background mutational rate is rarely exceeded, but genetic change via profound sporadic chromosomal rearrangements results in copy number variations and aberrant gene expression. In cancer, dysfunctional differentiation is imposed upon the normal epithelial lineage, with disruption/disappearance of the basement membrane, loss of the contiguous basal cell layer and expansion of the luminal population. An initiating role for androgen receptor (AR) is attractive, due to the luminal phenotype of the tumours, but alternatively a pool of CSCs, which express little or no AR, has also been demonstrated. Indolent and aggressive tumours may also arise from different stem or progenitor cells. Castrate resistant prostate cancer (CRPC) remains the inevitable final stage of disease following treatment. Time-limited effectiveness of second-generation anti-androgens, and the appearance of an AR-neuroendocrine phenotype imply that metastatic disease is reliant upon the plasticity of the CSC population, and indeed CSC gene expression profiles are most closely related to those identified in CRPCs