Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

2010 Mathematics Subject Classification: Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L07.

Equidistribution estimates for Fekete points on complex manifolds

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.

Equidistribution of Fekete Points on the Sphere

Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of Fekete points in the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz-Zygmund arrays and interpolating arrays, that have been studied recently.

Equidistribution of Fekete Points on the Sphere

Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of Fekete points in the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz-Zygmund arrays and interpolating arrays, that have been studied recently.

The bogomolov conjecture for totally degenerate Abelian varieties

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

Distribution, composition and seasonality of aquatic birds in the Nhecolandia sub-region of South Pantanal, Brazil

Despite remarkable significance of Pantanal for the conservation of aquatic birds, the status of their populations, the spatiotemporal patterns of distribution and habitat use and structure of communities are little known. Thus, we studied three aquatic environments (Negro river, bays and salines) from 2007 to 2009 in the Nhecolandia Pantanal to verify the distribution and composition of aquatic birds and also if there is significant seasonal influence on these aspects. We adopted the transect method (288 hours of sampling) and recorded 135 species (7.834 individuals). The Negro river showed the highest diversity, while the salines the lowest. The similarity of aquatic bird communities was higher between bays and salines, followed by Negro river and bays and lower between salines and Negro river. The equidistribution is more variable in the salines and more stable in the Negro river. The environments strongly differ from each other in aquatic bird composition in space (habitat use and distribution) and time (seasonal water fluctuations). The diversity of bird community in the dry season varies significantly in the salines, followed by the bays and more stable in the Negro river. The Negro river, regardless of large annual amplitude of flow, is more seasonally stable since its riparian vegetation is continuous (not isolated) and constant. These aspects provide better conditions to stay all year, contributing to decrease the seasonal nomadic tendencies of aquatic birds. Finally, all these data provide strong arguments to the preservation of all phytophysiognomies in the Nhecolandia sub-region of Pantanal, but with special attention to the salines widely used by many flocks of aquatic birds (mainly in the dry season) and migrant and/or rare species restricted to this habitat.