On the Grothendieck-Serre Conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$. We generalize some other related results from the simple simply-connected case to the case of arbitrary reductive group schemes.Comment: Final version to be published in the Journal of Algebra and Number Theory. We slightly change the isotropy condition as well as the terminology: see Definition 1.1 of strongly locally isotropic reductive groups. We remove the assumption that U is geometrically regular in Theorem 1 (regular is enough). Other minor improvement

On the Grothendieck-Serre conjecture on principal bundles in mixed characteristic

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split.Comment: Minor corrections and improvement

Algebraic and hamiltonian approaches to isostokes deformations

We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.Comment: 23 pages, minor corrections in the introduction, references expande

Generically isotropic reductive group schemes are locally isotropic

Let $R$ be a semilocal geometrically factorial Noetherian domain of characteristic zero. We show that a reductive $R$-group scheme is isotropic if it is generically isotropic. We derive various consequences, in particular for the Grothenieck-Serre conjecture and for homotopic invariance of torsors.Comment: Preliminary version. Comments are welcom

Estimating snow cover from publicly available images

In this paper we study the problem of estimating snow cover in mountainous regions, that is, the spatial extent of the earth surface covered by snow. We argue that publicly available visual content, in the form of user generated photographs and image feeds from outdoor webcams, can both be leveraged as additional measurement sources, complementing existing ground, satellite and airborne sensor data. To this end, we describe two content acquisition and processing pipelines that are tailored to such sources, addressing the specific challenges posed by each of them, e.g., identifying the mountain peaks, filtering out images taken in bad weather conditions, handling varying illumination conditions. The final outcome is summarized in a snow cover index, which indicates for a specific mountain and day of the year, the fraction of visible area covered by snow, possibly at different elevations. We created a manually labelled dataset to assess the accuracy of the image snow covered area estimation, achieving 90.0% precision at 91.1% recall. In addition, we show that seasonal trends related to air temperature are captured by the snow cover index.Comment: submitted to IEEE Transactions on Multimedi