A criterion for quadraticity of a representation of the fundamental group of an algebraic variety

13 pagesInternational audienceLet $\Gamma$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $\rho:\Gamma\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(\Gamma, G)$. It is known from the work of Goldman and Millson that if $\Gamma$ is the fundamental group of a compact Kähler manifold and $\rho$ has image contained in a compact subgroup then $\rho$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(\Gamma, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion holds

Vulnerability of the agricultural sector to climate change: The development of a pantropical Climate Risk Vulnerability Assessment to inform sub-national decision making

As climate change continues to exert increasing pressure upon the livelihoods and agricultural sector of many developing and developed nations, a need exists to understand and prioritise at the sub national scale which areas and communities are most vulnerable. The purpose of this study is to develop a robust, rigorous and replicable methodology that is flexible to data limitations and spatially prioritizes the vulnerability of agriculture and rural livelihoods to climate change. We have applied the methodology in Vietnam, Uganda and Nicaragua, three contrasting developing countries that are particularly threatened by climate change. We conceptualize vulnerability to climate change following the widely adopted combination of sensitivity, exposure and adaptive capacity. We used Ecocrop and Maxent ecological models under a high emission climate scenario to assess the sensitivity of the main food security and cash crops to climate change. Using a participatory approach, we identified exposure to natural hazards and the main indicators of adaptive capacity, which were modelled and analysed using geographic information systems. We finally combined the components of vulnerability using equal-weighting to produce a crop specific vulnerability index and a final accumulative score. We have mapped the hotspots of climate change vulnerability and identified the underlying driving indicators. For example, in Vietnam we found the Mekong delta to be one of the vulnerable regions due to a decline in the climatic suitability of rice and maize, combined with high exposure to flooding, sea level rise and drought. However, the region is marked by a relatively high adaptive capacity due to developed infrastructure and comparatively high levels of education. The approach and information derived from the study informs public climate change policies and actions, as vulnerability assessments are the bases of any National Adaptation Plans (NAP), National Determined Contributions (NDC) and for accessing climate finance

The structure of quasi-transitive graphs avoiding a minor with applications to the domino problem

An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. We prove that every locally finite quasi-transitive graph $G$ avoiding a minor has a tree-decomposition whose torsos are finite or planar; moreover the tree-decomposition is canonical, i.e. invariant under the action of the automorphism group of $G$. As applications of this result, we prove the following. (i) Every locally finite quasi-transitive graph attains its Hadwiger number, that is, if such a graph contains arbitrarily large clique minors, then it contains an infinite clique minor. This extends a result of Thomassen (1992) who proved it in the 4-connected case and suggested that this assumption could be omitted. (ii) Locally finite quasi-transitive graphs avoiding a minor are accessible (in the sense of Thomassen and Woess), which extends known results on planar graphs to any proper minor-closed family. (iii) Minor-excluded finitely generated groups are accessible (in the group-theoretic sense) and finitely presented, which extends classical results on planar groups. (iv) The domino problem is decidable in a minor-excluded finitely generated group if and only if the group is virtually free, which proves the minor-excluded case of a conjecture of Ballier and Stein (2018)