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**Diagrams and irregular connections on the Riemann sphere(****arXiv****)**

**Author : **Jean Douçot

**Abstract : **We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, generalizing previous constructions to the case when there are several irregular singularities. The construction relies on applying the Fourier-Laplace transform to reduce to the case where there is only one irregular singularity at infinity, and then using the definition of Boalch-Yamakawa in that case. We prove that the diagram is invariant under the symplectic automorphisms of the Weyl algebra, so that there are several readings of the same diagram corresponding to connections with different formal data, usually on different rank bundles.

**2. On the separatrix graph of a rational vector field on the Riemann sphere(****arXiv****)**

**Author :** Kealey Dias, Antonio Garijo

**Abstract :** We consider the rational flow ξR(z)=R(z)(d/dz) where R is given by the quotient of two polynomials without common factors on the Riemann sphere. The separatrix graph ΓR is the boundary between trajectories with different properties. We characterize the properties of a planar directed graph to be homeomorphic to the separatrix graph of a rational vector field on the Riemann sphere.

**3.The N-vortex Problem on a Riemann Sphere (****arXiv****)**

**Author : **Qun Wang

**Abstract : **This article investigates the dynamical behaviours of the n-vortex problem with vorticity Γ on a Riemann sphere S2equipped with an arbitrary metric g. From perspectives of Riemannian geometry and symplectic geometry, we study the invariant orbits and prove that with some constraints on vorticity Γ, the n-vortex problem possesses finitely many fixed points and infinitely many periodic orbits for generic g. Moreover, we verify the contact structure on hyper-surfaces of the vortex dipole, and exclude the existence of perverse symmetric orbits.

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