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Description

There are strong analogies between the problem of finding fixed points for group actions on a manifold, finding sections of a fibration of topological spaces, and finding solutions of systems of polynomial equations. In particular, every fibration over a curve whose fibers are path connected admits a section, as does every fibration over a surface whose fibers are simply connected. There are algebro-geometric analogues of path connected and simply connected: "rationally connected" and "rationally simply connected" complex projective manifolds. Work of Graber, Harris and myself and of de Jong, He and myself establishes analogues of the fibration result in algebraic geometry. Chenyang Xu and I combined this with beautiful work of H\'{e}l\`{e}ne Esnault to explain several classical results in number theory (overglobal function fields): the Tsen-Lang theorem, a theorem of Brauer-Hasse-Noether, and the split case of a theorem of Harder on Serre's "Conjecture II".