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Rémi Reboulet

Metric properties of the set of Hermitian norms on a complex vector space.
Jeudi, 18 Novembre, 2021 - 17:30 à 18:30
Résumé : 

We study some metric aspects of the space N(V) of Hermitian norms on a finite-dimensional complex vector space V. We explain how one can endow N(V) with a CAT(0) (i.e. non-positively curved) structure. The latter property implying geodesicity, we thus look at the boundary at infinity of N(V), and realize it as a space of non-Archimedean (or ultrametric) norms. We then see how positivity properties of convex functions can be read off simple non-Archimedean data, and briefly explain how this provides a heuristic for proving existence of solutions to geometric PDEs on complex manifolds, using variational tools.

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