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Topics in group actions on m.m. spaces

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Gerardo Sosa
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Tue, 2017-09-05 15:30 - 16:00
MPIM Lecture Hall

The goal is to give a short overview of topics in symmetric transformations on metric measure spaces. We will address the questions of when is the group of symmetries of a m.m. space well-behaved? And of what can be concluded concerning the induced geometry of a space with symmetries?

In the first part of the talk we study the existence of a differential structure on symmetry groups of metric measure spaces. For a class of m.m. spaces, we present a necessary and sufficient condition for the existence of such a structure. As a consequence, we  recover classical results and provide new examples of spaces with smooth symmetry groups; such is the case of some spaces satisfying synthetic lower Ricci curvature bounds in the Lott-Sturm-Villani sense.

Motivated by these results, in the remainder we consider symmetric groups acting on m.m. spaces which satisfy a curvature-dimension type condition. We show that these conditions are preserved under quotient maps and glance into various novel applications. This second part is based on a collaboration together with Galaz-García, Kell, and Mondino.

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