Speaker
Description
Pyramids are a generalisation of mm-spaces, introduced by Gromov in the study of sequences of spaces with unbounded dimension. The space of pyramids provides a compactification of the space of mm-spaces, but pyramids come as an abstract object without any geometric representation. Extended topological metric measure (emm-) spaces, introduced by Ambrosio, Gigli and Savaré, are geometric objects that, thanks to the decoupling between their metric and topological structures, allow to study infinite dimensional spaces such as the Wiener space.
We show how every pyramid can be represented by an emm-space in a natural way. This representation is not unique in general: we provide counterexamples but we prove that concentrated pyramids (a more regular subclass) admits a unique representative. We show that the convergence of concentrated pyramids can be studied geometrically through the study concentrated emm-spaces, their emm-representations, extending convergence in concentration for mm-spaces. Finally, we provide some example of (concentrated or not) emm-spaces and discuss some stability results.
This is a work in collaboration with N. Gigli and K. Suzuki.