Enrico Pasqualetto: Limits and ultralimits of metric-measure structures
In metric geometry and analysis on nonsmooth spaces, it is often convenient (or necessary) to take limits
of sequences of metric measure spaces, for example to define tangents or asymptotic cones. This amounts
to introducing a topology, or at least a notion of convergence, on the so-called "space of spaces".
In the first part of the talk, I will give an overview on several such notions of convergence, focussing
mostly on the pointed-measured-Gromov-Hausdorff topology (pmGH for short) and its variants, but briefly
mentioning also the concentration topology and the convergence in the sense of pyramids. The emphasis will
be on the role of these notions in the study of nonsmooth spaces verifying synthetic curvature-dimension bounds.
In the second part of the talk, I will discuss a joint work with Dr. Timo Schultz, where we constructed
ultralimits of pointed metric measure spaces and we investigated their relation with pmGH limits.
We also used the ultralimit machinery to obtain new results on the category of pointed metric measure spaces.
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