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mm-spaces; general theory and classification

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Anatoly Vershik
St. Petersburg Department of Steklov Institute of Mathematics
Fri, 2017-09-15 09:30 - 10:30
MPIM Lecture Hall

The $mm$-space (or admissible triple, or Gromov triple) is a metric space with probability measure with
some (very weak) condition on concordance between two structures.

The point of view of author is: to fix measure structure and to vary the metrics as the measurable functions
of two variables.

Then the space of continuous measure spaces is the cone of the metrics on the standard measure space
(Lebesgue space). The series of natural properties of the triples will be formulated in the talk.

The main fact (Gromov,Vershik) is the theorem about classification of the mm-spaces up to measure preserving
isometries. The complete invariant of $mm$-spaces is so called matrix distributions --- the measure on the matrices
of distances or random matrices. Analysis of $mm$-spaces is the theory of matrix distributions.

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