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Cours de l'IHÉS­­ 2014-2015

# Probability, Symmetry, Linearity (1/6)

## by Prof. Mikhael GROMOV (IHES)

mercredi 8 octobre 2014 de au (Europe/Paris)
at Université Pierre et Marie Curie ( Amphithéâtre 55 A )
4, Place Jussieu 75252 Paris
 Description I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.    1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.  2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.    I will start with a category theoretic view on probability and entropy. This includes representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set; definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF. (Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).   Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems. (Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps - Wolfgang Pauli Lectures, May 25, 2009.)   Also I say something about the von Neumann entropy.   Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.   (The above mentioned papers can be found on my web page at IHES in the section "recent"). Autres occasions Organisé par Emmanuel Ullmo Contact Email: cecile@ihes.fr