Mikhael Gromov - 3/6 Probability, symmetry, linearity
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2014-10-23

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 :
http://www.ihes.fr/~gromov/topics/recent.html