Doctor in mathematics (2019), Institut de Mathématiques de Jussieu – Paris Rives Gauche (IMJ-PRG), Université Paris Diderot.
The title of my dissertation is “Topology of Statistical Systems: A Cohomological Approach to Information theory”; my advisor was Daniel Bennequin.
The thesis focuses on the cohomological interpretation of entropy and other related quantities, e.g. multinomial coefficients. It shows that information has topological meaning and topology serves as a unifying framework. I developed a combinatorial analogue of the fundamental equation of information theory (which provides a new characterization of generalized multinomial coefficients) and introduced a cohomological characterization of differential entropy for gaussian variables (that generalizes the axiomatic approach of Shannon, Khinchin…).
With D. Bennequin, Olivier Peltre and Grégoire Sergeant-Perthuis, I have also studied the interactions between information theory, statistical mechanics and machine learning. Graphical models provide a unified language for these theories and some interesting relations: for example, belief propagation—an algorithm introduced in bayesian learning—computes the critical points of the Bethe free energy—an approximation introduced much before in statistical mechanics. Graphical models are also the main family of examples for the “information structures” used in information cohomology.
Universitè Paris Diderot – host of (one half of) the IMJ-PRG.