Groups, entropies and number theory

In this talk, we will show that there exists an intrinsic group-theoretical structure behind the notion of entropy. It comes from the requirement of composability of an entropy with respect to the union of two statistically independent subsystems. We propose a formulation of the celebrated Shannon-Khinchin axioms, in which a new composability axiom replaces the traditional additivity axiom. The theory of formal groups offers a natural language for our group-theoretical approach to generalized entropies.
At the same time, we will propose a simple universal class of trace-form admissible entropies. This class contains many well known examples of entropies and infinitely many new ones, a priori multi-parametric. Due to its specific relation with the Lazard universal formal group of algebraic topology, the new family of entropies introduced in this work will be called the universal group entropy.
We shall also prove that the celebrated Renyi entropy is the first example of a new family of non trace-form entropies, that we call the Z-entropies. They are all strictly composable, i.e. possess a composition law for any possible probability distribution. This property defines again a group-theoretical structure, which determines crucially the statistical and thermodynamical properties of the underlying entropies. A new class of divergences and applications to information theory are proposed.
The theory of group entropies will also be related with the theory of L-series and generalized Bernoulli polynomials.