Soutenance de thèse de Muhamet Ibrahimi (directrice de thèse E. Floriani, co-directeur M. Merkel)

https://college-doctoral.univ-amu.fr/soutenance/2569

The performance studies of the celebrated Lempel–Ziv coding algorithm have led to some deep insights into the specific entropy and relative entropy of stationary measures on shift spaces.
Notable among those is the characterisation of the specific entropy of a stochastic source in terms of the exponential asymptotics of recurrence times of a typical signal, and the related characterisation of the specific cross entropy in terms of waiting times. These and other related entropic estimators
have found diverse practical applications beyond information theory.

In this talk I will describe a research program dealing with refinements of the mathematical theory
of entropic estimators that originated in information theory, and to their theoretical and practical
applications.

Based on the joint works with G. Cristadoro, M. Degli Esposti, and R. Raquepas.

https://arxiv.org/pdf/2209.09716.pdf
https://arxiv.org/pdf/2209.09717.pdf

Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes.
It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian
framework, within a purely algebraic formalism. Like Riemannian geometry, NCG also has links with
physics. Indeed, NCG provided a powerful framework for the reformulation of the Standard Model of Particle
Physics (SMPP), taking into account general relativity, in a single "geometric" representation, based on Non-
Commutative Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient framework to
study various possibilities to go beyond the SMPP, such as Grand Unified Theories (GUTs). This thesis intends to
show an elegant method recently developed by Thierry Masson and myself, which proposes a general scheme to
elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs based on approximately finite
𝐶∗-algebras (AF-algebras), using either derivations of the algebra or spectral triples to build up the underlying
differential structure of the Gauge theory. The inductive sequence defining the AF-algebra is used to allow the
construction of a sequence of NCGFTs of Yang-Mills Higgs types, so that the rank 𝑛 + 1 can represent a grand
unified theory of the rank 𝑛.The main advantage of this framework is that it controls, using appropriate conditions,
the interaction of the degrees of freedom along the inductive sequence on the AF algebra. This suggests
a way to obtain GUT-like models while offering many directions of theoretical investigation to go beyond the
SMPP.

Je parlerai des récents développements en difféologie du point de vue mathématique mais aussi ayant un rapport avec la physique théorique, en espérant susciter la curiosité et des questions nouvelles.