This book offers a clear and accessible pathway into information geometry for advanced undergraduate and graduate students. Readers will be guided from the fundamentals of topology and differentiable manifolds to the more advanced concepts of probability geometry and Frobenius manifolds in an intuitive manner, allowing them to build their knowledge gradually. Divided into three main parts, the first provides a concise introduction to differential topology and geometry, emphasizing the role of smooth manifolds, connections, and curvature in the formulation of geometric structures. Part II is then devoted to probability, measures, and statistics, where the notion of a probability space is refined into a geometric object, thus paving the way for a deeper mathematical understanding of statistical models. Finally, the third part introduces Frobenius manifolds, revealing their surprising connection to exponential families of probability distributions and, more broadly, their role in the geometry of information. Throughout all the chapters, there are exercises to test understanding, as well as solutions to some of the more challenging problems to aid learning.
Introduction.- Foundations of General Topology.- Topological and Modeled
Manifolds.- Differentiability and Gateaux Derivatives.- Fiber
Bundles.- Connections, Parallel Transport, and Sheafs.- Probability Theory,
Measure Theory, and Statistics.- Categorical Structures in Probability and
Measure Theory.- Categorical and Geometric Structures in Statistical Manifold
Theory.- Frobenius Manifolds.- Unveiling the Hidden Geometry of Statistical
Manifolds.- Statistical Frobenius Manifolds and Learning.
Noémie Combe is a mathematician and research professor at the Léonard de Vinci University Pole. She has secured several prestigious research grants, serving as principal investigator and group leader of the renowned Polonez Bis 3 programmefunded by Horizon 2020at the University of Warsaw. Prior to this, she led a Minerva Group at the Max Planck Society for five years, where her research received significant recognition and sustained support. She earned her degrees from the University of Geneva, and completed her PhD at the University of AixMarseille and Sorbonne University (Paris 6).
Philippe Combe is an emeritus professor of Mathematical Physics at the University of Aix-Marseille, as well as a longstanding member of the Centre for Theoretical Physics (CPT). Over a distinguished career spanning more than four decades, he has made foundational contributions to functional analysis, group theory, operator theory, geometry, statistics, and stochastic process. His work is characterized by highimpact collaborations across Europe and beyond. In addition to his research, Philippe Combe has provided key leadership in the scientific community.
Hanna Nencka is a distinguished invited researcher in the Polonez Bis 3-sponsored Research Group at the University of Warsaw. She has held tenured professorships at Columbia University (USA), the Polish Academy of Science, and the University of Madeira (Portugal). Her prolific scholarship includes over one hundred peer-reviewed publications in leading journals. Her pioneering contributions to mathematical physics have earned her international recognition and multiple research awards, namely from the Polish Academy of Sciences for outstanding work.
