This project is a second edition of the textbook: Deformation Theory of Discontinuous Groups (De Gruyter 2022). It is devoted to studying various geometric and topological concepts related to the deformation and moduli spaces of discontinuous group actions, and building some interrelationships between these concepts. It presents full proofs of recent results, computes fundamental examples, and serves as an introduction and reference for students and researchers in Lie theory, discontinuous groups, and deformation spaces. A part of the first edition, the setting of affine actions is introduced and new ideas and methods are developed with full proofs. The setting of compact extensions is also re-written with new approaches and proofs. It also contains the most recent developments of the theory, extending from basic concepts to a comprehensive exposition, and highlighting the newest approaches and methods in deformation theory. It also includes the most recent solutions to many open questions over the last decades and brings related newest research results in this area. For specialists and beginners in deformation theory, the settings of Heisenberg and Threadlike cases are differently re-written with full details and proofs.

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Ali Baklouti is a Full Professor of Mathematics at the University of Sfax, Tunisia. He received his Ph.D. in Mathematics from the University of Metz (France) in 1995. He was elected as Vice-President of the University of Sfax since December 2020 and as the President of the Tunisian Mathematical Society for two consecutive terms (April 2016-March 2019 and April 2019-March 2022), and he has been the Deputy Director of the Mediterranean Institute of Mathematical Sciences since January 2012. He was also nominated as a member of the "Tunisian Academy of Sciences, Letters and Arts: Beit Elhikma" in December 2016.
He is the prize winner of the AMU-Pacom 2022, Awards & Medals, Category A in Mathematics.
He has recently been awarded the Royal Society for Africa Prize (previously known as Pfizer Prize) for 2024. This is a prestigious reward for mathematics, especially for harmonic and geometric analysis of homogeneous spaces and the deformation theory of discontinuous groups.
He is Co-Editor-in-Chief of the "Tunisian Journal of Mathematics (MSP, USA)" and the Editor-in-Chief of "Advances in Pure and Applied Mathematics (ISTE, UK)". He is a member of the editorial board of two other mathematics journals.
Hi is also the author of two research books: De Gruyter 2022 (individual) and Springer 2021 (jointly with H. Fujiwara and J. Ludwig).