**Logical reflections: Profinite monoids, propositional quantifiers, and temporal operators** **Habilitation à diriger des recherches, Sam van Gool** **[IRIF](https://www.irif.fr), [Université Paris Cité](https://www.u-paris.fr)** **Manuscript.** Available [here](https://www.samvangool.net/hdr.pdf). **Defense slides.** Available [here](https://samvangool.net/hdr-defense-slides.pdf). **Abstract.** My HDR manuscript surveys some of the research in algebra, topology, logic, and the foundations of computer science that I have contributed to since completing my PhD in 2014, and suggests directions for further research in this field. The work I report on falls into three research themes: (1) profinite monoids and their relationship to automata and regular languages; (2) uniform interpolation and its relationship to model-complete theories; (3) axiomatization and unifiability for temporal logics. A thematic coherence between these topics lies in the recurring appearance of techniques from infinite model theory and universal algebra for studying finitary phenomena, and the use of projective limits of finite structures. This report aims to show how these techniques help with solving problems in all three of these, a priori different, research themes. **Jury and reviewers.** + Valérie Berthé, Directrice de Recherche, CNRS, Université Paris Cité, examinatrice + Jean Goubault-Larrecq, Professeur des Universités, Université Paris-Saclay, rapporteur + Victoria Gould, University of York, rapportrice + Dexter Kozen, Professor, Cornell University, rapporteur + Paul-André Melliès, Directeur de Recherche, CNRS, INRIA, Université Paris Cité, examinateur + Carlos Simpson, Directeur de Recherche, CNRS, Université Côte d'Azur, examinateur + Christine Tasson, Professeure des Universités, Institut Supérieur de l'Aéronautique et de l'Espace, examinatrice **Defense.** The defense took place on **Friday, December 6th, 2024**, at **13h**, in room **2011** of the **Sophie Germain** building, 8 place Aurélie Nemours, Paris 13ème.