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In 2025, I was awarded a grant from Centre International de Mathématiques Pures et Appliquées (CIMPA) called "Research in pairs" with the objective of visiting my collaborator Professor Santi Spadaro of the University of Catania. Besides the visit to Catania, I was invited to give a mini-course on a topic I am expert on. Thus, I visited the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste and gave a total of four lectures, each with a duration of 2 hours, on the topic of homogeneous spaces.
Title: Some Notions of Topological Homogeneity
Abstract: A topological space X is homogeneous if for every x, y in X there is an autohomeomorphism h of X such that h(x)=y. Informally speaking, a space is homogeneous if it “looks” the same around every point. Homogeneity is a very natural notion in mathematics because most of the notable topological spaces are homogeneous: euclidean spaces, spheres, manifolds, the Cantor set and the Hilbert cube, among others. By replacing the points by countable dense sets in the definition of homogeneous space we obtain the definition of countable dense homogeneous space; CDH space, for short. Although this notion was explicitly named for the first time by Bennett in 1972, it appears in the work of Georg Cantor, the founder of Set Theory. A recently active area of research is the construction of CDH spaces that are not Polish (separable and completely metrizable). The objective of the course is to present the notion of homogeneity, give simple examples and progress towards the construction of non-Polish CDH spaces.
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