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Homogeneous structures in mathematics

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I will give an introduction to a subject broadly called `Fraïssé theory’ which studies and provides examples of homogeneous objects in many different areas of mathematics. Some examples include : the random (or Rado) graph from the category of graphs, Urysohn universal space from the category of metric spaces, Gurarii space from the category of Banach spaces, Poulsen simplex from the category of Choquet simplices, etc. It is a meeting point of ideas from combinatorics, group theory, topological dynamics and ergodic theory, representation theory, computer science, etc. I will talk about some of these connections and about problems people work on there.