It was noticed in 1980s that many properties of semi-algebraic sets could be derived from several simple axioms, which is now known as o-minimal structures. These structures provide a candidate of the framework to develop the notion of tame topology, as envisaged by Grothendieck in 1984, and the techniques have profound applications to number theory and arithmetic geometry. More recently, after the work of B. Bakker, Y. Brunebarbe, B. Klingler, and J. Tsimerman, o-minimal structures have become an important tool in the study of Hodge theory. In this expository talk I would start with basic definitions and examples of o-minimal structures, with a view from and towards algebraic geometry, then discuss the particular applications to Hodge theory.
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