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Friday, September 19, 2014

**Abstract:** Back in the day, Klein gave a famous talk where he addressed the question: what is a geometry? His proposed answer to this question was his Erlanger program. In this talk we will give a modern treatment of Klein's ideas with the notion of a $(G,X)$ structure. After developing some general theory, we will focus on flat geometries. Our main examples will be hyperbolic geometry and Lorentzian geometry. Flatness of $(G,X)$ structures on a topological space $M$ can also be packaged in the language of $G$-bundles on $M$ with flat connections; and so, is intimately related with representations $\pi_1(M)\rightarrow G$. This perspective gives some very nice tools to study things like moduli spaces of certain $(G,X)$ structures.