10th February – Giacomo: Characterising the Tangent Space

The tangent space to a smooth manifold is an important object in differential geometry. Depending on context, different ways of viewing this object can be more or less helpful. In the talk I will try to motivate the tangent space geometrically, defining it as derivations of smooth functions. I will then give two alternative definitions, in terms of germs of functions and velocities of curves and show the equivalence between these. If time allows I will briefly talk about connections and geodesics.