A glimpse into high dimensional convex geometry
Geometry of the plane and the 3-space was one of the first areas studied by mathematicians. The main inspiration that led to ancient results has not changed ever since: it is our natural visualisation of the space. However, as the number of dimensions grows large, basic features vary, and our intuition leads us astray more and more times.
In the talk, we shall see a couple of peculiar phenomena that match the above scheme. We are mainly interested in asymptotic results, i.e. the order of magnitude of quantities compared to the number of dimensions. We shall start with concentration results from probability, which, seen from a geometric viewpoint, give us information about the distribution of volume in particular convex bodies: balls and cubes. As it turns out, volume behaves like dust: a matter that is dense everywhere, yet it is impossible to grasp. Then, results about sections of convex bodies will be presented, containing Dvoretzky's theorem about almost ellipsoidal slices, as well as one of the main conjectures of the field: the slicing conjecture.