PreparationCoursDoctoral2020

Introduction to Integral Geometry and related Subjects

As a preparation to the course by Rémi Langevin, we will give some introductory talks to Integral Geometry, Convex Geometry etc.

This takes place on Mondays, 15:15-17:00 at EPFL, room CM09

March 16 :  Class is cancelled

March 9 : Marc Troyanov : Geometric Valuations continued

I will give more examples of geometric valuations, some application and I will prove Hadwiger's theorem. I also plan to disuss the Blaschke-Santalò  Kinematic Formula.

All the necessary definitions and statement will be recalled so this talk should be quite independant from the previous ones.

March 2 :  Marc Troyanov : Geometric Valuations and Hadwiger's Theorem

I will start the introduction to the theory of Geometric Valuations: I will give the basic definitions and examples, and state Hadwiger's characterisation theorem that describes all continuous valuation on the convex ring that are invariant under rigid motions. Time permiting I'll give a sketch of the proof.

Monday February 24 :  Marc Troyanov : What is the average shadow area of a cube ?

We consider a cube of edge 1. If one chose a random direction and (orthogonally) project the cube on a fixed plane, then we obtain a polygon (generically an hexagon) that we shall call the shadow of the cube in the given direction. The area of that polygon is then a random variable (since one projects the cube in a random direction). In this talk we will compute the expectation value of the shadow area by (at least) two different methods.

We will see that this problem has some relation with several notions in integral geometry such as the Cauchy and Crofton formulas, the Minkowski QuermassIntegralle, the Hadwiger Theorem etc.

Prerequisite : Nothing !  The talk should be accessible to any one with a bachelor's level.