PhD Course on Integral Geometry

The goal of this course is to introduce to various techniques from Integral Geometry and to apply them to global  problems in extrinsic global geometry, geometric probability etc.

The main lecturers are Rémi Langevin (Dijon) and Marc Troyanov (EPFL).

The course will take places on Monday, 15h15-17h (exact time to be set) at EPFL room CM09. It will be divided in two independent parts.


First part (from February 24 to April 6),  we will provide some background on topics such as : Basic concepts of Convex Geometry, The Brunn-Minkowski Inequality, Isoperimetric Inequality, the coarea formula, Cauchy and Crofton Formulas, Minkowski Functionals, the Hadwiger Theorem, etc. Participants will be encourage to give oral presentation to (related) subject of their choice.

The list of talks is here


Second part (from April 20 to June 20), the lectures will be given by Rémi-Langevin, below is a detailed  descriptiion.

Both parts will be independant and it is possible to follow only one of part (although it is recomended if possible to participate in both).

Note  that this course is also enlisted as  EPFL Doctoral School, MATH 731(2) Topics in geometric analysis II  and is worth 2 credits. Students who will need credits should contact Marc Troyanov.

                   Integral Methods in Extrinsic Geometry

  Rémi Langevin, Université de Bourgogne.

  EPFL,  Monday at 15:15   April 20 - June 15, 2020.  Room CM 09


Description. The goal of these lectures is to give an  introduction to the study of local and global geometric invariants of some classes of submanifolds in standard spaces such as the real (Euclidean) spaces, the spheres and the complex (Hermitan) spaces, through methods from Integral geometry.

Typical results we plan to cover in the course are :

If time permits we will also discuss bounds on the total curvature of codimension 1 foliations in Euclidean space and on the sphere, Dupin and Darboux cycides with some application to the geometric structure of isolated singularities in the complex plane and 3-space.

Prerequisite:  Although this course is a doctoral course, we will assume only some familiarity with basic differential geometry (curves and surfaces) and measure theory.

Key-words: Grassmannian, Exchange Theorem, Gauss Curvature, Lorentz spaces.


Some references