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# 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 https://wiki.epfl.ch/grtr/introtalks2020

**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 :

- Cauchy-Crofton formulas expressing the length of a curves as an integral on the set of intersecting hyperplanes.
- Fenchel-Fary-Milnor lower bound on the total curvature of a knot in Euclidean 3-space.
- Chern-Lashof’s lower bound on the total curvature of a closed surface in Euclidean 3-space.
- At least one result of extrinsic conformal geometry using a model of the space of sphères in the Lorentz space.

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**

- T. F. Banchoff Critical Points and Curvature for Embedded Polyhedral Surfaces, The American Mathematical Monthly, (1970) 475-485.
- T.E. Cecil Lie sphere geometry with applications to submanifolds, Springer (1992).
- T.E. Cecil and S.S Chern, Dupin submanifolds in Lie sphere geometry. Differential geometry and topology (Tianjin, 1986–87), 1–48, Springer Lecture

Notes in Math., 1369. - L. Cesari and S.S Chern, Global Differential Geometry (Studies in Mathematics, Vol 27) Mathematical Association of America, 1989.
- U. Hertrich-Jeromin. Introduction to Möbius Differential Geometry, London Math. Soc Lecture Note 300, Cambridge University Press (2003).
- M. do Carmo. Differential Geometry of Curves and Surfaces, Prentice-Hall (1976).
- Rémi Langevin, Integral geometry from Buffon to geometers of today. Cours Spécialisés , 23. Société Mathématique de France, Paris, 2015.
- L.A. Santalo. Integral geometry and geometric probability, Encyclopedia of mathematics and its applications. Addison Wesley (1976).