**DETAILED PROGRAM ** **HERE**

Geometry is a vast branch of mathematics concerned with questions of shape, size, and the properties of space. *Differential geometry*, in particular, uses techniques of calculus and linear algebra to study problems in geometry. It studies the structure of *differentiable manifolds* which, roughly speaking, formalize the concept of curve to higher dimension. This year our M2R focuses on *several contemporary subfields of geometry*.

A first fundamental course is devoted to the classical topic of **Riemannian geometry**. This is the branch of differential geometry that studies *manifolds* endowed with local notions of angle, length of curves, surface area and volume. A particular emphasis is given to the interactions with the theory of **optimal transportation** and curvature bounds.

Adding constraints on the admissible curves leads to **sub-Riemannian geometry**, the second fundamental course of the M2R. Its typical problems are tackled also with techniques coming from **optimal control theory**.

The third fundamental course is devoted to **contact and symplectic geometry**, and in particular to their interaction with *differential topology*. Both *symplectic* and *contact geometry* are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the odd-dimensional constant-energy hypersurface.

The second part of the M2R consists in *advanced courses*.

The first one is devoted to **geometric analysis**, a mathematical discipline at the interface of *differential geometry* and *differential equations*. A particular emphasis is given to heat equation on *Riemannian manifolds* and *Lie groups*, and the interaction between the the analytic properties of its solutions and the geometrical properties of the manifold.

The second one is a course in **random geometry**. This course has a probabilistic flavor, and applies techniques coming from topology and Riemannian geometry to study the *average* properties of the zero locus of random real polynomials (such as its size, shape and measure).

A third possible advanced course is an introduction to **mathematical general relativity** and in particular to the *Einstein equations* on which the whole theory is based. Solutions of the Einstein equations are 4 dimensional *Lorentzian manifolds*. We will discuss the local theory of the Einstein equations as well as some particular solutions describing black holes.

**DETAILED PROGRAM ** **HERE**

**Fundamental courses :**

- Riemannian geometry and optimal transportation (H. Pajot) (12 ECTS)
- Sub-Riemannian geometry and optimal control theory (G. Charlot, L. Rizzi) (12 ECTS)
- Introduction to contact and symplectic geometry (S. Courte, S. Guillermou) (12 ECTS)

**Advanced courses:**

- Geometric analysis on Riemannian manifolds and Lie groups (E. Russ) (6 ECTS)
- Random Geometry (D. Gayet) (6 ECTS)

**Supplementary possible courses (if approved):**

- Introduction to Mathematical General Relativity (D. Hafner) (6 ECTS)

**Note :** Students may replace one of these courses with another of a different Master 2 (this needs no timetable conflict and approval by the administration). See for instance the pages of MSIAM's Master 2 degree and Lyon's Master 2 degree.