We plan to run this summer course from July 4-July 12. The schedule will 10-12 am and 2-3 pm. This summer course will consist of two parts.

In the first part of this course, we focus on three geometric PDE's: the mean curvature flow, the isometric embedding equation, and the constraint equations (submanifold geometry). The topics arechosen so students who attend this summer course will be able to understand the basic materials covered in the research talks later on in follow-up activities. The follow-up activities will consist of student seminar on “stability of Schwarzschild spacetime”, a special activity on “the isometric embedding equation” and several research talks by experts like Pengzi Miao, Lan-Hsuan Huang, Damin Wu and Richard Schoen.
 

In order to make it accessible to undergraduate students, the first week will be dedicated to basic techniques which may include:

Riemannian geometry: Riemannian curvature tensor, orthonormal frames, Ricci tensors and Einstein equations. This should include a lot of exercise problems related to the calculations of spherical symmetry and warped product.

 

Geometric PDE: The introduction of the above three PDE's. Basic methods that include the implicit function theorem, the method of continuity, the Picard method, and the maximum principle.

 

Submanifold geometry: Gauss-Codazzi equations, tubular neighborhood theorem in normal geodesic coordinates, second variation formula, the constraint equation and the static equation.

 

The second part of the course will focus on "the geometry of black holes" in which we will discuss the basic geometry of black holes and their perturbations, with emphasis on the Schwarzschild spacetime.