Summer Graduate School

Soap bubble: equilibrium solution of the rescaled mean curvature flow and constant curvature surface.

This graduate summer school will introduce students to two important and inter-related fields of differential geometry: geometric flows and minimal surfaces.

Geometric flows have had far reaching influences on numerous branches of mathematics and other scientific disciplines. An outstanding example is the completion of Hamilton’s Ricci flow program by Perelman, leading to the resolution of the Poincare conjecture and Thurston’s geometrization conjecture for 3-manifolds. In this part of the summer school, students will be guided through basic topics and ideas in the study of geometric flows.

Since Penrose used variations of volume to formulate and study black holes in general relativity (in his Nobel prize-winning work), the intriguing connections between minimal surfaces and general relativity have been a strong driving force for the modern developments of both research areas. This part of the summer school will introduce students to the basic theory of minimal submanifolds and its applications in Riemannian geometry and general relativity.

The curriculum of this program will be accessible and will have a broad appeal to graduate students from a variety of mathematical areas, introducing some of the latest developments in each area and the remaining open problems therein, while simultaneously emphasizing their synergy.

School Structure

The school will consist of two lectures per day and two collaboration sessions, which will be led by the lecturer and teaching assistants.  The purpose of these sessions is to encourage and strengthen higher-level thinking about the material taught in the lectures and to direct further reading for interested students.   Interactive learning activities will be conducted. Students will be encouraged to participate in oral or poster sessions on their solutions or other material relevant to the course.

Prerequisites

Students are expected to have had courses in graduate real analysis and Riemannian geometry. A course in graduate-level partial differential equations is recommended.

• Differential Geometry. For example, Chapters 1-3 in Riemannian Geometry by P. Petersen.
• Partial Differential Equations. For example, Chapters 5-6 in Partial Differential Equations by L.C. Evans.

For eligibility and how to apply, see the Summer Graduate Schools homepage