Lecture | Date | Topics |
---|---|---|
1 | 1/25/2018 |
Hypersurface in Euclidean space: Quick review of linear algebra Implicit function theorem Submanifold in Euclidean space Tangent, cotangent bundles of submanifold Induced metric, second fundamental form of hypersurface Induced connection Intrinsic curvature Poincar\'e lemma Frobenius theorem Fundamental theorem of hypersurfaces in Euclidean space |
2 | 2/1/2018 |
Intrinsic geometry of a manifold: Intrinsic definition of manifold Intrinsic definitions of tangent, cotangent, and tensor bundles Intrinsic definition of metric, connection, and curvature Examples: Sphere, hyperbolic surface (sine-Gordon) Quick intro to tensors Riemann and Ricci curvature tensors, scalar curvature |
3 | 2/8/2018 |
Matrix groups: Basic examples Push-forward of vector field Left invariant vector fields and Lie algebra Lie bracket and Maurer-Cartan equations Group of rigid motions as subgroup of GL(n+1) Left invariant differential forms and Maurer-Cartan Moving frames on hypersurface in Euclidean space Frobenius theorem Fundamental theorem of hypersurfaces |
4 | 2/15/2018 |
Moving frames: Sphere, projective spaces, Grassmannians Moving frames on Riemannian manifold Moving frame approach to stereographic projection Principal bundle formulation |
5 | 2/22/2018 |
Geodesics: Energy functional Variational formulas, Jacobi fields Parallel transport and holonomy Geodesics and distance Jacobi equation, Sturm comparison theorem Applications of comparison theorem Closed geodesics and fundamental group |
6 | 3/1/2018 |
Geometric consequences of Ricci curvature bounds Laplacians Bochner formulas Volume comparison, proof of Bishop-Gromov Isoperimetric inequalities |
7 | 3/8/2018 |
PDEs on a Riemannian manifold Sobolev norms and inequalities Linear elliptic and parabolic PDEs on a Riemannian manifold Hodge theorem |
3/15/2018 | Spring Recess | |
8 | 3/22/2018 |
Nonlinear PDEs on a Riemannian manifold |
9 | 3/29/2018 |
Ricci curvature equations DeTurck trick Prescribed Ricci equation Einstein equation Ricci flow |
10 | 4/5/2018 | TBA |
11 | 4/12/2018 | TBA |
12 | 4/19/2018 | TBA |
13 | 4/26/2018 | TBA |
14 | 5/3/2018 | TBA |