Notes: Course notes
The real definition of a smooth manifold
| Instructor | Prof. John Loftin, 323 Smith, Phone: 5156 ext. 23. |
| loftin@ andromeda. rutgers. edu | |
| Website | www.andromeda.rutgers.edu/~loftin (this syllabus is attached to the website) |
| Text | Notes and references will be provided. Recommended text: Michael E. Taylor, Partial Differential Equations, Basic Theory (Springer) |
| Supplemental reading | TBA |
| Prerequisites | Real Analysis I (metric spaces, Lebesgue theory, basic Fourier analysis) |
| Overview | The main focus will be on differential equations. We will develop tools from functional analysis along the way as needed. |
| Topics | Ordinary differential equations, the calculus of variations, weak solutions and distributions, examples of evolution equations, topics in elliptic PDE |
| Homework | Homework will be collected about every 2 weeks. |
Tentative outline of topics:
1. Ordinary Differential Equations
a. Existence and Uniqueness; Contraction Mappings
b. (Another Application of Contraction
Mapping: Inverse and Implicit Function Theorems)
c. Some techniques for explicitly solving ODEs
2. The Calculus of Variations
a. Computations; Review of Integration by Parts
b. Example: Geodesics on Riemannian Manifolds
c. Example: The Minimal Surface Equation
d. The Direct Method: Finding a
Length-Minimizing Curve in a Free Homotopy Class
3. Constant Coefficient Evolution Equations; the Heat Kernel
4. Topics in Elliptic PDEs