MATH742: Geometric Analysis (Fall 2015)

This will be an advanced course in elliptic and parabolic PDE with a slant 
towards applications in geometry and geometric (as opposed to frequency or 
Fourier domain) methods such as ball coverings, the coarea formula, the 
isoperimetric inequality, and Hausdorff measure.

Prerequisites: MATH673 and MATH674, or permission of the instructor. 
MATH740 is not required. If needed I will review some basic differential 
geometry of submanifolds such as the first and second fundamental form of a hypersurface in R^n.

1) Scalar elliptic equations
        - Linear theory in Holder spaces (Riesz potentials, Simon's blowup method)
        - Linear theory in W^{k,p} spaces, p not equal to 2 (Calderon-Zygmund theory)
        - Linear theory with measurable coefficients (De Giorgi-Moser iteration)

2) Scalar parabolic equations
        - Basic properties of the heat kernel on R^n (uniqueness, growth, analyticity)
        - Heat kernel estimates and the logarithmic Sobolev inequality
        - Nash's entropy method and heat kernel regularity under measurable coefficients

3) Elliptic and parabolic systems
        - C^{0,alpha} and higher regularity of bounded weak solutions
        - unbounded and pathological weak solutions, regularity off of negligible sets
        - detailed analysis of a specific problem such as minimal surfaces, Yang-Mills, or Navier-Stokes

In 1) we will often be using the book by Gilbarg and Trudinger, which has plenty of 
exercises. There will also be options for student projects.