Advanced Lecture Course

Title for the Advanced Lecture Course: Nonlinear Diffusion

Lecturer: Toti Daskalopoulos, Columbia University

Teaching Assistant: Maria-Cristina Caputo, University of Texas, Austin

We will present some of the basic models of nonlinear diffusion which appear in geometric evolutions equations. Our discussion will include the porous medium u_t = Δu^m, with m>0, the evolution of a planar curve by its curvature (the curve shortening flow) and the evolution of a metric on a surface by its Ricci curvature (the Ricci flow on surfaces).

We will discuss the basic questions of the existence and the regularity of solutions, as well as the asymptotic behavior of solutions and their geometric consequences. We will give a brief overview of the important known results and we will present some of the basic PDE techniques which have been developed to study these equations, as they have a wider scope of applicability beyond the specific problems discussed. Such techniques include the maximum principle, local regularity estimates and Harnack type inequalities among others.

References:

1. Daskalopoulos, P. and Kenig, C. Degenerate diffusions, Initial value problems and local regularity theory; EMS Tracts in Mathematics, 1, European Mathematical Society (EMS), Zurich, 2007.

2. Daskalopoulos, P. and del Pino, M. On a singular diffusion equation; Comm. Anal. Geom. 3 (1995), no. 3-4, 523-542.

3. Evans, Lawrence C. Partial differential equations; Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.

4. Gage, M. and Hamilton, R.S. The heat equation shrinking convex plane curves; J. Diff. Geom. 23 (1996), 69-96.

5. Hamilton, R. The Ricci flow on surfaces; Mathematical and general relativity, 71 (1988), 237-261.

6. Lieberman, G. M. Second order parabolic differential equations; World Scientific, River Edge, NJ, 1996.

 

Title for Advanced Lecture course: PDE in Conformal Geometry

Lecturer: Alice Chang, Princeton University

Teaching Assistant: Yi Wang, Princeton University

We will discuss some model partial differential equations which appear in problems in differential geometry, in particular in conformal geometry. This includes:

• Gaussian curvature equation on surfaces, uniformization theorem of classifying compact surfaces.
• Non-compactness of Sobolev embedding, Yamabe problem.
• Q-curvature, some PDE and geometric aspects.

The course will essentially be self-contained, some basic knowledge in differential geometry at the undergraduate level will be helpful.

References:

1. Sun-Yung Alice Chang, "Conformal invariants and partial differential equations", Colloquium Lecture notes, AMS, Phoenix 2004, Bulletin AMS, 42, No. 3, 2005, pp. 365-393.

2. Sun-Yung Alice Chang, M. Eastwood, B. Orsted and P. Yang, "What is Q-curvature?" in Acta Applicandae Mathematicae: An International Survey, Journal on Applying Mathematics and Mathematical Application, 2007.

3. R. Schoen and S. T. Yau, "Lecture notes in differential geometry" International Press, Chapter 5.

 

 

Beginning Lecture Course

Title for the Beginning Lecture Course: Partial Differential Equations on Surfaces

Lecturer: Irina Mitrea, University of Virginia

Teaching Assistant: Katharine Ott, University of Kentucky

The goal of this mini-course is to provide a self-contained introduction for undergraduate students who are familiar with Multivariable Calculus into the area of Global Analysis/Partial Differential Equations on Manifolds.

As opposed to the standard setting in Differential Geometry when one deals with abstract Riemannian manifolds, we will be working with surfaces in R³. What we will systematically emphasize in this setting is the possibility of describing entities intrinsically assoicated with the surface (such as mean curvature, concept of gradient, concept of divergence, concept of Laplacian) in terms of the standard Euclidean coordinates of the ambient space. An attractive feature of this approach is that it provides a simple and intuitive description of basic concept in the geometry (curvature, surface area) and analysis (gradient, divergence, Laplacian) of the surface without having to resort to local coordinates on the surface.

We will begin by trying to understand what form the Divergence Theorem takes on a surface S and build up to the point when we will be able to solve boundary value problems on S (e.g., Δu=f on S and u|_∂S =0). This generalizes results that have been done in Multivariable Calculus when things were considered in the flat setting. One intriguing part of this mini-course is to monitor how the curvature of the surface S enters this analysis.

 

 

References:

1. An Introduction to Partial Differential Equations, by Y. Pinchover and J. Rubinstein, Cambridge University Press, Cambridge, 2005.

2. Differential geometry of curves and surfaces, by M. DoCarmo, Prentice Hall, Inc., Englewood Cliffs, NJ, 1976.

3. Elementary differential geometry, by A. Pressley, Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2001.