IMPRS Ringvorlesung "Partial Differential Equations"
Date and location: Wednesdays 13:30-15:00 in room G10 (MPI MIS)
- Weyl's isometric embedding theorem
- Lecturer: Dr. P. Hornung
- Date: 11, 18, 25. April and 2. May
- Abstract: We give a (very slightly modified) version of Nirenberg's celebrated proof of the Weyl's problem about the realization of smooth convex metrics on the 2-sphere.
- Discretisation of linear, instationary partial differential equations
- Lecturer: Prof. W. Hackbusch
- Date: 9., 16. and 23. May
- Abstract: The lecture considers the instationary difference equations obtained by discretising partial differential equations. We focus to the three properties: consistency, convergence, and stability. The equivalence theorem states that convergence and stability are (almost) equivalent. Therefore, it is of interest to describe sufficient and/or necessary criteria for stability.
- Variational methods for PDEs: existence of closed geodesics on spheres.
- Lecturer: Prof. E. Spadaro
- Date: 30. May and 6., 13. June
- Abstract:Partial differential equations can be often treated via variational methods. Some of these equations are, indeed, naturally associated to an energy and their solutions correspond to critical points of such an energy. In order to give a glimpse of these techniques, in this lecture we will review the min-max construction of a closed geodesic on a 2-dimensional sphere, as proved by Birkhoff in 1917. In particular, we will discuss the difference between finding minimizers and general stationary points of a functional and, time permitting, will put this result in perspective of recent developments.
- An invitation to quasiconformal maps.
- Lecturer: Prof. L. Szekelyhidi
- Date: 27. June and 4., 18. July. No lecture on 11. July.
- Abstract: Quasiconformal maps are a generalization of conformal maps and as such, evolved originally from complex function theory. Besides being fascinating objects in their own right, they have been an invaluable tool in many different fields, ranging from elliptic PDEs to Teichm\"uller theory, from nonlinear elasticity to low-dimensional topology. In the words of Juha Heinonen "they are flexible enough to be ubiquitous, yet they harbor enough analytic and geometric properties so as to be useful in a variety of contexts." In fact, the theory is much more real than complex analysis, using tools from geometric measure theory, singular integrals, PDE and the calculus of variations. In these lectures I will give a brief introduction to quasiconformal mappings from a primarily analytic point of view.