TU Darmstadt

One of the main questions in the theory of the linear transport equation is whether uniqueness of solutions to the Cauchy problem holds in the case the given vector field is not smooth. We will show that even for incompressible, Sobolev (thus quite “well-behaved”) vector fields, uniqueness of solutions can drastically fail. This result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem, and, more generally, it can be interpreted as an instance of the “flexibility vs. rigidity” duality, which is a common feature of PDEs appearing in completely different fields, such as differential geometry and fluid dynamics (joint with G. Sattig and L. Székelyhidi). 

I will explain what the Willmore Morse Index of unbranched Willmore spheres in Euclidean three-space is and how to compute it. It turns out that several geometric properties at the ends of complete minimal surfaces with embedded planar ends are related to the mentioned Morse index.
One consequence of that computation is that all unbranched Willmore spheres are unstable (except for the round sphere). This talk is based on work with Jonas Hirsch.

 

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs.

The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves.

We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides

1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and

2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid

such that global convergence to the solution of the infinite-dimensional problem is ensured.

We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems.

Numerical results are presented that illustrate the potential of the approach.

This is joint work with Jan Carsten Ziems.