Regularity Theory for mean curvature flows with Neumann free boundary conditions.
We consider mean curvature flow with Neumann free boundary conditions. That is, mean curvature flow with a boundary that is allowed to move freely along a fixed support surface provided that the flowing and support surfaces meet perpendicularly for as long as the flow is defined. We show parallels between the theory for mean curvature flow with Neumann free boundary conditions and the boundaryless mean curvature flow. We observe, in particular the first singular time of the flow and show that the Hausdorff n measure (where n is the dimension of the flowing surface) of the singularity set at the first singular time is zero.

Homoclinic orbits and Hopf points in forward-backward delay equations
Forward-backward delay equations have recently attracted much attention. They typically arise as traveling wave equations of lattice-differential equations. In contrast to pure delay equations forward-backward delay equations are not well-posed. This talk focuses on a bifurcation of a homoclinic orbit to an asymptotic equilibrium, which undergoes a Hopf bifurcation. Using invariant manifolds we can successfully detect bifurcating solutions near the primary homoclinic orbit. This is the first time that such a global bifurcation is analysed in the setting of forward-backward delay equations."

Global dynamics of blow-up profiles in one-dimensional reaction diffusion equations (joint work with Hiroshi Matano)
We consider one-dimensional prototype reaction diffusion equations on the interval. We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium. In particular, sign-changing solutions are included.

On the boundary behaviour of H-surfaces in a partially free boundary configuration
We study surfaces with precribed mean curvature (shortly H-surfaces), which are spannend into a partially free boundary configuration consisting of a two-dimensional support manifold and a Jordan arc. New asymptotic expansions for the H-surface and its derivatives near the meeting points of Jordan arc and support manifold will be presented. These representations extend known results by G. Dziuk for minimal surfaces. The difficulties in the H-surface case arise from the non-perpendicular intersection of surface and support manifold.

Mean curvature flow for triple junctions of surfaces
For a class of networks of surfaces in three-dimensional space, consider the geometric motion described informally as follows: the cells are parametrized by a disk or an annulus, and their interiors move by mean curvature flow. Each boundary component parametrizes either a `liquid edge' or a `free boundary'. Along each `liquid edge', three surfaces meet making constant 120 degree angles, while on the `free boundaries', the surfaces intersect a fixed support surface orthogonally. I'll discuss a proof of short-time existence of classical solutions. This is analogous to a well-known geometric evolution for curves, but the existence proof for that case does not translate directly to surfaces.

Randomization of deterministic dynamics
A simple scheme will be presented allowing us to randomize the dynamics related to both discrete (periodic orbits) as well as continuous (periodic solutions) dynamical systems. Some simple examples of its application will also be given.

1/4 pinched manifolds are space forms
We describe recent joint work with Richard Schoen on the Ricci flow in higher dimensions. We show that the notion of positive isotropic curvature (PIC) leads to new invariant curvature conditions for the Ricci flow. Using these ideas, we obtain a new convergence result for a class of manifolds that includes all manifolds with 1/4-pinched sectional curvatures.