ABSTRACT:
In this talk I discuss recent work with Matthias Roeger, Yves van  
Gennip, and Marco Veneroni on curvature-penalizing energies that arise  
in models of pattern formation. These nonlocal energies contain  
competing terms, penalizing both rapid variation and large-scale  
aggregation, and this competition results in structures with a  
preferred length scale.

In various different settings we find that in a more subtle way - or  
more precisely, at a higher order in $\epsilon$ - the energy also  
penalizes curvature. We show how this phenomenon can be understood,  
and we formalize it in different settings by Gamma-convergence results.

The findings give insight into the role of curvature in stabilizing  
some naturally-occurring structures, such as the lipid bilayers that  
bound every living cell and such as striped patterns in various  
biological and chemical systems.