The existence problem for Steiner networks in strictly convex domains
  We consider the existence problem for `Steiner networks' (trivalent graphs with 120 degree angles at each junction) in strictly convex domains, with `Neumann' boundary conditions (orthogonal intersection with the domain boundary.) For each of the three possible combinatorial possibilities, sufficient conditions on the domain are derived for existence; in addition, in each case explicit examples of nonexistence are given. 
Alex Freire (University of Tennessee Knoxville)