Regular and chiral polytopes

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Most of my research deals with the interplay between geometry, combinatorics and group theory.  In this context I study abstract polytopes, which are combinatorial structures that generalize the classical polytopes i.e. higher dimensional polyhedra.  As of recently I also study symmetric and semisymmetric graphs.  Graph theory owes many powerful ideas to geometry and now we explore a way in which graphs can be embedded in polytopes.  Of particular interest are chiral polytopes, which have maximal symmetry by rotation (in contrast with regular polytopes which have maximal symmetry by reflection) and hence come in two enantiomorphic forms, that is in left- and right-handed versions. These are fundamental geometric objects, which in my opinion have great potential for applications in crystallography, biology, chemistry and physics.  Because of their high degree of symmetries they also have great aesthetic appeal.