Regular and chiral polytopes

我的大部分研究涉及几何、组合学和群论之间的相互作用。在这种情况下，我研究抽象多面体，它是一种组合结构，推广了经典多面体，即高维多面体。最近我也在研究对称和半对称图。图论有许多强大的思想来自于几何学，现在我们探索一种图可以嵌入多面体的方法。特别令人感兴趣的是手性多面体，它通过旋转具有最大对称性（与通过反射具有最大对称性的规则多面体相反），因此有两种对纯形式，即左手和右手版本。这些是基本的几何对象，在我看来，它们在晶体学、生物学、化学和物理学方面有很大的应用潜力。由于它们的高度对称，它们也有很大的审美吸引力。

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.