Symmetries of polyhedral structures

我的大部分研究涉及数学中对称性的各个方面。 特别是，我探索具有许多对称性的数学对象。 我使用的技术涉及几何学、组合学和代数学之间的相互作用。 我对多面体和更高维的多面体结构非常感兴趣，这些结构被称为多面体。 正多面体自古以来就被研究。 我也研究图，特别是那些在某些方面与多面体有关的图。 由于它们的高度对称性，我所研究的图和多面体也具有很大的美学吸引力。 多年来，我一直参与研究手性多面体的性质。 手性多面体具有最大的旋转对称性（与反射对称性最大的规则多面体相反），在晶体学，生物学，化学和物理学中有很大的应用潜力。

Most of my research deals with various aspects of symmetry in mathematics. In particular, I explore mathematical objects that have many symmetries. The techniques I employ involve an interplay between geometry, combinatorics and algebra. Of great interest to me are polyhedra and higher dimensional polyhedra-like structures called polytopes. Regular polyhedra have been studied since antiquity. I also study graphs, especially those that are in some ways related to polytopes. Because of their high degree of symmetries, the graphs and polytopes I explore also have great aesthetic appeal. For many years I have been involved with investigating properties of chiral polytopes. Chiral polytopes have maximal symmetry by rotation (in contrast with regular polytopes which have maximal symmetry by reflection) and have great potential for applications in crystallography, biology, chemistry and physics.