Topological and Geometric Aspects of Tiling Dynamical Systems

洛伦佐·萨登博士将分析有关瓷砖空间的拓扑和动力学的几个问题。其中一个问题涉及计算平铺空间的切赫上同调群。另一个问题是找出替换平铺空间上的自然平移作用(或这种作用的变形)何时在拓扑上混合的标准。第三个问题是考虑双曲空间的平铺，其中的自然群作用是非阿贝尔的，因为群是不可修改的。这些问题是由准晶引起的，准晶是一类高度有序但不是晶体的固体。准晶是由非周期的晶格来模拟的，遍历理论将准晶的整体性质(如它的衍射光的能力、它的电导和它的硬度)与可能的准晶空间或等价的晶格空间中的某些平均值联系在一起。我们越好地理解瓷砖空间的抽象数学性质，我们就越好地理解准晶和其他材料的实际物理性质。

Dr. Lorenzo Sadun will analyze several problems on the topology and dynamics of tiling spaces. One problem concerns computing the Cech cohomology groups of tiling spaces. Another problem is to find criteria for when the natural translation action on a substitution tiling space (or a deformation of such an action) is topologically mixing. A third problem is to consider tilings of hyperbolic space, where the natural group action is nonabelian and in that the group is not amenable.These problems are motivated by quasicrystals, a class of solids that are highly ordered but are not crystals. Quasicrystals are modeled by aperiodic tilings, and ergodic theory relates the bulk properties of a quasicrystal (such as its ability to diffract light, its electrical conductance, and its rigidity) to certain averages taken over a space of possible quasicrystals, or equivalently a space of tilings. The better we understand the abstract mathematical properties of tiling spaces, the better we understand the down-to-earth physical properties of quasicrystals and of other materials.