Topological dynamics of tilings

Lorenzo Sadun博士将研究非周期平铺空间的拓扑和动力学性质。他的重点将是这些空间的结构作为有限CW复形的逆极限，以及它们的Cech上同调。虽然已经计算了许多平铺空间的上同调，但关于Cech上同调的函数性(例如，当以规定的方式改变平铺空间时上同调会发生什么)以及上同调告诉我们关于潜在的平铺的上同调，我们知之甚少。在此过程中，他将研究具有连续旋转对称性的瓷砖和缺乏有限局部复杂性的瓷砖。这两类平铺目前都还不清楚，但萨登博士和他的合作者正在开发的技术应该可以让他将关于平移有限平铺的结果推广到这些其他类别。像彭罗斯瓷砖这样的非周期瓷砖已经被用来对准晶等物理材料进行建模。瓷砖空间的抽象数学性质与所模拟材料的具体物理性质密切相关。这些特性包括衍射谱、导电性和材料的抗剪切能力。萨登博士的目标是进一步发展这种对应关系，既可以通过计算以前无法理解的瓷砖空间的拓扑性质，也可以通过跟踪瓷砖空间的拓扑结构如何反映底层瓷砖的变化。

Dr. Lorenzo Sadun will investigate topological and dynamical properties of spaces of nonperiodic tilings. His emphasis will be on the structure of these spaces as inverse limits of finite CW complexes, and on their Cech cohomology. Although the cohomologies of many tiling spaces have been computed, little is known about the functorial properties of the Cech cohomology (e.g., what happens to the cohomology when the tiling space is changed in a prescribed way), and what the cohomology tells us about the underlying tilings. Along the way, he will study tilings with continuous rotational symmetry and tilings that lack finite local complexity. Neither class of tilings is currently understood, but the techniques that Dr. Sadun and his collaborators are developing should allow him to extend results about translationally finite tilings to these other categories. Nonperiodic tilings, such as the Penrose tiling, have been used to model physical materials such as quasicrystals. Abstract mathematical properties of a space of tilings are closely related to concrete physical properties of the material being modeled. These properties include the diffraction spectrum, the electrical conductivity, and the ability of the material to resist shears. Dr. Sadun's goal is to develop this correspondence further, both by calculating topological properties of tiling spaces that have previously defied understanding, and by tracking how the topology of a tiling space reflects changes to the underlying tilings.