Topological dynamics of tilings

Lorenzo Sadun博士将研究非周期性平铺空间的拓扑和动力学性质。 他的重点将是对这些空间的结构作为逆限制的有限CW复合物，并对他们的切赫上同调。 虽然已经计算了许多平铺空间的上同调，但对Cech上同调的函子性质知之甚少（例如，当平铺空间以规定的方式改变时，上同调会发生什么），以及上同调告诉我们关于底层平铺的什么。沿着，他将研究平铺连续旋转对称和平铺，缺乏有限的局部复杂性。这两类镶嵌目前都不被理解，但Sadun博士和他的合作者正在开发的技术应该允许他将关于有限镶嵌的结果扩展到其他类别。非周期性镶嵌，如彭罗斯镶嵌，已被用来模拟物理材料，如准晶。 摘要平铺空间的抽象数学性质与被建模材料的具体物理性质密切相关。这些特性包括衍射光谱、导电性和材料的抗剪切能力。 Sadun博士的目标是进一步发展这种对应关系，既通过计算以前无法理解的平铺空间的拓扑特性，也通过跟踪平铺空间的拓扑如何反映底层平铺的变化。

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.