非可換幾何学による物質のトポロジー相の解明

我们以前已经确定Delone集是一个在低温下有用的原子组态模型，并且不需要对材料的周期性或结构做任何假设。我们现在已经开发了一个框架来理解材料科学和超材料中系统的拓扑相，可以用Delone组态来建模。在最近的工作中，我们能够证明所谓的Delone材料的体复拓扑相的量子化。这一结果给出了对Delone和非周期晶格的电导性质的预测，似乎是新的和新颖的。特别地，我们提供了一个数学定理和有力的数值证据，证明了在非周期和非晶态金属，例如金属玻璃中，量子霍尔效应是可能的。未来有趣的研究将是调查这些特性是否可以通过实验实现。这篇论文于2017年12月提交，目前正在接受同行评审。我们还在完成使用K-理论更全面地刻画Delone格和拓扑相的拓扑性质的工作。这包括具有反线性对称的拓扑相，例如时间反转对称。我们还证明了体-边界对应，并研究了Delone拓扑材料的边缘性质。我们预计在未来几周内完成这项工作。

We have previously identified Delone sets as a useful model for atomic configurations at low temperature and without any assumptions on the periodicity or structure of the material. We have now developed a framework to understand the topological phases of systems in material science and meta-materials that can be modelled by a Delone configuration. In recent work, we were able to prove the quantization of the so-called bulk complex topological phases of Delone materials. This result gives a prediction of conductivity properties of Delone and aperiodic lattices that appears to be new and novel. In particular, we provide a mathematical theorem and strong numerical evidence that a quantum Hall like effect is possible in aperiodic and amorphous metals, e.g. metallic glass. Interesting future research would be to investigate whether such properties can be experimentally realized. This paper was submitted in December 2017 and is currently under peer review.We are also finalizing work that more comprehensively characterizes the topological properties of Delone lattices and topological phases using K-theory. This includes topological phases with anti-linear symmetries such as time reversal symmetry. We also prove the bulk-boundary correspondence and study edge properties of Delone topological materials. We expect to complete this work within the coming weeks.