Research about supersymmetric higher dimensional non-commutative geometry and topologial quantum many-body states

建立了超对称高维非交换几何，并将其数学应用于拓扑问题。在2011年，我们探索了超对称高维球体和双曲面的非交换几何。此外，我们将数学应用于超对称量子自旋系统和非厄米量子力学，并利用了它们的拓扑结构，如边缘态和隐藏秩序。2012年，我们进一步研究了超对称量子自旋系统，特别是最近提出的量子纠缠。2013年，我们将非交换几何应用于拓扑绝缘子（特别是a类和aiii类），并分析了它们的物理性质。

We established supersymmetric higher dimensional non-commutative geomery and applied its mathematics to the topological matters. In 2011, we explored non-commutative geometry for supersymmetric higher dimensional spheres and hyperboloids. Besides, we applied the mathematics to the supersymmetric quantum spin systems and non-Hermitian quantum mechanics and exploited their topological structures such as edge states and hidden order. In 2012, we further investigated the supersymmetric quantum spin systems especially about the recently proposed quantum entanglement. In 2013, we applied the non-commutative geometry to the topological insulators (in particular A-class nad AIII-class) and analyzed their physical properties.