Applied index theory for quantum and classical systems

量子和经典系统的应用指数理论

• 批准号: 281672735
• 负责人: Professor Dr. Hermann Schulz-Baldes
• 金额: --
• 依托单位: Department Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/281672735?language=en
• 关键词: Applied; index; theory; quantum; classical

The first goal of index theory is to relate topological invariants to indices of Fredholm operators. The most famous result in this direction is the Atiyah-Singer index theorem, but there exist far reaching non-commutative generalizations. While there is a general theory, such index theorems have to be established case by case in applications. The second goal of index theory is to connect invariants and indices of problems related via exact sequences. For example, this allows to read off the topology of boundary states or point defects from bulk invariants. The proposal aims to implement this program in situations which have not been tackled before like interacting spin systems, photonic crystals and lattices of classical springs, and also to further develop the index approach to scattering systems and topological materials.

指数论的第一个目标是将拓扑不变量与Fredholm型算子的指数联系起来。在这个方向上最著名的结果是Atiyah-Singer指数定理，但也存在影响深远的非对易推广。虽然有一个一般的理论，但在应用中必须逐个建立这样的指数定理。指标理论的第二个目标是通过正合列将相关问题的不变量和指标联系起来。例如，这允许从整体不变量中读出边界状态或点缺陷的拓扑。该方案的目的是在以前没有解决的情况下，如相互作用的自旋系统、光子晶体和经典弹簧的晶格，实现这一程序，并进一步发展散射系统和拓扑材料的指数方法。