Applied index theory for quantum and classical systems

指标论的第一个目标是将拓扑不变量与Fredholm算子的指标联系起来。在这个方向上最著名的结论是Atiyah-Singer指标定理，但也存在意义深远的非交换推广。虽然有一个一般的理论，但这些指标定理必须在应用中逐个建立。指标论的第二个目标是把通过精确序列相关的问题的不变量和指标联系起来。例如，这允许从批量不变量中读取边界状态或点缺陷的拓扑结构。本课题旨在将该方案应用于自旋相互作用系统、光子晶体和经典弹簧晶格等尚未解决的问题，并进一步发展散射系统和拓扑材料的指数方法。

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.