中科院数学与系统科学研究院

数学研究所

学术报告

几何拓扑研讨班

报告人：Guo Chuan Thiang (Peking University）

题  目： Large-scale geometry obstructs localization.

时  间：2024.10.22（周二）10:30-11:30

地  点：晨兴 410

摘  要: Given a Riemannian manifold, we may choose a discretization and find a localized basis for the Hilbert space of square-integrable functions. However, there exist natural Hilbert subspaces which do not admit such a discretely-localized description. This is based on the concept of topological insulators in physics, as quantum systems with no atomic limit. I will explain that the obstruction to localizing a Hilbert subspace is a coarse-geometric invariant, which can be computed by an index formula with concrete physical meaning. The lesson is that discretization of space has a very different meaning in quantum mechanics, with invariants going beyond the classical topological ones associated to triangulations of the manifold.