Floquet theory provides a powerful framework for studying periodic graph operators by reducing them to families of finite-dimensional matrices with Laurent polynomial entries. This reduction, rooted in the underlying periodicity of the graph, enables us to use tools from algebraic geometry, complex analysis, and combinatorics. The resulting dispersion polynomial encodes the spectral information of operators. In this talk, I will highlight recent results on the algebraic properties of dispersion polynomials, with particular emphasis on their factorization. I will then show how these advances in algebraic properties shed light on spectral phenomena such as flat bands, embedded eigenvalues, and quantum ergodicity.

Tencent Meting ID: 914-948-441