Abstract : Topological materials are a class of quantum materials whose properties are preserved under topological transformations. The delicate structures of these materials admit novel and subtle propagating wave patterns which are immune to backscattering from disorder and defects. Recent years have witnessed vast of new experiments and theories about the wave phenomena in topological materials. The goal of this minisymposium is to bring together theoretical and applied researchers in these areas to discuss recent advances in the mathematical theories and physical applications. Topics will include analysis of the underlying governing equations, numerical methods on computing edge states, and experimental realizations.
Organizer(s) : Hailong Guo, Emmanuel Lorin, Xu Yang
[03694] Computation of phononic crystals using the PG finite element method
Format : Talk at Waseda University
Author(s) :
Liqun Wang (China University of Petroleum-Beijing)
Abstract : Phononic crystals are composite materials with periodic distribution of two or more media. The difficulty of computing the band structure of the phononic crystals lies in capturing the complex geometry and jump conditions effectively on the interface between the scatterer and the matrix. This talk will present the Petrov-Galerkin Finite Element Method for the band structure computation of phononic crystals, and the properties of various materials are also discussed.
[05337] Conically degenerate spectral points of the periodic Schrödinger operator
Format : Talk at Waseda University
Author(s) :
Yi Zhu (Tsinghua University)
Abstract : Conical spectral points on the dispersion bands are the origin of many novel topological phenomena, including various topological phases. I will first review recent mathematical theories on these points, especially Fefferman & Weinstein's results (JAMS 2012) on 2D Dirac points, which paved the way for rigorous justifications of such points. Then I will focus on our recent progress in constructing 3-fold Weyl points at which two energy bands intersect conically with an extra band sandwiched in between. We give the existence of such points in the spectrum of the 3-dimensional Schrödinger operator H = −Δ+V (x) with V (x) being in a large class of periodic potentials. This is the first rigorous result on the existence of 3-fold Weyl points for a broad family of 3D continuous Schrödinger equations. Our result extends Fefferman-Weinstein's pioneering work to higher dimensions and multiplicities.
[02726] Frozen Gaussian sampling for wave equations
Format : Talk at Waseda University
Author(s) :
Lihui Chai (Sun Yat-sen University)
Ye Feng (Sun Yat-sen University)
Zhennan Zhou (Peking University)
Abstract : We introduce the frozen Gaussian sampling (FGS) algorithm to solve the wave equation in the high-frequency regime. The FGS algorithm is a Monte Carlo sampling strategy based on the frozen Gaussian approximation, which greatly reduces the computation workload in wave propagation and reconstruction. We propose feasible and detailed procedures to implement the FGS algorithm, and we analyze the error caused by the sampling algorithm with Gaussian initial conditions and WKB initial conditions respectively.
[05595] Unfitted Computation of edge modes in photonic graphene
Author(s) :
Hailong Guo (The University of Melbourne )
Yi Zhu (Tsinghua University)
Xu Yang (University of California Santa Barbara)
Abstract : Photonic graphene, a photonic crystal with honeycomb structures, has been intensively studied in both theoretical and applied fields. In this paper, we propose a new unfitted Nitsche's method of computing edge modes in photonic graphene with some defect. The unique feather of the methods is that they can arbitrary handle high contrast with geometric unfitted meshes. We establish the optimal convergence of methods.
