Abstract: The last decade has witnessed tremendous advances in the fabrication of two-dimensional (2D) materials with novel electronic structures. Celebrated examples of such materials include graphene and black phosphorus.
The surface conductivity in these systems in the infrared frequency regime permits the propagation of fine-scale electromagnetic waves called surface plasmon-polaritons (SPPs).
In this talk, I will discuss macroscopic consequences of the optical conductivity of 2D materials via solutions of classical Maxwell's equations. I will formally discuss the following topics:
(I) Edges of anisotropic 2D materials act as induced sources of SPPs.
(II) Periodic structures made of 2D materials intercalated in conventional dielectrics may allow for the propagation
of homogenized, slowly varying waves with nearly no phase delay (epsilon-near-zero behavior).
(III) The curvature of 2D materials may generate further confinement of SPPs.
(IV) Nonlinearities of the 2D material and the ambient media cause non-intuitive dispersion of SPPs.
Part of this work is jointly with: A. Andreeva (U. Minnesota), E. Kaxiras (Harvard), T. Low (U Minnesota), M. Luskin (U. Minnesota), M. Maier (Texas A&M), A. Mellet (U MD)
Abstract: In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime. This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.
Abstract: Every organism transmits the information for making a similar organism across bottleneck stages that are considered generational boundaries. The bottleneck stage is minimally a single cell, which has two interdependent but distinct stores of information. One store is the well-understood linear DNA sequence that is replicated during cell divisions. The other is a three-dimensional arrangement of molecules that cycles during development such that it is nearly recreated at the start of each generation. Together they form the cell code for making an organism â a union of âgeneticâ and âepigeneticâ information stores that coevolve. I will discuss the implications of this perspective for our understanding of living systems and the beginnings of a framework for the joint consideration of all the information that is transmitted across generations to perpetuate life.
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