Tianyuan Mathematical Center in Northeast China will open the short Course “Mathematical theory of plasmon/polariton resonances and applications” between July 18 and July 21, 2022. It is held by Prof. Hongyu Liu from City University of Hong Kong, Hong Kong (China) and Prof. Hongjie Li from The Chinese University of Hong Kong, Hong Kong (China).

一、About the short course

1、Instructors：

Prof. Hongyu Liu, City University of Hong Kong, Hong Kong (China)

Prof. Hongjie Li, The Chinese University of Hong Kong, Hong Kong (China)

2、Course name：Mathematical theory of plasmon/polariton resonances and applications

3、Period：July 18, 19, 20, 21, 2022, 9:00-12:00 am

4、Prerequisite knowledge：Ordinary Differential Equations (ODE), Partial Differential Equations (PDE), Functional Analysis.

5、Textbooks：

[1] Y. Deng and H. Liu, Spectral Theory of Localized Resonances and Applications, preprint.

[2] H. Li, H. Liu and J. Zou, Elastodynamical resonances and cloaking of negative material structures beyond quasistatic approximation, arXiv:2203.08432. 

[3] H. Li, H. Liu and J. Zou, Minnaert resonances for bubbles in soft elastic materials, SIAM J. Appl. Math. , 82(2022), 119--141. 

[4] Y. Deng, H. Li and H. Liu, Spectral properties of Neumann-Poincare operator and anomalous localized resonance in elasticity beyond quasi-static limit, Journal of Elasticity, 140(2020), 213--242. 

[5] Y. Deng, H. Li and H. Liu, Analysis of surface polariton resonance for nanoparticles in elastic system, SIAM J. Math. Anal., 52 (2020), no. 2, 1786–1805. 

[6] E. Blasten, H. Li, H. Liu and Y. Wang, Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincare eigenfunctions, ESAIM: Math. Model. Numer. Anal., 54 (2020), no. 3, 957-976, https://doi.org/ 10.1051/m2an/2019091. 

[7] H. Li and H. Liu, On anomalous localized resonance and plasmonic cloaking beyond the quasistatic limit, Proceedings of the Royal Society A, 474: 20180165, http://doi.org/10.1098/rspa.2018.0165. 

[8] H. Li, S. Li, H. Liu and X. Wang, Analysis of electromagnetic scattering from plasmonic inclusions at optical frequencies and applications, ESAIM: Math. Model. Numer. Anal., 53 (2019), no. 4, 1351--1371. 

[9] H. Li, J. Li and H. Liu, On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonance, Journal de Mathematiques Pures et Appliquees, 120 (2018), 195--219. 

[10] Y. Deng, H. Li and H. Liu, On spectral properties of Neumann-Poincare operator and plasmonic cloaking in 3D elastostatics, Journal of Spectral Theory, 9 (2019), no. 3, 767--789. 

[11] H. Li and H. Liu, On anomalous localized resonance for the elastostatic system, SIAM J. Math. Anal., 48 (2016), no. 5, 3322--3344. 

[12] H. Li and H. Liu, On three-dimensional plasmon resonance in elastostatics, Annali di Matematica Pura ed Applicata, 196 (2017), no. 3, 1113--1135.

[13] H. Li, J. Li and H. Liu, On quasi-static cloaking due to anomalous localized resonance in R3, SIAM J. Appl. Math., 75 (2015), 1245--1260. 

[14] H. Li, Recent progress on the mathematical study of anomalous localized resonance in elasticity, Electronic Research Archive, 28 (2020), no. 3, 1257--1272. 

二、Content and Schedule

    In this series of lectures, we will discuss several topics related to the mathematical theory of metamaterials and their applications, including invisibility cloaking and super-resolution imaging. The emphasis is on various plasmon/polariton resonances induced by metamaterial structures, which form the fundamental basis for the cutting-edge applications.  

This course will be structured in 4 parts: 

• PART I: Introduction to mathematical theory of metamaterials. We will give a short introduction to the mathematical theory of metamaterials and provide a concise review on the state-of-the-art development.

• PART II: Plasmon resonance and its cloaking effect in acoustic and electromagnetic scattering. We will discuss two principles to achieve the cloaking effect, i.e. the variational perspective and the spectral perspective. Then the critical Neumann-Poincare (N-P) operator will be discussed, especially on the spectral properties.

• PART III: Polariton resonance and its cloaking effect in elastodynamics. We will discuss the cloaking effect caused by the polariton resonance for the elastic system. Both the variational approach and the spectral method will be introduced. At last, we will analyze the super-resolution based on the metamaterials.

• PART IV: Atypical resonance beyond the quasi-static approximation. We will introduce the theory on achieving the cloaking effect beyond the quasi-static approximation in the finite regime. Both the acoustic system and the elastic system shall be considered. 

三、About the Lecturers

1、 Prof. Hongyu Liu, Professor and Associate Head at the Department of Mathematics, City University of Hong Kong. Research Interests include Inverse Problems and Imaging, Spectral Theory and Scattering Theory, Analysis and PDEs, Mathematical Materials Science and Numerical Analysis & Scientific Computing. 

2、Prof. Hongjie Li, Research Assistant Professor at the Department of Mathematics, The Chinese University of Hong Kong. Fields of Interest include Inverse Problems and Wave Imaging, Partial differential Equations, Asymptotic and Spectral Analysis and Finite Element Method.