会议时间：2021年5月13日—2021年5月14日

会议地点：数学院145报告厅

线上报告网址：腾讯会议，https://meeting.tencent.com/s/CeKNfq3ZtnWW

会议ID：949 666 413

报告题目:Mathematical analysis of plasmon resonances for curved nanorods

报告人: 郑光辉 湖南大学

报告时间：5月13日9:00-10:00

报告地点：数学院145报告厅

报告摘要：In this talk, the plasmon resonances for curved nanorods which present anisotropic geometries is considered. We analyze quantitative properties of the plasmon resonance and its relationship to the metamaterial configurations and the anisotropic geometries of the nanorods. Based on delicate and subtle asymptotic and spectral analysis of the layer potential operators, particularly the Neumann-Poincare operators, associated with anisotropic geometries, we derive sharp asymptotic formulae of the corresponding scattering field in the quasi-static regime. By carefully analyzing the asymptotic formulae, we establish sharp conditions that can ensure the occurrence of the plasmon resonance. The resonance conditions couple the metamaterial parameters, the wave frequency and the nanorod geometry in an intricate but elegant manner. We provide thorough resonance analysis by studying the wave fields both inside and outside the nanorod. Furthermore, our quantitative analysis indicates that different parts of the nanorod induce varying degrees of resonance. Specically, the resonant strength at the two end-parts of the curved nanorod is more outstanding than that of the facade-part of the nanorod.

(Joint work with Youjun Deng (CSU) and Hongyu Liu (CityUHK))

报告人简介：郑光辉，湖南大学数学学院，副教授，硕士生导师。2012年博士毕业于兰州大学数学与统计学院，曾先后访问巴黎高师数学系，香港浸会大学数学系进行科研合作。主要从事偏微分方程反问题的理论及算法、贝叶斯统计反演与推断、等离子共振及超分辨成像等方面的研究。相关研究成果发表在《Inverse Problems》、《J. Math. Pures Appl.》、《SIAM J. Numer. Anal.》、《J. Differential Equations》、《Adv. Comput. Math.》等多个SCI杂志上。主持国家自然科学青年基金1项和湖南省面上项目1项。

报告题目: Inverse Elastic Scattering for a Random Potential

报告人: 李建樑 长沙理工大学

报告时间：5月13日10:00-11:00

报告地点：数学院145报告厅

报告摘要：Consider the inverse scattering problem for the time-harmonic elastic wave equation in R^n (n=2,3) with an inhomogeneous and anisotropic mass density given by a compactly supported potential. The potential is assumed to be a microlocally isotropic generalized Gaussian random function with a classical pseudo-differential operator describing the covariance. The purpose of this paper is to reconstruct the principal symbol of the covariance operator from the scattered wave measured in a domain away from the potential region. For such a distributional potential, the well-posedness of the direct problem is derived by constructing an equivalent Lippmann-Schwinger integral equation which has a unique solution. For the inverse problem, we show that, with probability one, the principal symbol of the covariance operator can be uniquely determined by the amplitude of $n$ scattered waves averaged over the frequency band, produced by a single realization of the random potential. Here, the $n$ scattered fields are associated with $n$ incident point sources with $n$ directions which are perpendicular to each other. The analysis is carried out by employing the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of the Fourier integral operators.

报告人简介：李建樑，2009年6月本科毕业于中国农业大学理学院，2014年7月获中国科学院大学理学博士学位。2017年11月－2018年11月受国家留学基金委资助赴美国普渡大学数学系访问一年。主要研究领域为反散射问题的理论与数值方法、随机反散射问题的理论研究。主持国家自然科学基金青年项目1项，湖南省教育厅一般项目1项。在《Inverse Problems in Science and Engineering》、《Applicable Analysis》、《Computers and Mathematics with applications》、《SIAM Journal on Imaging Sciences》、《SIAM Journal on Mathematical Analysis》、《Communications in Partial Differential Equations》、《Inverse Problems and Imaging》发表论文9篇。

报告题目: Variational optimization of enstrophy for probing fundamental bounds in fractional Burgers equation

报告人:运东方 中南大学

报告时间：5月13日11:00-12:00

报告地点：数学院145报告厅

报告摘要：This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blowup in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.

报告人简介：运东方，中南大学数学与统计学院讲师。2009和2012年在 东南大学和山东大学获得学士和硕士学位，2015年在香港城市大学获得哲学博士学位。随后在加拿大菲尔兹数学科学研究所和麦克马斯特大学从事博士后研究，2020年8月到中南大学工作。主要研究领域包括Navier-Stokes方程解的奇异性（爆破）和偏微分方程的无网格数值法。主要成果发表在Journal of Fluid Mechanics, Journal of Nonlinear Science, Engineering Analysis with Boundary Elements等国际知名期刊上。

报告题目: Invisibility enables super-visibility

报告人: 刘宏宇 香港城市大学

报告时间：5月13日14:00-15:00

报告地点：线上报告

报告摘要：Invisibility and super-resolution imaging seem mutually exclusive. In this talk, I shall show how invisibility enables us to develop a novel super-resolution imaging scheme. When invisibility occurs, the wave generates certain interior resonant modes inside the object. Those so-called transmission resonant modes are shown to carry the geometrical and topological information of the underlying object. Moreover, they can be reconstructed from the observable patterns. Our study provides a new perspective on inverse problems.

报告人简介：刘宏宇，香港城市大学教授。刘宏宇教授于2007年在香港中文大学数学系取得博士学位，曾任职于香港浸会大学（2014-2020，教授，数学系副主任），美国北卡罗来纳大学（2011-2014），英国雷丁大学（2010/2011），美国华盛顿大学（2007--2010）。刘宏宇教授的研究领域为应用数学和计算数学，包括反问题，波成像，数学物理方程，超材料的数学理论，散射理论和谱理论。刘宏宇教授在上述领域取得了一系列创新性的研究成果,在国际高水平学术期刊发表论文130余篇，其中9篇论文被相关杂志评为年度亮点论文、特色论文或高被引论文。所获荣誉包括获国际反问题协会Calderon Prize、香港数学会青年学者奖及反问题领域MediaV Prize。

报告题目:On generalized Holmgren’s principle to the Lame operator with applications to inverse elastic problems

报告人: 刁怀安 东北师范大学

报告时间：5月13日15:00-16:00

报告地点： 线上报告

报告摘要：Consider the Lamé operatorL(u):=μ∆u+(λ+μ)∇(∇·u)that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lamé eigenfunctionu, namely−L(u)=κuwithκ∈R+andu∈L^2(Ω)^2, Ω⊂R^2. We introduce the so-called homogeneous line segments ofuin ,Ω on whichu, its traction or their combination via an impedance parameter is vanishing. We give a comprehensive study on characterizing the presence of one or two such line segments and its implication to the uniqueness ofu. The results can be regarded as generalizing the classical Holmgren’s uniqueness principle for the Lamé operator in two aspects. We establish the results by analyzing the development of analytic microlocal singularities ofuwith the presence of the aforesaid line segments. Finally, we apply the results to the inverse elastic problems in establishing two novel unique identifiability results. It is shown that a generalized impedance obstacle as well as its boundary impedance can be determined by using at most four far-field patterns. Unique determination by a minimal number of far-field patterns is a longstanding problem in inverse elastic scattering theory.

报告人简介：刁怀安，博士毕业于香港城市大学，目前任职东北师范大学数学与统计学院副教授，博士生导师，研究兴趣为数值代数、随机化算法、微分算子谱理论、波散射问题，在Journal de Mathématiques Pures et Appliquées、Calculus of Variations and Partial Differential Equations、Communications in Partial Differential Equations、Mathematics of Computation、SIAM Journal on Mathematics Analysis、Inverse Problems，以及机器学习领域顶级会议NeurIPS 2019，等国际主流期刊发表科研论文40余篇。出版学术专著一本。主持并完成国家自然科学基金青年基金、数学天元基金与教育部博士点基金新教师基金项目各一项。曾多次受邀访问国内外高校进行合作研究与学术交流。

报告题目: Mathematical study on plasmon resonance beyond quasi-static approximation

报告人: 李宏杰 香港中文大学

报告时间：5月13日16:00-17:00

报告地点：线上报告

报告摘要：This talk discusses the mathematical progress on the plasmon resonances and their application to invisibility cloaking for optics and linear elasticity. First, I shall briefly discuss the major results obtained in the quasi-static regime. Then I shall focus on talking about our recent study beyond the quasi-static approximation for the Lame system.

报告人简介：李宏杰，男，1990年出生，香港中文大学科研助理教授。2019年获得香港浸会大学数学专业博士学位，然后到香港中文大学从事博士后研究工作，现任科研助理教授。主要研究范围为偏微分方程、反问题、数值计算、隐形、等离子共振、渐进分析。

报告题目: Fourier method for reconstructing elastic body force from the coupled-wave field

报告人: 汪贤超 香港城市大学

报告时间：5月13日17:00-18:00

报告地点：线上报告

报告摘要：This paper is concerned with the inverse source problem of the time-harmonic elastic waves. A novel non-iterative reconstruction scheme is proposed for determining the elastic body force by using the multi-frequency Fourier expansion. The key ingredient of the approach is to choose appropriate admissible frequencies and establish an relationship between the Fourier coefficients and the coupled-wave field of compressional wave and shear wave. Both theoretical justifications and numerical examples are presented to verify the validity and robustness of the proposed method.

报告人简介：汪贤超，1990.09，男，香港城市大学，博士后。2019年毕业于哈尔滨工业大学数学学院计算数学专业，获理学博士学位。于2019年6月入选“香江学者计划”，并在香港城市大学开展为期2年的博士后研究。目前研究领域为数学物理反问题，主要方向为波动方程反散射问题的数值分析与计算。相关研究可应用于手势识别，超分辨率成像，以及共振与隐形。目前已在“Inverse Problems”、“Journal of Elasticity”和“SIAM Journal on Imaging Sciences”等期刊发表SCI检索学术论文10余篇。其中1篇论文入选反问题领域国际著名期刊“Inverse Problems”的2017年度“亮点论文”。目前主持国家自然青年基金项目1项和“香江学者计划”项目1项，参与面上项目2项。曾先后应邀访问香港浸会大学和南方科技大学并进行科研合作。目前担任美国数学评论《Mathematical Reviews》评论员，中国仿真学会不确定系统分析与仿真专委会委员，以及SCI期刊“Inverse Problems and Imaging”审稿专家。