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Topics in Material Science: Mathematics of Linear Wave Phenomena
SEC livestreamed graduate course
Math 73841
Louisiana State University and Auburn University
Fall Semester 2018 and Spring Semester 2019
Prof. Stephen Shipman (LSU) and Prof. Junshan Lin (Auburn)
Registration:
Each student must register for the course at their home institution.
Livestreaming: Tuesday and Thursday 1:552:55 Central Time
The class will be livestreamed via Zoom. The software can be downloaded at this link. More information about the use of software can be found at this link.
A course website for participants at Auburn University is forthcoming.
For participants at LSU for Spring 2019:
Place: Lockett Hall 381
Time: 1:302:50 Tuesday and Thursday
Office: Room 314 of Lockett Hall
Telephone: 225/5781674
Email: shipman@lsu.edu
Office Hours: By appointment
Course Description
This course will be taught jointly by Profs. Stephen Shipman (LSU) and Junshan Lin (Auburn University) in a livestreaming setting. Students from any SEC university may enroll. It is intended to be a twosemester course, although the first semester may be taken in isolation. The second semester run in the spring of 2019.
The material in this course builds toward open problems in mathematical physics centered around wave dynamics and scattering in electromagnetics and acoustics. The mathematical topics form a coherent body of theory and techniques. The main components are the partial differential equations (PDE) of electromagnetics and acoustics and other derived phenomena in complex media; integral equations and boundaryintegral representations of solutions to PDE; Fourier analysis and the residue calculus for the study of scattering and resonance; spectral theory of differential and integral operators illustrated and motivated by examples; and asymptotic analysis. The second semester of the course will concentrate on specific problems that are motivated by modern scientific and mathematical research. A goal is that students will be able to understand open problems in this area and be equipped with the basic mathematical tools to begin trying to solve them.
Prerequisite
The prerequisites are linear algebra and basic complex variables (good undergraduate courses are sufficient) and a standard graduate analysis course that includes measure and integration and standard function spaces. Helpful but not required is a background in partial differential equations, abstract functional analysis, and/or numerical analysis.
Course topics and notes for Spring 2019
The Auburn website for this course is here.
There, you will find the list of topics and Prof. Lin's notes for the course.
Course topics for Fall 2018

Preliminary mathematical material
 Measure and integration on the line: Increasing functions and the Helly Selection Theorem; functions of bounded variation; the RiemannStieljes integral and the Riesz Representation Theorem; signed measures and Jordan decomposition; the LebesgueStieljes integral; the Helly convergence theorem
 Fourier transforms: discrete, continuous, and finite; complexanalytic extension; Floquet transform
 Finitedimensional spectral theory: selfadjoint operators and the spectral resolution; normal operators
 Spectral resolution for differential operators through the Fourier transform
 Resolvent and spectral measures

Basic models of scattering and resonance: the generalized Fourier transform
 Quantum: The Schrödinger equation on the line with a potential well
 Classical: The Resonant Lamb model: An oscillator attached to a string
 Modified, or scattered, waves
 Eigenvalues and bound states
 Preservation of continuous spectrum for wave scattering
 Eigenvalues embedded in the continuous spectrum and resonance

Applications and open problems
 The equations of acoustics and electromagnetics
 Infinite periodic quantum graphs
 Bound states in the continuum for open periodic waveguides
 Embedded eigenvalues, resonance, and asymmetry
 Plasmoninduced resonance and field enhancement
Course topics for spring 2019
The course will mainly focus on mathematical studies of plasmonic resonances. These include plasmon for nano particles, surface plasmons, resonance in extraordinary optical transmission through nanoholes. The mathematical studies will be mainly based on integral equations, and their asymptotic analysis and numerical solution, homogenization theory. If time allows, inverse problems arising in acoustic and electromagnetics may also be introduced.
Literature
We will not use any one source as a text book. The lectures will draw from our notes and various items of literature.
There is a bibliography of relevant works below, with links to some PDF files of excerpts.
Evaluation
Evaluation is based on participation in the course and a goodfaith effort to learn the material.
Fullfilling these basic requirements warrants a grade of A in the course. An A+ is reserved for students who submit excellent solutions to the exercises in the notes.
Lecture Notes and Problem Sets
Course Notes
Bibliography

Michael Reed and Barry Simon,
Methods of Modern Mathematics:
Vol. I Functional Analysis,
Vol. II Fourier Analysis and SelfAdjointness,
Vol. III Scattering Theory,
Vol. IV Analysis of Operators,
excerpts from Vol. I (spectral theorem),
Vol. II (selfadjoint extensions),
Academic Press, 1980.

N. I. Akhiezer and I. M. Glazman,
Theory of Linear Operators in Hilbert Space,
Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.

A. N. Kolmogorov and S. V. Fomin,
Introductory Real Analysis, §36: The Stieljes integral, 1968 (translation by Silverman, Dover 1975).

Gerald B. Folland,
Real Analysis: Modern Techniques and Their Applications, Second Edition,
Wiley 1999.

Maciej Zworski, Resonances in Physics and Geometry, Notices of the AMS, 1999.

Maciej Zworski, Mathematical Theory of Scattering Resonances Lecture notes, Version 0.1, 2016.

Maciej Zworski Mathematical Study of Scattering Resonances 2016.

Michael Reed and Barry Simon,
Methods of Modern Mathematics:
Vol. I Functional Analysis,
Vol. II Fourier Analysis and SelfAdjointness,
Vol. III Scattering Theory,
Vol. IV Analysis of Operators,
excerpts from Vol. I (spectral theorem),
Vol. II (selfadjoint extensions),
Academic Press, 1980.

Stephen P. Shipman and Aaron T. Welters,
Resonant electromagnetic scattering in anisotropic layered media,
J. Math. Phys. 54(10) 103511140 (2013).
pdf

Gayan S. Abeynanda and Stephen P. Shipman,
Dynamic Resonance in the HighQ and NearMonochromatic Regime,
2016

Shanhui Fan and John D. Joannopoulos,
Analysis of guided resonances in photonic crystal slabs,
Physical Review B, Vol 65, 235112 (2002).

Stephen P. Shipman and Stephanos Venakides,
An Exactly Solvable Model for Nonlinear Resonant Scattering, Nonlinearity, Vol. 25, No. 9 (2012) 24732501.

N. I. Akhiezer and I. M. Glazman,
Theory of Linear Operators in Hilbert Space,
Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.

Gerald B. Folland,
Introduction to Partial Differential Equations,
Second Edition, Princeton University Press, 1995.
Excerpt: Chapter 0.

G. B. Whitham,
Linear and Nonlinear Waves,
WileyInterscience, 1973.
