Need advice or assistance for son who is in prison. His interest is scattering theory The letter below is written by my son. I have been sending him text books and looking for answers on the internet to keep his interest up. He has progressed so far on his own and now he needs direction and assistance from a professional in mathematics. Any advice or assistance you can provide is greatly appreciated.
My name is ---, I'm 25, I've been in prison for the past 6 years, and I'm self-taught in mathematics. I began with a list of courses required in a standard undergraduate curriculum and studied the required texts from each course. I covered the basics in this way before branching off into my own interests, beginning with partial differential equations and eventually landing in scattering theory.
I began studying mathematics because it was fun and interesting (and passed the time), but it has since become so much more. The progress that I've made, combined with the observation that I am capable of at least understanding research in my fields of interest, has compelled me to take the next step into conducting research of my own, and my current goal is to make advances of publishable value. I am just beginning in this process, yet already I have made progress studying scattering resonances. At the moment, I'm working on a number of problems related to resonance counting. In particular, my primary focus is on "inverse resonance counting": By assuming an asymptotic formula for the resonance counting function (as well as some other results concerning distribution), my goal is to determine properties of the potential. Similarly, in the case of a surface with hyperbolic ends, the goal is to determine properties of the surface from knowledge of an exact asymptotic formula for the counting function. My primary resources at present are Mathematical Theory of Scattering Resonances by Dyatlov and Zworski, and Spectral Theory of Infinite-Area Hyperbolic Surfaces by Borthwick.
I'm not sure what I'm asking for here, I just know that I am ready for the next step and seek some guidance as I enter the world of research mathematics. I encounter many problems when it comes to research, such as staying up to date on current topics, finding open problems which suit my skills and interests, and finding papers on topics I need to study more deeply. For example, right now I am in need of results on how resonances change under smooth, small changes in the potential. One of my texts mentioned the paper of P. D. Stefanov, Stability of Resonances Under Smooth Perturbations of the Boundary (1994), but I need more, and that paper makes no citation to papers of the same content. How do I find papers which are similar, or even cite this one?
In short, without direct access to the internet or fellow researchers, I hit many roadblocks which are not math-related, and that can be frustrating. I'm looking for ways to make my unconventional research process go a little more smoothly. If anyone has any suggestions, please let me know here.
And thanks in advance.
 A: I would look up colleges and universities that offer masters & PhD level degrees in math/scattering theory.  Most schools will list the text books required for their classes.  Those are the books I'd get for him.  The other option would be to go to Google Scholar and look up scattering theory to see the latest/greatest books & other publications in this area.
A: I have received a response back from my son he said. "I took Calculus my first and only year at Michigan State University, prior to my incarceration. That is the highest course I have taken formally. Shortly after beginning my sentence, I asked my father for a multivariable calculus textbook and he sent me one. I studied it deeply, and enjoyed it so much that I asked my dad if he could find anything online about what's after calculus in a standard undergraduate curriculum. He found MIT's opencourseware website, and sent me screenshots of a page listing which courses were required of an undergrad math major at MIT, some sample undergrad course loads, as well as pages listing course titles, with descriptions and prerequisites. From there I would ask my dad for all the info on a given course, including required textbooks, lecture notes, and problem sets. Many courses even gave dates on which the problems were due. He'd order me the book(s), and print out and mail me the problems and notes. Starting with linear algebra and a course on ordinary differential equations, I proceeded this way for a couple of years. Often, I would also study the chapters in the books which weren't required by the course and at least attempt every problem. Here is a sample of some of the books required with the courses:
Strang - Introduction to Linear Algebra
Zill - A First Course in Differential Equations
Pinter - A Book of Abstract Algebra
Rudin - Principles of Mathematical Analysis
Ahlfors - Complex Analysis
do Carmo - Differential Geometry of Curves and Surfaces
Simmons - Introduction to Topology and Modern Analysis
Lee - Introduction to Smooth Manifolds
As I matured mathematically, I stopped using the opencourseware site, and began studying in areas which had interested me. I no longer read textbooks linearly, nor do I try to digest every concept in every book I obtain. But I have many, many books. Among them are about 4 on PDE's in general, a handful on more technical but related topics, like perturbations, scaling, dimensional analysis, waves. I own Hormander's, The Analysis of Linear Partial Differential Operators, Vols 1-4, and Reed and Simon's, Methods of Modern Mathematical Physics, Vols 1-3. I have 2 on semiclassical and microlocal analysis, 2 on analytic number theory. Of course in addition to Volume 3 of the Reed and Simon series, the books mentioned in the original post are my resources on scattering theory. I also have a handful of introductory physics texts. Aside from books, I have a few research articles in scattering related to my recent attempts at research. Scattering is the only field in which I've made a serious attempt at research. Hopefully this answers your question, and if not, please follow up!"
A: The papers which cite this work of Stefanov can be found using Google Scholar. There are 4 of them. For more information on the subject one can write to Plamen Stefanov,
whose e-mail address is available in public domain.
A: As your son is interested in scattering theory and inverse problems and knows the recent book by Dyatlov and Zworski, I'd recommend him to read publications by the French scattering theory community which is very strong and has been very active in recent years. For example, there are the notes
C. Guillarmou, Scattering for the geodesic flow on surfaces with boundary, Contemporary Math 700 (2017)
which are freely available here and feature a rich list of suggested additional references at the end.
A: This doesn't answer your question, but I wanted to mention that your son might find inspiration in historical examples of mathematical achievements of prisoners. It's rare but it does happen. The first answer to this question lists some major examples.
(One person not mentioned there is Paul Turan, who created the subject of extremal graph theory while imprisoned in a concentration camp. Obviously that's a different kind of example.)
A: It sounds like your son could really benefit from having a mentor, and if he happens to be in North America, someone has set up a math resources network:
https://www.prisonmathproject.org/
A: 
In a follow up to my previous question my son has asked me to send him
this, S. V. Petras, On the Continuous Dependence of the Poles of the
Scattering Matrix on the Coefficients of an Elliptic Operator,
Proc. Steklov Inst. Math. 159 (1983) pgs. 135-139. I have searched the
web including Google Scholar which only has a version in Russian.

I found the English version of the article in our library, here is a link to a photocopy.
(apologies for the poor quality of the photocopy, I was not able to make it myself because of the lockdown restrictions; I trust it is still usable.)

Follow-up:

I'm getting close and these two short papers sound like they might be
what I need:
• E. Korotyaev, Stability for Inverse Resonance Problem, Int. Math. Res.
Not. 73 (2004), pgs. 3927 - 3936
• E. Korotyaev, Inverse Resonance Scattering on the Real Line, Inverse
Problems 21 (2005), no. 1, pgs. 325 - 341.

Dan: I have access to both these papers. Could you tell me your email address (contact info in my profile), so that I can email the pdf's to you? (I'd avoid posting them here, because of copyright restrictions.)
A: In a follow up to my previous question my son has asked me to send him this, "S. V. Petras, On the Continuous Dependence of the Poles of the Scattering Matrix on the Coefficients of an Elliptic Operator. In Proc. Steklov Inst. Math. 159 (1983) pgs. 135-139" I have searched the web including google scholar which only has a version in Russian. Does anyone know if this is available for free on the web? If not I will purchase the text book and send it to him. Thank you again for all your help. The response to my first post was more than I imagined and my son is very excited to know he has support in your community.
