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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.

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    $\begingroup$ Welcome to the site, Dan. I'd like to, potentially, put you in contact with one of the authors you mention -- could you email me at daa399@nyu.edu ? $\endgroup$ – sharpend Jan 23 at 19:50
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    $\begingroup$ Could you describe what your son has studied, including the book titles, before he got to scattering theory and whether he took a course or did it on his own? $\endgroup$ – Deane Yang Jan 23 at 21:52
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    $\begingroup$ My experience with autodidacts is that, although they are usually quite smart, they are not always careful enough to do the math rigorously, starting from the definitions. It would be worthwhile for someone to act as a mentor and verify that this student has indeed learned the foundations solidly. If not, it's usually easy enough to help him develop the skills needed and fill in the holes. After that, it would be good to have an expert in scattering theory take over from there. There's no real limit to how far he can go. $\endgroup$ – Deane Yang Jan 24 at 21:53
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    $\begingroup$ Inmates in Michigan corrections facilities are allowed to take correspondence courses at their own expense. I don't know if your son can afford this, but, if so, it's worth pursuing. Working with a professor and other students can be invaluable in confirming that you are doing things correctly and progressing well. michigan.gov/documents/corrections/05_02_119_181368_7.pdf $\endgroup$ – Deane Yang Jan 24 at 22:04
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    $\begingroup$ The reactions to this post shows how supporting and welcoming this community can be. This is MO at its best. $\endgroup$ – Francesco Polizzi Jan 25 at 10:07
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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!"

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    $\begingroup$ If he can do MIT Courseware and does all the homework, that would be fantastic. If he has any specific math questions, you can post them here or on math.stackexchange.com. Actually, if you start with that, we’ll be able to tell whether there are any serious gaps in his understanding of math and help him with that. And if he’s solid, he’ll get great answers here. $\endgroup$ – Deane Yang Jan 25 at 1:19
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    $\begingroup$ Dan, you are a good father. $\endgroup$ – Nik Weaver Jan 25 at 1:37
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    $\begingroup$ It appears your son is more advanced in math than I assumed. Until he finds a mentor, I recommend that you post his questions on MathOverflow. If any of his questions are not suitable for this site, I'm sure someone will offer guidance on a better place to ask it. $\endgroup$ – Deane Yang Jan 25 at 15:54
  • $\begingroup$ Dan, Tell to your boy, Your idea and decide is fantastic. and ask him (for advice to me) that how he made this decision? $\endgroup$ – C.F.G Jan 25 at 17:22
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    $\begingroup$ Responding to the edit: this looks like a very solid background, with no obvious gaps. $\endgroup$ – Nik Weaver Jan 26 at 1:44
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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.

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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.

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    $\begingroup$ This may be the most helpful comment yet; it needs more upvotes. $\endgroup$ – Nik Weaver Jan 28 at 3:01
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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.)

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    $\begingroup$ theconversation.com which is a web magazine close I would say to quantamagazine.com in quality has an article not about mathematicians in jail but about an inmate in a similar situation to OP's son working his way trough learning math and even being able to publish a paper, the paper is in another area but the article is not long so maybe it could printed and hopefully sent to OP's in case it could be useful to him $\endgroup$ – Daniel D. Jan 26 at 1:41
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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/

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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.)

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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.

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    $\begingroup$ The book is here in WorldCat worldcat.org/title/… you should be able to have your local library request an interlibrary loan and then scan the copies. You may also want to directly contact the libraries as they may be willing to copy the article for you (in my experience librarians love to help). $\endgroup$ – RBega2 Jan 26 at 0:40
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    $\begingroup$ Thank you. I was able to send him screen shots of the Russian version with the correct symbols and then I used google translate to send him a corresponding copy to read along side. In the mean time I am requesting a version from the library. $\endgroup$ – Dan Cunningham Jan 26 at 12:47
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    $\begingroup$ Dan, I'm not sure if you're familiar with it, but there's something called 'scihub' and another thing called 'libgen'. It's probably verboten to talk about it on here, but feel free to shoot me an e-mail and I can explain a bit more. $\endgroup$ – Harry Gindi Jan 26 at 14:33
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    $\begingroup$ our library has this journal in English, it is not digitized (only volumes since 2006 are); I will make an effort to get it and if succesful (not trivial in these pandemic times) I will email you a scan. $\endgroup$ – Carlo Beenakker Jan 27 at 20:32
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    $\begingroup$ got it: ilorentz.org/beenakker/MO/Petras.pdf $\endgroup$ – Carlo Beenakker Feb 1 at 16:57
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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.

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